FROM
PREHISTORY TO
THE
INVENTION
Georges Ifrah
THE UNIVERSAL HISTORY OF
NUMBERS
FROM PREHISTORY TO THE
INVENTION OF THE COMPUTER
GEORGES IFRAH
Translated from the French
by David Bellos, E. F. Harding, Sophie Wood, and Ian Monk
John Wiley & Sons, Inc.
New York • Chichester • Weinheim • Brisbane • Singapore • Toronto
For you, my wife,
the admirably patient witness of the joys and agonies that this hard labour has
procured me, or to which you have been subjected, over so many years.
For your tenderness and for the intelligence of your criticisms.
For you, Hanna, to whom this book and its author owe so much.
And for you, Gabrielle and Emmanuelle,
my daughters, my passion.
* *
This book is printed on acid-free paper. ©
Published by John Wiley & Sons, Inc., in 2000.
Published simultaneouly in Canada
First published in France with the title Histoire universelle des chiffres
by Editions Robert Laffont, Paris, in 1994.
First published in Great Britain in 1998 by The Harvill Press Ltd
Copyright © 1981, 1994 by Editions Robert Laffont S.A., Paris
Translation copyright © 1998 by The Harvill Press Ltd
This translation has been published with the financial support of the European Commission
and of the French Ministry of Culture.
All illustrations, with the exception of Fig. 1.30-36 and 2.10 by Lizzie Napoli,
have been drawn, or recopied, by the author.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under
Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the
Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center,
222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for
permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue,
New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, e-mail: PERMREQ@WILEY.COM.
This publication is designed to provide accurate and authoritative information in regard to the subject matter covered.
It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional
advice or other expert assistance is required, the services of a competent professional person should be sought.
Library of Congress Cataloging- in-Publication Data :
Ifrah, Georges
[Histoire universelle des chiffres. English!
The universal history of numbers : from prehistory to the invention of the computer /
Georges Ifrah ; [translated by David Bellos, E. F. Harding, Sophie Wood, and Ian Monk].
p. cm.
Includes bibliographical references and index.
ISBN 0-471-39340-1 (paper)
1. Numeration — History I. Title.
QA 141. 1 3713 2000
513.2 21— dc21 99-045531
Printed in the United States of America
10 98765432
SUMMARY TABLE OF CONTENTS
Foreword v
List of Abbreviations vi
Introduction xv
Where “Numbers" Come From
CHAPTER 1 Explaining the Origins: Ethnological and Psychological
Approaches to the Sources of Numbers 3
CHAPTER 2 Base Numbers and the Birth of Number-systems 23
CHAPTER 3 The Earliest Calculating Machine - The Hand 47
CHAPTER 4 How Cro-Magnon Man Counted 62
CHAPTER 5 Tally Sticks: Accounting for Beginners 64
CHAPTER 6 Numbers on Strings 68
CHAPTER 7 Number, Value and Money 72
CHAPTER 8 Numbers of Sumer 77
CHAPTER 9 The Enigma of the Sexagesimal Base 91
CHAPTER 10 The Development of Written Numerals in Elam and
Mesopotamia 96
CHAPTER 11 The Decipherment of a Five-thousand-year-old System 109
CHAPTER 12 How the Sumerians Did Their Sums 121
CHAPTER 13 Mesopotamian Numbering after the Eclipse of Sumer 134
CHAPTER 14 The Numbers of Ancient Egypt 162
CHAPTER 15 Counting in the Times of the Cretan and Hittite Kings 178
CHAPTER 16 Greek and Roman Numerals 182
CHAPTER 17 Letters and Numbers 212
CHAPTER 18 The Invention of Alphabetic Numerals 227
CHAPTER 19 Other Alphabetic Number-systems 240
CHAPTER 20 Magic, Mysticism, Divination, and Other Secrets 248
CHAPTER 21 The Numbers of Chinese Civilisation 263
CHAPTER 22 The Amazing Achievements of the Maya 297
CHAPTER 23 The Final Stage of Numerical Notation 323
CHAPTER 24 PART I Indian Civilisation: the Cradle of
Modern Numerals 356
CHAPTER 24 PART II Dictionary of the Numeral Symbols
of Indian Civilisation 440
CHAPTER 25 Indian Numerals and Calculation in the Islamic World 511
CHAPTER 26 The Slow Progress of Indo-Arabic Numerals in
Western Europe 577
CHAPTER 27 Beyond Perfection 592
Bibliography 601
Index of Names and Subjects 616
FOREWORD
The main aim of this two-volume work is to provide in simple and
accessible terms the full and complete answer to all and any questions that
anyone might want to ask about the history of numbers and of counting,
from prehistory to the age of computers.
More than ten years ago, an American translation of the predecessor of
The Universal History of Numbers appeared under the title From One to Zero ,
translated by Lowell Bair (Viking, 1985). The present book - translated
afresh - is many times larger, and seeks not only to provide a historical
narrative, but also, and most importantly, to serve as a comprehensive,
thematic encyclopaedia of numbers and counting. It can be read as a whole,
of course; but it can also be consulted as a source-book on general topics
(for example, the Maya, the numbers of Ancient Egypt, Arabic counting, or
Greek acrophonics) and on quite specific problems (the proper names of
the nine mediaeval apices, the role of Gerbert of Aurillac, how to do a long
division on a dust-abacus, and so on).
Two maps are provided in this first volume to help the reader find what
he or she might want to know. The Summary Table of Contents above gives a
general overview. The Index of names and subjects, from p. 616, provides a
more detailed map to this volume.
The bibliography has been divided into two sections: sources available
in English; and other sources. In the text, references to works listed in
the bibliography give just the author name and the date of publication, to
avoid unnecessary repetition. Abbreviations used in the text, in the captions
to the many illustrations, and in the bibliography of this volume are
explained below.
LIST OF ABBREVIATIONS
Where appropriate, cross-references to fuller information in the Bibliography are given in the form: “see: author
AA
The American Anthropologist
Menasha, Wisconsin
AAN
American Antiquity
AANL
Atti dell’Accademia Pontificia de’Nuovi Lined Rome
AAR
Acta Archaeologica
Copenhagen
AAS
Annales archeologiques syriennes
Damascus
AASOR Annual of the American School of Oriental
Research
Cambridge, MA
AAT
Aegypten und altes Testament
ABSA
Annual of the British School in Athens
London
ACII
Appendice al Corpus Inscriptionum
Italicorum
see: gamurrini
ACLHU Annals of the Computation Laboratory of
Harvard University
Cambridge, MA
ACT
Astronomical Cuneiform Texts
See: neugebauer
ACOR
Acta Orientalia
Batavia
ADAW
Abhandlungen der deutschen Akademie der
Wissenschaften zu Berlin
ADFU
Ausgrabungen der deutschen
Forschungsgemeinscha.fi in Uruk-Warka
Berlin
ADO
Annals of the Dudley Observatory
Albany, NY
ADOGA Ausgrabungen derdeutschen Orient-
Gesellschaft in Abusir
ADP
Archives de Psychologic
Geneva
ADSM
Album of Dated Syriac Manuscripts
see: hatch
AEG
Aegyptus. Rivista italiana di egittologia
e papirologia
Milan
AESC
Annales. Economies, Societes, Civilisations
Paris
AFD
Annales d’une famille de Dilbat
see: gautier
AFO
Archiv fur Orienforschung
Graz
AGA
Aegyptologische Abhandlungen
Wiesbaden
AGM
Abhandlungen zur Geschichte der Mathematik Leipzig
AGMNT
Archiv fir Geschichte der Mathematik, der
Naturwissenschaften und der Technik
AGW
Abhandlungen der Gesellschaft der
Wissenschaften
Gottingen
AHC
Annals of the History of Computing
IEEE, New York
AHES
Archive for the History of the Exact Sciences
AI
Arad Inscriptions
see: aharoni
AIEE
American Institute of Electrical Engineers
New York, NY
AIHS
Archives internationales d'histoire des sciences
Paris
AIM
Artificial Intelligence Magazine
AJ
Accountants Journal
AJA
American Journal of Archaeology
New York, NY
AJPH
American Journal of Philology
New York, NY
AJPS
American Journal of Psychology
New York, NY
AJS
American Journal of Science
New York, NY
AJSL
American Journal of Semitic Languages and
Literature
Chicago, IL
AKK
Akkadika
Brussels
AKRG
Arbeiten der Kaiserlichen Russischen
Gesandschaft zu Peking
Berlin
AM
American Machinist
AMA
Asia Major
Leipzig & Lond<
AMI
Archaeologische Mitteilungen aus Iran
Berlin
AMM
American Mathematical Monthly
AMP
Archiv der Mathematik und Physik
ANS
Anatolian Studies
London
ANTH
Anthropos
Goteborg
ANTHR
Anthropologie
Paris
AOAT
Alter Orient und Altes Testament
Neukirchen-Vlu
AOR
Analecta Orientalia
Rome
Vll
LIST OF ABBREVIATIONS
AOS
American Oriental Series
New Haven, CT
BAB
Bulletin de TAcademie de Belgique
Brussels
APEL
Arabic Papyri in the Egyptian Library
see: grohmann
BAE
Bibliotheca Aegyptica
Brussels
ARAB
Arabica. Revue d etudes arabes
Leyden
BAMNH
Bulletin of the American Museum of Natural
ARBE
Annual Report of the American Bureau of
History
New York, NY
Ethnology
Washington, DC
BAMS
Bulletin of the American Mathematical Society
ARBS
Annual Report of the Bureau of the
BAPS
Bulletin de TAcademie polonaise des Sciences
Warsaw
Smithsonian Institution
Washington, DC
BARSB
Bulletin de TAcademie royale des sciences et
ARC
Archeion
Rome
belles-lettres de Bruxelles
Brussels
ARCH
Archeologia
Rome
BASOR
Bulletin of the American School of Oriental
ARCHL
Archaeologia
London
Research
Ann Arbor, MI
ARCHN
Archaeology
New York, NY
BCFM
Bulletin du Club frangais de la medaille
Paris
ARM
Armenia
BCMS
Bulletin of the Calcutta Mathematical Society
Calcutta
ARMA
Archives royales de Mari
Paris
BDSM
Bulletin des sciences mathematiques
Paris
AROR
Archiv Orientalni
Prague
BEFEO
Bulletin de TEcole frangaise d’Extreme-Orient
Paris & Hanoi
ARYA
see: shukla &
BEPH
Beitrage zur englischen Philologie
Leipzig
SARMA
BFT
Blatter fur Technikgeschichte
Vienna
AS
Automata Studies
Princeton, NJ, 1956
BGHD
Bulletin de geographic historique et descriptive
Paris
ASAE
Annales du service de Tantiquite de I’Egypte
Cairo
BHI
Bulletin hispanique
Bordeaux
ASB
Assyriologische Bibliothek
Leipzig
BHR
Bibliotheque d’humanisme et de Renaissance
Geneva
ASE
Archaeological Survey of Egypt
London
BIFAO
Bulletin de I’lnstitut frangais d’antiquites
ASI
Archaeological Survey of India
New Delhi
orientales
Cairo
ASMF
Annali di scienze matematiche efisiche
Rome
BIMA
Bulletin of the Institute of Mathematical
ASNA
Annuaire de la societe frangaise de
Applications
numismatique et d’archeologie
Paris
BJRL
Bulletin of the John Rylands Library
Manchester, UK
ASOR
American School of Archaeological Research
Ann Arbor, Ml
BLPM
Bulletin de liaison des professeurs de
ASPN
Annales des sciences physiques et naturelles
Lyon
mathematiques
Paris
ASR
Abhandlungen zum schweizerischen Recht
Bern
BLR
Bell Laboratories Record
Murray Hill, NJ
ASS
Assyriological Studies
Chicago, 1L
BMA
Biblioteca Mathematica
ASTP
Archives suisses des traditions populaires
BMB
Bulletin for Mathematics and Biophysics
ASTRI
LAstronomie indienne
see: billard
BMET
Bulletin du Musee d’ethnologie du Trocadero
Paris
AT
Annales des Telecommunications
Paris
BMFRS
Biographical Memoirs of Fellows of the Royal
ATU
Archaische Texte aus LJruk
see: falkenstein
Society
London
ATU2
Zeichenliste der Archaischen Texte aus Uruk
see: green &
BMGM
Bulletin of the Madras Government Museum
Madras
NISSEN
BRAH
Boletin de la Real Academia de la Historia
Madrid
AUT
Automatisme
Paris
BSA
Bulletin de la Societe d'anthropologie
Paris
BSC
Bulletin scientifique
Paris
THE UNIVERSAL HISTORY OF NUMBERS
VII)
BSEIN
Bulletin de la Societe d’encouragement pour
C1G
l Industrie nationale
Paris
BSFE
Bulletin de la Societe franyaise d’Egyptologie
Paris
CII
BSFP
Bulletin de la Societe franyaise de philosophic
Paris
CIIN
BSPF
Bulletin de la Societe prehistorique franyaise
Paris
BSI
Biblioteca Sinica
Paris
CIL
BSM
Bulletin de la Societe mathematique de France Paris
BSMA
Bulletin des sciences mathematiques et
CIS
astronomiques
Paris
CJW
BSMF
Bollettino di bibliografia e di storia delle
CNAE
scienze matematiche e fisiche
Rome
CNP
BSMM
Bulletin de la Societe de medecine mentale
Paris
COWA
BSNAF
Bulletin de la Societe nationale des
antiquaires de France
Paris
CPH
BSOAS
Bulletin of the School of Oriental and
rptM
African Studies
London
v_r iiN
BST
Bell System Technology
Murray Hill, NJ
CR
CRAI
CAA
Contributions to American Archaeology
Washington, DC
CAAH
Contributions to American Anthropology
CRAS
and History
Washington, DC
CAH
The Cambridge Ancient History
Cambridge, UK,
CRCIM
1963
CAP1B
Corpus of Arabic and Persian Inscriptions
CRGL
of Bihar
Patna
CAW
Carnegie Institution
Washington, DC
CRSP
CDE
Chronique d’Egypte, in: Bulletin periodique
de la Fondation egyptienne de la reine
CSKBM
Elisabeth
Brussels
CENT
Centaurus
Copenhagen
CSMBM
CETS
Comparative Ethnographical Studies
CGC
A Catalogue of Greek Coins in the British
CTBM
Museum
see: poole
CHR
China Review
CIC
Corpus des Inscriptions du Cambodge
see: coedes
D
CIE
Corpus inscriptionum etruscarum
1970
DAA
Corpus inscriptionum graecarum
Corpus inscriptionum ludaicorum
Corpus inscriptionum Indicarum
Corpus inscriptionum latinarum
Corpus inscriptionum semiticarum
Coins of the Jewish War
Contributions to North American Ethnology
Corpus Nummorum Palaestiniensium
Relative Chronologies in Old World
Archaeology
Classical Philology
Le Cabinet des poinyons de I’Imprimerie
national
Classical Review
Comptes-rendus des seances de I’Academie
des Inscriptions et Belles-Lettres
Comptes-rendus des seances de I’Academie
des Sciences
Comptes-rendus du Deuxieme Congres
international de Mathematiques de Paris
Comptes-rendus du Groupe linguistique
detudes hamito-semitiques
Comptes-rendus de la Societe imperiale
orthodoxe de Palestine
Catalogue of Sanskrit Buddhist Manuscripts
in the British Museum
Catalogue of Syriac Manuscripts in the
British Museum
Cuneiform Texts from Babylonian Tablets
in the British Museum
Le Temple de Dendara
Denkmaler aus Aegypten und Aethiopien
see: boeckh,
FRANZ, CURTIUS &
KIRKHOFF
see: frey
London, Benares &
Calcutta, 1888-1929
Leipzig & Berlin,
1861-1943
Paris, 1889-1932
see: kadman
Washington, DC
Jerusalem
see: erichsen
Chicago, IL
Paris, 1963
Paris
Paris
see: duporck
Paris
London
see: wright
London, 1896
see: chassinat
see: lepsius
IX
LIST OF ABBREVIATIONS
DAB
Dictionnaire archeologique de la Bible
Paris: Hazan, 1970
DAC
Dictionnaire de I'Academie franqaise
DAE
Deutsche Aksoum-Expedition
Berlin
DAFI
Cahiers de la Delegation archeologique
franqaise en Iran
Paris
DAGR
Dictionnaire des antiquites grecques et
romaines
see: daremberg &
SAGLIO
DAR
Denkmaler des Alten Reiches im Museum
von Kairo
see: borchardt
DAT
Dictionnaire archeologique des techniques
Paris: L’Accueil, 1963
DCI
Dictionnaire de la civilisation indienne
see: Frederic
DCR
Dictionnaire de la civilisation romaine
see: fredouille
DG
Demotisches Glossar
see: erichsen
DgRa
De Gestis regum Anglorum libri
see: Malmesbury
DI
Der Islam
DJD
Discoveries in the Judaean Desert of Jordan
Clarendon Press,
Oxford
DMG
Documents in Mycenean Greek
see: ventris &
CHADWICK
DR
Divination et rationalite
Paris: Le Seuil, 1974
DS
Der Schweiz
DSB
Dictionary of Scientific Biography
see: gillespie
DTV
Dictionnaire de Trevoux
Paris, 1771
EJ
Encyclopaedia Judaica
Jerusalem
EMDDR
Entwicklung der Mathematik in der DDR
Berlin, 1974
EMW
Enquetes du Musee de la vie wallone
ENG
Engineering
Paris
EP
Encyclopedic de la Pleiade
Paris
EPP
L’Ecriture et la psychologie des peoples
Paris: A. Colin, 1963
ERE
Encyclopaedia of Religions and Ethics
Edinburgh & New
York, 1908-1921
ESIP
Ecritures. Systemes ideographiques et
pratiques expressives
see: christin
ESL
L’Espace et la lettre
Paris: UGE, 1977
ESM
Encyclopedic des sciences mathematiques
Paris, 1909
EST
Encyclopedic internationale des sciences et
des techniques
Paris, 1972
EUR
Europe
EXP
Expedition
Philadelphia, PA
FAP
Fontes atque Pontes. Eine Festgabe fur
Helmut Brunner
see: AAT 5 (1983)
FEHP
Facsimile of an Egyptian Hieratic Papyrus
see: birch
FIH
Das Mathematiker-Verzeichnis im Fihrist
des Ibn Abi Jakub an Nadim
see: suter
FMAM
Field Museum of Natural History
Chicago, IL
FMS
Fruhmitelalterliche Studien
Berlin
E
Le Temple d’Edfou
see: chassinat
EA
Etudes asiatiques
Zurich
EBR
Encyclopaedia Britannica
London
EBOR
Encyclopedic Bordas
Paris
EC
Etudes cretoises
Paris
EE
Epigrafia etrusca
see: buonamici
EEG
Elementa epigraphica graecae
see: franz
EENG
Electrical Engineering
EG
Epigraphia greca
see: guarducci
El
Epigraphia Indica
Calcutta
EIS
Encyclopedic de T Islam
Leyden, 1908-1938
GIES
Glasgow Institute of Engineers and
Shipbuilders in Scotland
GKS
Das Grabdenkmal des Kdnigs S’ahu-Re
see: borchardt
GLA
De sex arithmeticae practicae specibus
Henrici Glareani epitome
Paris, 1554
GLO
Globus
GORILA Recueil des inscriptions en lineaireA
see: godart &
OLIVIER
GT
Ganitatilaka, by Shripati
see: kapadia
GTSS
Ganitasarasamgraha by Mahavira
see: rangacarya
THE UNIVERSAL HISTORY OF NUMBERS
HAN
Hindu-Arabic Numerals
see: smith &
KARPINSKI
HESP
Hesperis. Archives berberes et Bulletin de
llnstitut des Hautes Etudes marocaines
HF
Historical Fragments
see: legrain
HG
“Hommages a H. G. Giiterboch" in Anatolian
Studies
Istanbul, 1974
HGE
Handbuch der griechischen Epigraphik
see: larfeld
HGS
Histoire generate des sciences
see: taton
HLCT
Haverford Library Collection of Cuneiform
Tablets
New Haven, CT
HMA
Historia mathematica
HMAI
Handbook of Middle American Indians
Austin, TX
HNE
Handbuch der Nordsemitischen Epigraphik
see: lidzbarski
HOR
Handbuch der Orientalistik
Leyden & Cologne
HP
Hieratische Palaographie
see: moller
HPMBS
The History and Palaeography of Maury an
Brahmi Script
see: upasak
HUCA
Hebrew Union College Annual, ed.
S. H. Blank
IA
Indian Antiquary
Bombay
IDERIC
Institut d etudes et de recherches
interethniques et interculturelles
Nice
IEJ
Israel Exploration Journal
Jerusalem
IESIS
Indian Epigraphy and South Indian Scripts
see: sivaramamurti
IHE
Las Inscripcidnes hebraicas de Espaha
see: cantera &
MILLAS
IHQ
Indian Historical Quarterly
Calcutta
IJES
International Journal of Environmental
Studies
IJHS
Indian Journal of History of Science
IMCC
Listes generates des Inscriptions et
Monuments du Champa et du Cambodge
see: coedes &
PARMENTIER
INEP
Indian Epigraphy
see: sircar
INM
Indian Notes and Monographs
INSA
Die Inschriften Asarhaddons, Konig von
Assyrien
see: borger
IOS
Israel Oriental Studies
IP
Indische Palaeographie
see: buhler
IR
Inscription Reveal, Documents from the Time
of the Bible, theMishna and the Talmud
Jerusalem, 1973
ISCC
Inscriptions sanskrites du Champa et du
Cambodge
see: barth &
BERGAIGNE
IS
Isis, revue d’histoire des sciences
JA
Journal asiatique
Paris
JAI
Journal of the Anthropological Institute of
Great Britain
JAOS
Journal of the American Oriental Society
Baltimore, MD
JAP
Journal of Applied Psychology
JASA
Journal of the American Statistical Association
JASB
Journal of the Asiatic Society of Bengal
Calcutta
JB
Jinwen Bian
see: rong ren
JBRAS
Journal of the Bombay branch of the Royal
Asiatic Society
Bombay
JCS
Journal of Cuneiform Studies
New Haven, CT
JEA
Journal of Egyptian Archaeology
London
JF1
Journal of the Franklin Institute
JFM
Jahrbuch iiber die Fortschritte der
Mathematik
JHS
Journal of Hellenic Studies
London
JIA
Journal of the Institute of Actuaries
JJS
Journal of Jewish Studies
London
JNES
Journal of Near Eastern Studies
Chicago, IL
JPAS
Journal and Proceedings of the Asiatic Society
of Bengal
Calcutta
JRAS
Journal of the Royal Asiatic Society
London
JRASB
Journal of the Royal Asiatic Society of Bengal
JRASI
Journal of the Royal Asiatic Society of Great
Britain and Ireland
London
London
x i
JRSA Journal of the Royal Society of the Arts
JRSS Journal of the Royal Statistical Society
JSA Journal de la societe des americanistes Paris
JSI Journal of Scientific Instruments London
JSO Journal de la societe orientale d’Allemagne
KAI
Kanaanaische und Aramaische Inschriften
see: donner &
ROLLIG
KAV
The Kashmirian Atharva-Veda
Baltimore, MD, 1901
KR
The Brooklyn Museum Aramaic Papyri
see: kraeling
KS
Keilschriften Sargons, Konig von Assyrien
see: lyon
LAA
Annals of Archaeology and Anthropology
Liverpool
LAL
Lalitavistara Sutra
see: lal litra
LAT
Latomus
Brussels
LAUR
Petri Laurembergi Rostochiensis Institutiones
arithmeticae
Hamburg, 1636
LBAT
Late Babylonian Astronomical and Related
Texts
see: pinches &
STRASMAIER
LBDL
Late Old Babylonian Documents and Letters
see: finkelstein
LEV
Levant
LIL
Lilavati by Bhaskata
see: dvivedi
LOE
The Legacy of Egypt
see: Harris
LOK
Lokavibhaga
see: anonymous
MA Mathematische Annalen
MAA Les Mathematiques arabes see: youshketvitch
MACH Machriq Baghdad
MAF Memorial de I’artilleriefrangaise Paris
MAGW Mitteilungen der Anthropologischen
Gesellschaft in Wien Vienna
MAPS Memoirs of the American Philosophical
Society Philadelphia, PA
MAR Die Mathematiker und Astronomen der
Araher und ihre Werke
see: suter
LIST OF ABBREVIATIONS
MARB
Memoires de I’Academie royale de Bruxelles
Brussels
MARI
Mari. Annales de recherches
interdisciplinaires
Paris
MAS
Memoirs of the Astronomical Society
MCM
Memoirs of the Carnegie Museum
Washington, DC
MCT
Mathematical Cuneiform Texts
see: neugebauer
& SACHS
MDP
Memoires de la delegation archeologique en
Susiane (vols. 1-5), continued as: Memoires
de la Delegation en Perse (vols. 6-13),
Memoires de la mission archeologique en Perse
(vols. 14-30), Memoires de la mission
archeologique en Iran (vols. 31-40),
Memoires de la delegation archeologique en
Iran (vols.41-)
MDT
Memoires de Trevoux
MFO
Melanges de la Faculte orientale
Beirut
MG
Morgenlandische Gesellschaft
MGA
Mathematical Gazette
MIOG
Mitteilungen des Instituts fur osterreichische
Geschichtsforschung
Innsbruck
MM
Mitteilungen fur Miinzsammler
Frankfurt /Main
MMA
Memoirs of the Museum of Anthropology
Ann Arbor, MI
MMO
Museum Monographs
Philadelphia, PA
MNRAS Monthly Notes of the Royal Astronomical
Society
MP
Michigan Papyri
Ann Arbor, MI
MPB
Mathematisch-physikalischeBibliothek
Leipzig
MPCI
Memoire sur la propagation des chiffres
indiens
see: woepke
MSA
Memoires de la societe d’anthropologie
Paris
MSPR
Mitteilungen aus der Sammlung der Papyrus
Rainer
MT
Mathematics Teacher
MTI
Mathematik Tijdschrift
MUS
Melanges de I'universite Saint -Joseph
Beirut
THE UNIVERSAL HISTORY OF NUMBERS
N
Le Nabateen
see: cantineau
NA
Nature
London
NADG
Neues Archiv der Gesellschaft fur dltere
deutsche Geschichtskunde
Hanover
NAM
Nouvelles Annales de Mathematiques
Paris
NAT
La Nature
Paris
NAW
Nieuw Archief voor Wiskunde
NAWG
Nachrichten der Akademie der Wissenschajten
zu Gottingen
Gottingen
NC
Numismatic Chronicle
London
NCEAM Notices sur les caracteres etrangers anciens
et modernes
see: fossey
NEM
Notices et Extraits des Manuscrits de la
Bibliotheque nationale
Paris
NMM
National Mathematics Magazine
NNM
Numismatical Notes and Monographs
New York
Nott
Christophori Nottnagelii Professoris
Wittenbergensis Institutionum
mathematicarum Wittenberg, 1645
NS New Scientist London
NYT New York Times
NZ Numismatische Zeitschrift Vienna
OED Oxford English Dictionary
OIP Oriental Institute Publications Chicago, IL
OR Orientalia Rome
PA Popular A stronomy
PEQ Palestine Exploration Quarterly London
PFT Persepolis Fortification Tablets see: hallock
PGIFAO Papyrus grecs de 1 'Institut franyais
d’Archeologie orientale Cairo
PGP Palaographie der griechischen Papyri see: s eider
PHYS Physis Buenos Aires
PI The Paleography of India see: ojha
PIB Paleographia Iberica
see: burnam
PLMS
Proceedings of the London Mathematical
Society
London
PLO
Porta Linguarum Orientalum
Berlin
PM
The Palace of Minos
see: evans
PMA
Periodico matematico
PMAE
Papers of the Peabody Museum
Cambridge, MA
PPS
Proceedings of the Prehistoric Society
PR
Physical Review
PRMS
Topographical Bibliography
see: porter
and moss
PRS
Proceedings of the Royal Society
London
PRU
Le Palais royal d’Ugarit
see: schaeffer
PSBA
Proceedings of the Society of Biblical
Archaeology
London
PSREP
Publications de la societe royale egyptienne
de papyrologie
Cairo
PTRSL
Philosophical Transactions of the Royal
Society
London
PUMC
Papyri in the University of Michigan
Collection
see: garett-winter
QSG
Quellen und Studien zur Geschichte der
Mathematik, Astronomie und Physik
Berlin
RA
Revue d’Assyriologie et d'Archeologie orientale
Paris
RACE
Real Academia de Ciencias Exactas,
FisicasyNaturales
Madrid
RAR
Revue archeologique
Paris
RARA
Rara Arithmetica
see: d. e. smith
RB
Revue biblique
Saint-Etienne
RBAAS
Report of the British Society for the
Advancement of Science
London
RCAE
Report of the Cambridge Anthropological
Expedition to the Torres Straits
Cambridge, 1907
RdSO
Revista degli Studi Orientali
Rome
RE
Revue d’Egyptologie
Paris
REC
Revue des Etudes Celtiques
Paris
REG
Revue des Etudes Grecques
Paris
REI
Revue des Etudes Islamiques
Paris
RES
Repertoire depigraphie semitique
Paris
RFCB
Reproduction fac similar
see: seler
RFE
Recueil de facsimiles
see: prou
RH
Revue historique
Paris
RHA
Revue de Haute-Auvergne
Aurillac
RHR
Revue de VHistoire des Religions
Paris
RHS
Revue d’Histoire des Sciences
Paris
RHSA
Revue d’Histoire des Sciences et de leurs
applications
Paris
RMM
Revue du Monde musulman
Paris
RN
Revue numismatique
Paris
RRAL
Rendiconti della Reale Accademia dei Lined
Rome
RSS
Rivista di Storia della Scienza
Florence
RTM
The Rock Tombs ofMeir
see: blackman
S
Aramaische Papyrus und Ostraka
see: sachau
SAOC
Studies in Ancient Oriental Civilizations
Chicago, IL
SC
Scientia
SCAM
Scientific American
New York
SE
Studi etruschi
Florence
SEM
Semitica
Paris
SGKIO
Studien zur Geschichte und Kultur des
islamischen Orients
Berlin
SHAW
Sitzungsberichte der Heidelberger Akademie
der Wissenschaften
Heidelberg
SHM
Sefer ha Mispar
see: silberberg
SIB
Scripta Pontificii Instituti Biblici
Rome
SIP
Elements of South Indian Paleography
see: burnell
SJ
Science Journal
SKAW
Sitzungsberichte der kaiserlichen Akademie
der Wissenschaften
Vienna
SMI
Scripta Minoa, 1
see: evans
LIST OF ABBREVIATIONS
SM2
Scripta Minoa, 2
see: evans & myres
SMA
Scripta Mathematica
SME
Studi medievali
Turin
SMS
Syrio-Mesopotamian Studies
Los Angeles, CA
SPA
La scrittura proto-elamica
see: meriggi
SPRDS
Scientific Proceedings of the Royal Dublin
Society
Dublin
SS
Schlern Schriften
Innsbruck
STM
Studia Mediterranea
Pavia
SUM
Sumer
Baghdad
SVSN
Memoires de la societe vaudoise des sciences
naturelles
Lausanne
SWG
Schriften der Wissenschaftlichen Gesellschaft
in Strassburg
Strasburg
TA
Tablettes Albertini
see: courtois,
LESCHI, PERRAT &
SAUMAGNE
TAD
Turk Arkeoloji Dergisi
TAPS
Transactions of the American Philosophical
Society
TASJ
Transactions of the Asiatic Society of Japan
Yokohama
TCAS
Transactions published by the Connecticut
Academy of Arts and Sciences
New Haven, CT
TDR
Tablettes de Drehem
see: genouillac
TEB
Tablettes de Tepoque babylonienne ancienne
see: birot
TH
Theophanis Chronographia
Paris, 1655
TIA
Thesaurus Inscriptionum Aegypticum
see: brugsch
TLE
Testimonia Linguae Etruscae
1968
TLSM
Transactions of the Literary Society of Madras Madras
TMB
Textes mathematiques de Babylone
see: th ureau-
DANGIN
TMIE
Travaux et memoires del'Institut d'Ethnologie
de Paris
Paris
TMS
Textes mathematiques de Suse
see: bruins &
RUTTEN
TRAR
Trattati d’Aritmetica
see: boncompagni
THE UNIVERSAL HISTORY OF NUMBERS
TRIA
Transactions of the Royal Irish Academy
Dublin
TSA
Tablettes sumeriennes archaiques
see: genouillac
TSM
Taylor's Scientific Memoirs
London
TTKY
Tiirk Tarih Kurumu Yayinlarindan
Ankara
TUTA
Tablettes d’Uruk
see: thureau-
DANGIN
TZG
Trierer Zeitschrift zur Geschichte und Kunst
des Trierer Landes
Trier
UAA
Urkunden des Aegyptischen Altertums
see: steindorff
UCAE
University of California Publication of
American Archaeology and Ethnology
Berkeley, CA
UMN
Unterrichtsblatterfur Mathematik und
Naturwissenschaften
URK
Hieroglyphischen Urkunden der
griechischen-romischen Zeit
see: sethe
URK.I
Urkunden des Alten Reichs
see: sethe
URK.IV
Urkunden derl8.ten Dynastie
see: sethe &
HELCK
UVB
Vorlaufiger Bericht iiber die Ausgrabungen in
Uruk-Warka
Berlin
xiv
WKP
Wochenschrift fur klassische Philologie
WM
World of Mathematics
YI
Xiao dun yin xu wenzi:yi bian
see: dong zuobin
YOS
Yale Oriental Series
New Haven, CT
ZA
Zeitschrift fur Assyriologie
Berlin
ZAS
Zeitschrift fur Aegyptische Sprache und
Altertumskunde
Berlin
ZDMG
Zeitschrift der Deutschen
Morgenlandischen Gesellschaft
Wiesbaden
ZDP
Zeitschrift des Deutschen
Palastina-Vereins
Leipzig & Wiesbaden
ZE
Zeitschrift fiir Ethnologie
Braunschweig
ZKM
Zeitschrift fiir die Kunde des
Morgenlandes
Gottingen
ZMP
Zeitschrift fiir Mathematik und Physik
ZNZ
Zbornik za Narodni Zivot i Obicaje juznih
Slavena
Zagreb
ZOV
Zeitschrift fiir Osterreichische Volkskunde
Vienna
ZRP
Zeitschrift fiir Romanische Philologie
Tubingen
VIAT Viator. Medieval and Renaissance Studies Berkeley, CA
XV
TEACHER LEARNS A LESSON
INTRODUCTION
Where “Numbers” Come From
TEACHER LEARNS A LESSON
This book was sparked off when I was a schoolteacher by questions
asked by children. Like any decent teacher, I tried not to leave any
question unanswered, however odd or naive it might seem. After all, a
curious mind often is an intelligent one.
One morning, 1 was giving a class about the way we write down
numbers. I had done my own homework and was well-prepared to
explain the ins and outs of the splendid system that we have for
representing numbers in Arabic numerals, and to use the story to show
the theoretical possibility of shifting from base 10 to any other base
without altering the properties of the numbers or the nature of the
operations that we can carry out on them. In other words, a perfectly
ordinary maths lesson, the sort of lesson you might have once sat
through yourself - a lesson taught, year in, year out, since the very
foundation of secondary schooling.
But it did not turn out to be an ordinary class. Fate, or Innocence,
made that day quite special for me.
Some pupils - the sort you would not like to come across too often,
for they can change your whole life! - asked me point-blank all the
questions that children have been storing up for centuries. They were
such simple questions that they left me speechless for a moment:
“Sir, where do numbers come from? Who invented zero?”
Well, where do numbers come from, in fact? These familiar symbols
seem so utterly obvious to us that we have the quite mistaken
impression that they sprang forth fully formed, as gods or heroes are
supposed to. The question was disconcerting. I confess I had never
previously wondered what the answer might be.
“They come . . . er . . . they come from the remotest past,” I fumbled,
barely masking my ignorance.
But I only had to think of Latin numbering (those Roman numerals
which we still use to indicate particular kinds of numbers, like
sequences of kings or millionaires of the same name) to be quite sure
that numbers have not always been written in the same way as they
are now.
“Sir!” said another boy, “Can you tell us how the Romans did their
sums? I’ve been trying to do a multiplication with Roman numerals for
days, and I’m getting nowhere with it!”
“You can’t do sums with those numerals,” another boy butted in.
“My dad told me the Romans did their sums like the Chinese do today,
with an abacus.”
That was almost the right answer, but one which I didn’t even
possess.
“Anyway,” said the boy to the rest of the class, “if you just go into a
Chinese restaurant you’ll see that those people don’t need numbers or
calculators to do their sums as fast as we do. With their abacuses, they
can even go thousands of times faster than the biggest computer in
the world.”
That was a slight exaggeration, though it is certainly true that
skilled abacists can make calculations faster than they can be done on
paper or on mechanical calculating machines. But modern electronic
computers and calculators obviously leave the abacus standing.
I was fortunate and privileged to have a class of boys from very varied
backgrounds. I learned a lot from them.
“My father’s an ethnologist,” said one. “He told me that in Africa and
Australia there are still primitive people so stupid that they can’t even
count further than two! They’re still cavemen!”
What extraordinary injustice in the mouth of a child! Unfortunately,
there used to be plenty of so-called experts who believed, as he did, that
“primitive” peoples had remained at the first stages of human evolution.
However, when you look more closely, it becomes apparent that
“savages” aren’t so stupid after all, that they are far from being devoid of
intelligence, and that they have extraordinarily clever ways of coping
without numbers. They have the same potential as we all do, but their
cultures are just very different from those of “civilised” societies.
But I did not know any of that at the time. I tried to grope my way
back through the centuries. Before Arabic numerals, there were Roman
ones. But does “before” actually mean anything? And even if it did,
what was there before those numerals? Was it going to be possible
to use an archaeology of numerals and computation to track back to
that mind-boggling moment when someone first came up with the
idea of counting?
Several other allegedly naive questions arose as a result of my pupils’
curious minds. Some concerned “counting animals” that you some-
times see at circuses and fairs; they are supposed to be able to count
(which is why some people claim that mathematicians are just
circus artistes!) Other pupils put forward the puzzle of “number 13”,
INTRODUCTION
X V 1
alternately considered an omen of good luck and an omen of bad luck.
Others wondered what was in the minds of mathematical prodigies,
those phenomenal beings who can perform very complex operations in
their heads at high speed - calculating the cube root of a fifteen-digit
number, or reeling off all the prime numbers between seven million and
ten million, and so on.
In a word, a whole host of horrendous but fascinating questions
exploded in the face of a teacher who, on the verge of humiliation, took
the full measure of his ignorance and began to see just how inadequate
the teaching of mathematics is if it makes no reference to the history
of the subject. The only answers I could give were improvised ones,
incomplete and certainly incorrect.
I had an excuse, all the same. The arithmetic books and the school
manuals which were my working tools did not even allude to the history
of numbers. History textbooks talk of Hammurabi, Caesar, King Arthur,
and Charlemagne, just as they mention the travels of Marco Polo and
Christopher Columbus; they deal with topics as varied as the history of
paper, printing, steam power, coinage, economics, and the calendar, as
well as the history of human languages and the origins of writing and of
the alphabet. But I searched them in vain for the slightest mention of the
history of numbers. It was almost as if a conspiracy of obviousness
aimed to make a secret, or, even worse, just to make us ignorant of one
of the most fantastic and fertile of human discoveries. Counting is what
allowed people to take the measure of their world, to understand it
better, and to put some of its innumerable secrets to good use.
These questions had a profound impact on me, beginning with this
lesson in modesty: my pupils, who were manifestly more inquisitive
than I had been, taught me a lesson by spurring me on to study the
history of a great invention. It turned out to be a history that I quickly
discovered to be both universal and discontinuous.
THE QUEST FOR THE MATHEMATICAL GRAIL
I could not now ever let go of these questions, and they soon drew me
into the most fascinating period of learning and the most enthralling
adventure of my life.
My desire to find the answers and to have time to think about them
persuaded me, not without regrets, to give up my teachingjob. Though I
had only slender means, I devoted myself full-time to a research project
that must have seemed as mad, in the eyes of many people, as the
mediaeval quest for the Holy Grail, the magical vessel in which the
blood of Christ on the cross was supposed to have been collected.
Lancelot, Perceval, and Gawain, amongst many other valiant knights of
Christendom, set off in search of the grail without ever completing
their quest, because they were not pure enough or lacked sufficient faith
or chastity to approach the Truth of God.
I couldn’t claim to have chastity or purity either. But faith and calling
led me to cross the five continents, materially or intellectually, and to
glimpse horizons far wider than those that the cloistered world of
mathematics usually allows. But the more my eyes opened onto the
wider world, the more I realised the depth of my ignorance.
Where, when and how did the amazing adventure of the human
intellect begin? In Asia? In Europe? Or somewhere in Africa? Did it take
place at the time of Cro-Magnon man, about thirty thousand years
ago, or in the Neanderthal period, more than fifty thousand years ago?
Or could it have been half a million years ago? Or even - why not? - a
million years ago?
What motives did prehistoric peoples have to begin the great
adventure of counting? Were their concerns purely astronomical (to do
with the phases of the moon, the eternal return of day and night, the
cycle of the seasons, and so on)? Or did the requirements of communal
living give the first impulse towards counting? In what way and after
what period of time did people discover that the fingers of one hand and
the toes of one foot represent the same concept? How did the need for
calculation impose itself on their minds? Was there a chronological
sequence in the discovery of the cardinal and ordinal aspects of the
integers? In which period did the first attempts at oral numbering
occur? Did an abstract conception of number precede articulated
language? Did people count by gesture and material tokens before doing
so through speech? Or was it the other way round? Does the idea of
number come from experience of the world? Or did the idea of number
act as a catalyst and make explicit what must have been present already
as a latent idea in the minds of our most distant ancestors? And finally,
is the concept of number the product of intense human thought, or is it
the result of a long and slow evolution starting from a very concrete
understanding of things?
These are all perfectly normal questions to ask, but most of the answers
cannot be researched in a constructive way since there is no longer
any trace of the thought-processes of early humans. The event, or, more
probably, the sequence of events, has been lost in the depths of pre-
historic time, and there are no archaeological remains to give us a clue.
However, archaeology was not necessarily the only approach to
the problem. What other discipline might there be that would allow
at least a stab at an answer? For instance, might psychology and
XVII
ethnology not have some power to reconstitute the origins of number?
The Quest for Number? Or a quest for a wraith? That was the
question. It was not easy to know which it was, but I had set out on it
and was soon to conquer the whole world, from America to Egypt, from
India to Mexico, from Peru to China, in my search for more and yet
more numbers. But as I had no financial backer, I decided to be my
own sponsor, doing odd jobs (delivery boy, chauffeur, waiter, night
watchman) to keep body and soul together.
As an intellectual tourist I was able to visit the greatest museums in
the world, in Cairo, Baghdad, Beijing, Mexico City, and London (the
British Museum and the Science Museum); the Smithsonian in
Washington, the Vatican Library in Rome, the libraries of major
American universities (Yale, Columbia, Philadelphia), and of course
the many Paris collections at the Musee Guimet, the Conservatoire
des arts et metiers, the Louvre, and the Bibliotheque nationale.
I also visited the ruins of Pompeii and Masada. And took a trip to the
Upper Nile Valley to see Thebes, Luxor, Abu Simbel, Gizeh. Had a look
at the Acropolis in Athens and the Forum in Rome. Pondered on
time’s stately march from the top of the Mayan pyramids at Quirigua
and Chichen Itza. And from here and from there I gleaned precious
information about past and present customs connected with the history
of counting.
When I got back from these fascinating ethno-numerical and
archaeo-arithmetical expeditions I buried myself in popularising and
encyclopaedic articles, plunged into learned journals and works of
erudition, and fired off thousands of questions to academic specialists in
scores of different fields.
At the start, I did not get many replies. My would-be correspondents
were dumbfounded by the banality of the topic.
There are of course vast numbers of oddballs forever pestering
specialists with questions. But I had to persuade them that I was serious.
It was essential for me to obtain their co-operation, since I needed to be
kept up to date about new and recent discoveries in their fields, however
apparently insignificant, and as an amateur I needed their help in avoid-
ing misinterpretations. And since I was dealing with many specialists
who were far outside the field of mathematics, I had not only to
persuade them that I was an honest toiler in a respectable field, but also
to get them to accept that “numbers” and “mathematics” are not quite
the same thing. As we shall see . . .
All this work led me to two basic facts. First, a vast treasure-house of
documentation on the history of numbers does actually exist. I owe a
great deal to the work of previous scholars and mention it frequently
THE QUEST FOR THE MATHEMATICAL GRAII.
throughout this book. Secondly, however, the articles and monographs
in this store of knowledge each deal with only one specialism, are
addressed to other experts in the same field, and are far from being
complete or comprehensive accounts. There were also a few general
works, to be sure, which I came across later, and which also gave me
some help. But as they describe the state of knowledge at the time they
were written, they had been long overtaken by later discoveries in
archaeology, psychology, and ethnography.
No single work on numbers existed which covered the whole of
the available field, from the history of civilisations and religions to the
history of science, from prehistoric archaeology to linguistics and philol-
ogy, from mythical and mathematical interpretation to ethnography,
ranging over the five continents.
Indeed, how can one successfully sum up such heterogeneous
material without losing important distinctions or falling into the trap of
simplification? The history of numbers includes topics as widely
divergent as the perception of number in mammals and birds, the
arithmetical use of prehistoric notched bones, Indo-European and
Semitic numbering systems, and number-techniques among so-called
primitive populations in Australia, the Americas, and Africa. How can
you catch in one single net things as different as finger-counting and
digital computing? counting with beads and Amerindian or Polynesian
knotted string? Pharaonic epigraphy and Babylonian baked clay
tablets? How can you talk in the same way about Greek and Chinese
arithmetic, astronomy and Mayan inscriptions, Indian poetry and
mathematics, Arabic algebra and the mediaeval quadrivium? And all of
that so as to obtain a coherent overall vision of the development through
time and space of the defining invention of modern humanity, which is
our present numbering system? And where do animals fit into what
is already an enormously complex field? Not to mention human
infants . . .
What I had set out to do was manifestly mad. The topic sat at the
junction of all fields of knowledge and constituted an immense universe
of human intellectual evolution. It covered a field so rich and huge that
no single person could hope to grasp it alone.
Such a quest is by its nature unending. This book will occupy a
modest place in a long line of outstanding treatises. It will not be the last
of them, to be sure, for so many more things remain undiscovered or not
yet understood. All the same, I think I have brought together practically
everything of significance from what the number-based sciences, of
the logical and historical kinds, have to teach us at the moment.
Consequently, this is also probably the only book ever written that gives
NTRODUCTION
a more or less universal and comprehensive history of numbers and
numerical calculation, set out in a logical and chronological way, and
made accessible in plain language to the ordinary reader with no prior
knowledge of mathematics.
And since research never stands still, I have been able to bring new
solutions to some problems and to open up other, long-neglected areas
of the universe of numbers. For example, in one of the chapters you will
find a solution to the thorny problem of the decipherment of Elamite
numbering, used nearly five thousand years ago in what is now Iran.
I have also shown that Roman numbering, long thought to have been
derived from the Greek system, was in fact a “prehistoric fossil”,
developed from the very ancient practice of notching. There are also
some new contributions on Mesopotamian numbering and arithmetic,
as well as a quite new way of looking at the fascinating and sensitive
topic of how “our" numbers evolved from the unlikely conjunction of
several great ideas. Similarly, the history of mechanical calculation
culminating in the invention of the computer is entirely new.
A VERY LONG STORY
If you wanted to schematise the history of numbering systems, you
could say that it fills the space between One and Zero, the two concepts
which have become the symbols of modern technological society.
Nowadays we step with careless ease from Zero to One, so confident
are we, thanks to computer scientists and our mathematical masters,
that the Void always comes before the Unit. We never stop to think for a
moment that in terms of time it is a huge step from the invention of the
number “one”, the first of all numbers even in the chronological sense,
to the invention of the number “zero”, the last major invention in the
story of numbers. For in fact the whole history of humanity is spread
out backwards between the time when it was realised that the void
was “nothing” and the time when the sense of “oneness” first arose, as
humans became aware of their individual solitude in the face of life and
death, of the specificity of their species as distinct from other living
beings, of the singularity of their selves as distinct from others, or of the
difference of their sex as distinct from that of their partners.
But the story is neither abstract nor linear, as the history of
mathematics is sometimes (and erroneously) imagined to be. Far from
being an impeccable sequence of concepts each leading logically to the
next, the history of numbers is the story of the needs and concerns of
enormously diverse social groupings trying to count the days in the year,
to make deals and bargains, to list their members, their marriages, their
bereavements, their goods and flocks, their soldiers, their losses, and
even their prisoners, trying also to record the date of the foundation of
their cities or of one of their victories.
Goatherds and shepherds needed to know when they brought their
flocks back from grazing that none had been lost; people who kept
stocks of tools or arms or stood guard over food supplies for a commu-
nity needed to know whether the complement of tools, arms or supplies
had remained the same as when they last checked. Or again, com-
munities with hostile neighbours must have been concerned to know
whether, after each military foray, they still had the same number of
soldiers, and, if not, how many they had lost in the fight. Communities
that engaged in trading needed to be able to “reckon” so as to be able to
buy or barter goods. For harvesting, and also in order to prepare in time
for religious ceremonies, people needed to be able to count and to
measure time, or at the very least to develop some practical means of
managing in such circumstances.
In a word, the history of numbers is the story of humanity being led by
the very nature of the things it learned to do to conceive of needs that
could only be satisfied by “number reckoning”. And to do that, everything
and anything was put in service. The tools were approximate, concrete,
and empirical ones before becoming abstract and sophisticated,
originally imbued with strange mystical and mythological properties,
becoming disembodied and generalisable only in the later stages.
Some communities were utilitarian and limited the aims of their
counting systems to practical applications. Others saw themselves in the
infinite and eternal elements, and used numbers to quantify the heavens
and the earth, to express the lengths of the days, months and years since
the creation of the universe, or at least from some date of origin whose
meaning had subsequently been lost. And because they found that they
needed to represent very large numbers, these kinds of communities did
not just invent more symbols, but went down a path that led not only
towards the fundamental rule of position, but also onto the track of
a very abstract concept that we call “zero”, whence comes the whole
of mathematics.
THE FIRST STEPS
No one knows where or when the story began, but it was certainly a very
long time ago. That was when people were unable to conceive of
numbers as such, and therefore could not count. They were capable, at
most, of the concepts of one, two, and many.
As a result of studies carried out on a wide range of beings, from
XIX
THE EARLIEST COUNTING MACHINES
crows to humans as diverse as infants, Pygmies, and the Amerindian
inhabitants of Tierra del Fuego, psychologists and ethnologists have
been able to establish the absolute zero of human number-perception.
Like some of the higher animals, the human adult with no training at all
(for example, learning to recognise the 5 or the 6 at cards by sight,
through sheer practice) has direct and immediate perception of the
numbers 1 to 4 only. Beyond that level, people have to learn to count. To
do that they need to develop, firstly, advanced number-manipulating
skills, then, for the purposes of memorisation and of communication,
they need to develop a linguistic instrument (the names of the
numbers), and, finally, and much later on, they need to devise a scheme
for writing numbers down.
However, you do not have to “count” the way we do if what you want
to do is to find the date of a ceremony, or to make sure that the sheep
and the goats that set off to graze have all come back to the byre. Even in
the complete absence of the requisite words, of sufficient memory, and
of the abstract concepts of number, there are all sorts of effective
substitute devices for these kinds of operation. Various present-day
populations in Oceania, America, Asia, and Africa whose languages
contain only the words for one, two, and many, but who nonetheless
understand one-for-one parities perfectly well, use notches on bones or
wooden sticks to keep a tally. Other populations use piles or lines of
pebbles, shells, knucklebones, or sticks. Still others tick things off by
the parts of their body (fingers, toes, elbows and knees, eyes, nose,
mouth, ears, breasts, and chest).
THE EARLIEST COUNTING MACHINES
Early humanity used more or less whatever came to hand to manage in a
quantitative as well as a qualitative universe. Nature itself offered every
cardinal model possible: birds with two wings, the three parts of a
clover-leaf, four-legged animals, and five-fingered hands . . . But as
everyone began counting by using their ten fingers, most of the
numbering systems that were invented used base 10. All the same,
some groups chose base 12. The Mayans, Aztecs, Celts, and Basques,
looked down at their feet and realised that their toes could be counted
like fingers, so they chose base 20. The Sumerians and Babylonians,
however, chose to count on base 60, for reasons that remain mysterious.
That is where our present division of the hour into 60 minutes of 60
seconds comes from, as does the division of a circle into 360 degrees,
each of 60 minutes divided into 60 seconds.
The very oldest counting tools that archaeologists have yet dug up are
the numerous animal bones found in western Europe and marked with
one or more sets of notches. These tally sticks are between twenty
thousand and thirty-five thousand years old.
The people using these bones were probably fearsome hunters, and,
for each kill, they would score another mark onto the tally stick.
Separate counting bones might have been used for different animals -
one tally for bears, another for bison, another for wolves, and so on.
They had also invented the first elements of accounting, since what
they were actually doing was writing numbers in the simplest notation
known.
The method may seem primitive, but it turned out to be remarkably
robust, and is probably the oldest human invention (apart from fire)
still in use today. Various tallies found on cave walls next to animal
paintings leave us in little real doubt that we are dealing with an
animal-counting device. Modern practice is no different. Since time
immemorial, Alpine shepherds in Austria and Hungary, just like Celtic,
Tuscan, and Dalmatian herdsmen, have checked off their animals by
scoring vertical bars, Vs and Xs on a piece of wood, and that is still how
they do it today. In the eighteenth century, the same “five-barred gate”
was used for the shelf marks of parliamentary papers at the British
House of Commons Library; it was used in Tsarist Russia and in
Scandinavia and the German-speaking countries for recording loans
and for calendrical accounts; whereas in rural France at that time,
notched sticks did all that present-day account books and contracts do,
and in the open markets of French towns they served as credit “slates”.
Barely twenty years ago a village baker in Burgundy made notches
in pieces of wood when he needed to tot up the numbers of loaves
each of his customers had taken on credit. And in nineteenth-century
Indo-China, tally sticks were used as credit instruments, but also as
signs of exclusion and to prevent contact with cholera victims. Finally,
in Switzerland, we find notched sticks used, as elsewhere, for credit
reckonings, but also for contracts, for milk deliveries, and for recording
the amounts of water allocated to different grazing meadows.
The long-lasting and continuing currency of the tally system is all the
more surprising for being itself the source of the Roman numbering
system, which we also still use alongside or in place of Arabic numerals.
The second concrete counting tool, the hand, is of course even older.
Every population on earth has used it at one stage or another. In various
places in Auvergne (France), in parts of China, India, Turkey, and the
former Soviet Union, people still do multiplication sums with their
fingers, as the numbers are called out, and without any other tool or
device. Using joints and knuckles increases the possible range, and it
INTRODUCTION
XX
allowed the Ancient Egyptians, the Romans, the Arabs and the Persians
(not forgetting Western Christians in the Middle Ages) to represent
concretely all the numbers from 1 to 9,999. An even more ingenious
variety of finger-reckoning allowed the Chinese to count to 100,000 on
one hand, and to one million using both hands!
But the story of numbers can be told in other ways too. In places as
far apart as Peru, Bolivia, West Africa, Hawaii, the Caroline Islands, and
Ryu-Kyu, off the Japanese coast, you can find knotted string used to
represent numbers. It was with such a device that the Incas sorted the
archives of their very effective administration.
A third system has a far from negligible role in the history of
arithmetic - the use of pebbles, which really underlies the beginning
of calculation. The pebble-method is also the direct ancestor of the
abacus, a device still in wide use in China, Japan and Eastern Europe.
But it is the very word calculation that sends us back most firmly to the
pebble-method: for in Latin the word for pebble is calculus.
THE FIRST NUMBERS IN HISTORY
The pebble-method actually formed the basis for the first written
numbering system in recorded history. One day, in the fourth millen-
nium BCE, in Elam, located in present-day Iran towards the Persian
Gulf, accountants had the idea of using moulded, unbaked clay tokens
in the place of ordinary or natural pebbles. The tokens of various shapes
and sizes were given conventional values, each different type represent-
ing a unit of one order of magnitude within a numbering system: a stick
shape for 1, a pellet for 10, a ball for 100, and so on. The idea must have
been in the air for a long time, for at about the same period a similarly
clay-based civilisation in Sumer, in lower Mesopotamia, invented an
identical system. But since the Sumerians counted to base 60 (sexagesi-
mal reckoning), their system was slightly different: a small clay cone
stood for 1, a pellet stood for 10, a large cone for 60, a large perforated
cone stood for 600, a ball meant 3,600, and so on.
These civilisations were in a phase of rapid expansion but remained
exclusively oral, that is to say without writing. They relied on the rather
limited potential of human memory. But the accounting system that
was developed from the principles just explained turned out to be very
serviceable. In the first development, the idea arose of enclosing the
tokens in a spherical clay case. This allowed the system not only to serve
for actual arithmetical operations, but also for keeping a record of
inventories and transactions of all kinds. If a check on past dealings
was needed, the clay cases could be broken open. But the second
development was even more pregnant. The idea was to symbolise on the
outside of the clay case the objects that were enclosed within it: one
notch on the case signified that there was one small cone inside, a pellet
was symbolised by a small circular perforation, a large cone by a thick
notch, a ball by a circle, and so on. Which is how the oldest numbers in
history, the Sumerian numerals, came into being, around 3200 BCE.
This story is obviously related to the origins of writing, but it must
not be confused with it entirely. Writing serves not only to give a visual
representation to thought and a physical form to memory (a need felt by
all advanced societies), but above all to record articulated speech.
THE COMMON STRUCTURE OF THE HUMAN MIND
It is extraordinary to see how peoples very distant from each other in
time and space used similar methods to reach identical results.
All societies learned to number their own bodies and to count on
their fingers; and the use of pebbles, shells and sticks is absolutely
universal. So the fact that the use of knotted string occurs in China,
in Pacific island communities, in West Africa, and in Amerindian
civilisations does not require us to speculate about migrations or long-
distance travellers in prehistory. The making of notches to represent
number is just as widespread in historical and geographical terms. Since
the marking of bone and wood has the same physical requirements and
limitations wherever it is done, it is no surprise that the same kinds of
lines, Vs and Xs are to be seen on armbones and pieces of wood found
in places as far apart as Europe, Asia, Affica, Oceania and the Americas.
That is also why these marks crop up in virtually identical form in
civilisations as varied as those of the Romans, the Chinese, the Khas
Boloven of Indo-China, the Zuni Indians of New Mexico, and amongst
contemporary Dalmatian and Celtic herdsmen. It is therefore not at
all surprising that some numbers have almost always been represented
by the same figure: 1, for instance, is represented almost universally by
a single vertical line; 5 is also very frequently, though slightly less
universally, figured by a kind of V in one orientation or another, and 10
by a kind of X or by a horizontal bar.
Similarly, the Ancient Egyptians, the Hittites, the Greeks, and the
Aztecs worked out written numbering systems that were structurally
identical, even if their respective base numbers and figurations varied
considerably. Likewise the common system of Sumerian, Roman, Attic,
and South Arabian numbering. Several family groupings of the same
kind can be found in other sets of unrelated cultures. There is no need to
hypothesise actual contact between the cultures in order to explain the
XXI
NUMBERS AND LETTERS
similarities between their numbering systems.
So it would seem that human beings possess, in all places and at all
times, a permanent capacity to repeat an invention or discovery already
made elsewhere, provided only that the society or individual involved
encounters cultural, social, and psychological conditions similar to
those that prevailed when the invention was first made.
This is what explains why in modern science, the same discovery is
sometimes made at almost the same time by two different scientists
working in complete isolation from each other. Famous examples of
such coincidences of invention include the simultaneous development
of analytical geometry by Descartes and Fermat, of differential calculus
by Newton and Leibnitz, of the physical laws of gasses by Boyle and
Mariotte, and of the principles of thermodynamics by Joule, Mayer, and
Sadi Carnot.
NUMBERS AND LETTERS
Ever since the invention of alphabetic writing by the Phoenicians (or at
least, by a northwestern Semitic people) in the second millennium
BCE, letters have been used for numbers. The simplicity and ingenuity
of the alphabetic system led to its becoming the most widespread
form of writing, and the Phoenician scheme is at the root of nearly
every alphabet in the world today, from Ffebrew to Arabic, from Berber
to Hindu, and of course Greek, which is the basis of our present
(Latin) lettering.
Given their alphabets, the Greeks, the Jews, the Arabs and many other
peoples thought of writing numbers by using letters. The system consists
of attributing numerical values from 1 to 9, then in tens from 10 to 90,
then in hundreds, etc., to the letters in their original Phoenician order
(an order which has remained remarkably stable over the millennia).
Number-expressions constructed in this way worked as simple
accumulations of the numerical values of the individual letters. The
mathematicians of Ancient Greece rationalised their use of letter-
numbers within a decimal system, and, by adding diacritic signs to the
base numbers, became able to express numbers to several powers of 10.
In poetry and literature, however, and especially in the domains of
magic, mysticism, and divination, it was the sum of the number-values
of the letters in a word that mattered.
In these circumstances, every word acquired a number-value, and
conversely, every number was “loaded” with the symbolic value of one
or more words that it spelled. That is why the number 26 is a divine
number in Jewish lore, since it is the sum of the number-values of the
letter that spell YAHWEH, the name of God:
mm =5+6+5+10
The Jews, Greeks, Romans, Arabs (and as a result, Persians and
Muslim Turks) pursued these kinds of speculation, which have very
ancient origins: Babylonian writings of the second millennium BCE
attribute a numerical value to each of the main gods: 60 was associated
with Anu, god of the sky; 50 with Enlil, god of the earth; 40 with Ea, god
of water, and so forth.
The device also allowed poets like Leonidas of Alexandria to compose
quite special kinds of work. It is also the basis for the art of the
chronogram (verses that express a date simultaneously in words and
in numbers) that can be found amongst the poets and stone-carvers of
North Africa, Turkey, and Iran.
From ancient times to the present, the device has given a rich field to
cabbalists, Gnostics, magicians, soothsayers, and mystics of every hue,
and innumerable speculations, interpretations, calculations and predic-
tions have been built on letter-number equivalences. The Gnostics, for
example, thought they could work out the "formula” and thus the true
name of God, which would enable them to penetrate all the secrets of
the divine. Several religious sects are based on beliefs of this kind (such
as the Hurufi or “Lettrists” of Islam) and they still have many followers,
some of them in Europe.
The Greeks and Jews who first established a number-coded alphabet
certainly could not have imagined that fifteen hundred or two thousand
years later a Catholic theologian called Petrus Bungus would churn out
a seven-hundred page numerological treatise “proving” (subject to a few
spelling improvements!) that the name of Martin Luther added up to
666. It was a proof that the “isopsephic” initiates knew how to read,
since according to St John the Apostle, 666 was the number of the “Beast
of the Apocalypse”, that is to say the Antichrist. Bungus was neither the
first nor the last to make use of these methods. In the late Roman
Empire, Christians tried to make Nero’s name come to 666; during
World War II, would-be numerological prophets managed to “prove”
that Hitler was the real "Beast of the Apocalypse”. A discovery that
many had already made without the help of numbers.
THE HISTORY OF A GREAT INVENTION
Logic was not the guiding light of the history of number-systems. They
were invented and developed in response to the concerns of accoun-
tants, first of all, but also of priests, astronomers, and astrologers, and
INTRODUCTION
only in the last instance in response to the needs of mathematicians.
The social categories dominant in this story are notoriously conser-
vative, and they probably acted as a brake on the development and
above all on the accessibility of numbering systems. After all, knowledge
(however rudimentary it may now appear) gives its holders power and
privilege; it must have seemed dangerous, if not irreligious, to share it
with others.
There were also other reasons for the slow and fragmentary develop-
ment of numbers. Whereas fundamental scientific research is pursued
in terms of scientists’ own criteria, inventions and discoveries only get
developed and adopted if they correspond to a perceived social need in
a civilisation. Many scientific advances are ignored if there is, as people
say, no “call” for them.
The stages of mathematical thought make a fascinating story.
Most peoples throughout history failed to discover the rule of
position, which was discovered in fact only four times in the history
of the world. (The rule of position is the principle of a numbering
system in which a 9, let’s say, has a different magnitude depending on
whether it comes in first, second, third . . . position in a numerical
expression.) The first discovery of this essential tool of mathematics
was made in Babylon in the second millennium BCE. It was then
rediscovered by Chinese arithmeticians at around the start of the
Common Era. In the third to fifth centuries CE, Mayan astronomers
reinvented it, and in the fifth century CE it was rediscovered for the
last time, in India.
Obviously, no civilisation outside of these four ever felt the need
to invent zero; but as soon as the rule of position became the basis for
a numbering system, a zero was needed. All the same, only three of
the four (the Babylonians, the Mayans and the Indians) managed to
develop this final abstraction of number: the Chinese only acquired it
through Indian influences. However, the Babylonian and Mayan zeros
were not conceived of as numbers, and only the Indian zero had roughly
the same potential as the one we use nowadays. That is because it is
indeed the Indian zero, transmitted to us through the Arabs together
with the number-symbols that we call Arabic numerals and which are in
reality Indian numerals, with their appearance altered somewhat by
time, use and travel.
Our knowledge of the history of numbers is of course only
fragmentary, but all the pieces converge inexorably towards the system
that we now use and which in recent times has conquered the
whole planet.
XXII
COMPUTATION, FIGURES, AND NUMBERS
Arithmetic has a history that is by no means limited to the history of
the figures we use to represent numbers. In this history of computation,
figures arose quite late on; and they constitute only one of many
possible ways of representing number-concepts. The history of numbers
ran parallel to the history of computation, became part of it only when
modern written arithmetic was invented, and then separated out again
with the development of modern calculating machines.
Numbers have become so integrated into our way of thinking that
they often seem to be a basic, innate characteristic of human beings, like
walking or speaking. But that is not so. Numbers belong to human culture,
not nature, and therefore have their own long history. For Plato, numbers
were “the highest degree of knowledge” and constituted the essence of
outer and inner harmony. The same idea was taken up in the Middle
Ages by Nicholas Cusanus, for whom “numbers are the best means of
approaching divine truths”. These views all go back to Pythagoras, for
whom “numbers alone allow us to grasp the true nature of the universe”.
In truth, though, it is not numbers that govern the universe. Rather,
there are physical properties in the world which can be expressed in
abstract terms through numbers. Numbers do not come from things
themselves, but from the mind that studies things. Which is why the
history of numbers is a profoundly human part of human history.
IN CONCLUSION
Once a person’s curiosity, on any subject, is aroused it is surprising
just how far it may lead him in pursuit of its object, how readily it
overcomes every obstacle. In my own case my curiosity about, or rather
my absolute fascination with, numbers has been well served by a number
of assets with which I set out: a Moroccan by birth, a Jew by cultural
heritage, I have been afforded a more immediate access to the study of
the work of Arab and Hebrew mathematicians than I might have obtained
as a born European. I could harmonise within myself the mind-set of
Eastern metaphysics with the Cartesian logic of the West. And I was
able to identify the basic rules of a highly complex system. Moreover I
possessed a sufficient aptitude for drawing to enable me to make simple
illustrations to help clarify my text. I hope that the reader will recognise
in this History that numbers, far from being tedious and dry, are charged
with poetry, are the very vehicle for traditional myths and legends - and
the finest witness to the cultural unity of the human race.
THE UNIVERSAL HISTORY OF NUMBERS
3
CAN ANIMALS COUNT?
CHAPTER 1
EXPLAINING THE ORIGINS
Ethnological and Psychological Approaches
to the Sources of Numbers
WHEN THE SLATE WAS CLEAN
There must have been a time when nobody knew how to count. All we can
surmise is that the concept of number must then have been indissociable
from actual objects - nothing very much more than a direct apperception
of the plurality of things. In this picture of early humanity, no one would
have been able to conceive of a number as such, that is to say as an abstrac-
tion, nor to grasp the fact that sets such as “day-and-night”, a brace of hares,
the wings of a bird, or the eyes, ears, arms and legs of a human being had
a common property, that of “being two”.
Mathematics has made such rapid and spectacular progress in what are
still relatively recent periods that we may find it hard to credit the existence
of a time without number. However, research into behaviour in early
infancy and ethnographic studies of contemporary so-called primitive
populations support such a hypothesis.
CAN ANIMALS COUNT?
Some animal species possess some kind of notion of number. At a rudi-
mentary level, they can distinguish concrete quantities (an ability that must
be differentiated from the ability to count numbers in abstract). For want
of a better term we will call animals’ basic number-recognition the seme
of number. It is a sense which human infants do not possess at birth.
Humans do not constitute the only species endowed with intelligence:
the higher animals also have considerable problem-solving abilities. For
example, hungry foxes have been seen to “play dead” so as to attract the
crows they intend to eat. In Kenya, lions that previously hunted alone
learned to hunt in a pack so as to chase prey towards a prepared ambush.
Monkeys and other primates, of course, are not only able to make tools but
also to learn how to manipulate non-verbal symbols. A much-quoted
example of the first ability is that of the monkey who constructed a long
bamboo tube so as to pick bananas that were out of reach. Chimpanzees
have been taught to use tokens of different shapes to obtain bananas,
grapes, water, and so on, and some even ended up hoarding the tokens
against future needs. However, we must be careful not to be taken in by the
kind of “animal intelligence” that you can see at the circus and the
fairground. Dogs that can “count” are examples of effective training or
(more likely) of clever trickery, not of the intellectual properties of canine
minds. However, there are some very interesting cases of number-sense in
the animal world.
Domesticated animals (for instance, dogs, cats, monkeys, elephants)
notice straight away if one item is missing from a small set of familiar
objects. In some species, mothers show by their behaviour that they know
if they are missing one or more than one of their litter. A sense of number
is marginally present in such reactions. The animal possesses a natural
disposition to recognise that a small set seen for a second time has
undergone a numerical change.
Some birds have shown that they can be trained to recognise more
precise quantities. Goldfinches, when trained to choose between two differ-
ent piles of seed, usually manage to distinguish successfully between three
and one, three and two, four and two, four and three, and six and three.
Even more striking is the untutored ability of nightingales, magpies and
crows to distinguish between concrete sets ranging from one to three or
four. The story goes that a squire wanted to destroy a crow that had made
its nest in his castle’s watchtower. Each time he got near the nest, the crow
flew off and waited on a nearby branch for the squire to give up and go
down. One day the squire thought of a trick. He got two of his men to go
into the tower. After a few minutes, one went down, but the other stayed
behind. But the crow wasn’t fooled, and waited for the second man to go
down too before coming back to his nest. Then they tried the trick with
three men in the tower, two of them going down: but the third man could
wait as long as he liked, the crow knew that he was there. The ploy only
worked when five or six men went up, showing that the crow could not
discriminate between numbers greater than three or four.
These instances show that some animals have a potential which is
more fully developed in humans. What we see in domesticated animals
is a rudimentary perception of equivalence and non-equivalence between
sets, but only in respect of numerically small sets. In goldfinches, there is
something more than just a perception of equivalence - there seems to be
a sense of “more than” and “less than”. Once trained, these birds seem to
have a perception of intensity, halfway between a perception of quantity
(which requires an ability to numerate beyond a certain point) and
a perception of quality. However, it only works for goldfinches when the
“moreness” or “lessness" is quite large; the bird will almost always confuse
five and four, seven and five, eight and six, ten and six. In other words,
goldfinches can recognise differences of intensity if they are large enough,
but not otherwise.
EXPLAINING THE ORIGINS
4
Crows have rather greater abilities: they can recognise equivalence and
non-equivalence, they have considerable powers of memory, and they can
perceive the relative magnitudes of two sets of the same kind separated in
time and space. Obviously, crows do not count in the sense that we do,
since in the absence of any generalising or abstracting capacity they cannot
conceive of any “absolute quantity”. But they do manage to distinguish
concrete quantities. They do therefore seem to have a basic number-sense.
NUMBERS AND SMALL CHILDREN
Human infants have few innate abilities, but they do possess something
that animals never have: a potential to assimilate and to recreate stage by
stage the conquests of civilisation. This inherited potential is only brought
out by the training and education that the child receives from the adults
and other children in his or her environment. In the absence of permanent
contact with a social milieu, this human potential remains undeveloped -
as is shown by the numerous cases of enfants sauvages. (These are children
brought up by or with animals in the wild, as in Francois Truffaut’s film, The
Wild Child. Of those recaptured, none ever learned to speak and most died
in adolescence.)
We should not imagine a child as a miniature adult, lacking only judge-
ment and knowledge. On the contrary, as child psychology has shown,
children live in their own worlds, with distinct mentalities obeying their
own specific laws. Adults cannot actually enter this world, cannot go back
to their own beginnings. Our own childhood memories are illusions, recon-
structions of the past based on adult ways of thinking.
But infancy is nonetheless the necessary prerequisite for the eventual
transformation of the child into an adult. It is a long-drawn-out phase of
preparation, in which the various stages in the development of human
intelligence are re-enacted and reconstitute the successive steps through
which our ancestors must have gone since the dawn of time.
According to N. Sillamy (1967), three main periods are distinguished:
infancy (up to three years of age), middle childhood (from three to six or
seven); and late childhood, which ends at puberty. However, a child’s intel-
lectual and emotional growth does not follow a steady and linear pattern.
Piaget (1936) distinguishes five well-defined phases:
1. a sensory -motor period (up to two years of age) during which the
child forms concepts of “object" out of fragmentary perceptions
and the concept of “self” as distinct from others;
2. a pre-operative stage (from two to four years of age), charac-
terised by egocentric and anthropomorphic ways of thinking
(“look, mummy, the moon is following me!");
3. an intuitive period (from four to six), characterised by intellec-
tual perceptions unaccompanied by reasoning; the child performs
acts which he or she would be incapable of deducing, for example,
pouring a liquid from one container into another of a different
shape, whilst believing that the volume also changes;
4. a stage of concrete operations (from eight to twelve) in which,
despite acquiring some operational concepts (such as class, series,
number, causality), the child’s thought-processes remain firmly
bound to the concrete;
5. a period (around puberty) characterised by the emergence of
formal operations, when the child becomes able to make hypothe-
ses and test them, and to operate with abstract concepts.
Even more precisely: the new-born infant in the cradle perceives the
world solely as variations of light and sound. Senses of touch, hearing and
sight slowly grow more acute. From six to twelve months, the infant
acquires some overall grasp of the space occupied by the things and people
in its immediate environment. Little by little the child begins to make
associations and to perceive differences and similarities. In this way the
child forms representations of relatively simple groupings of beings and
objects which are familiar both by nature and in number. At this age,
therefore, the child is able to reassemble into one group a set of objects
which have previously been moved apart. If one thing is missing from a
familiar set of objects, the child immediately notices. But the abstraction
of number - which the child simply feels, as if it were a feature of the
objects themselves - is beyond the child’s grasp. At this age babies do not
use their fingers to indicate a number.
Between twelve and eighteen months, the infant progressively learns to
distinguish between one, two and several objects, and to tell at a glance the
relative sizes of two small collections of things. However, the infant’s
numerical capabilities still remain limited, to the extent that no clear
distinction is made between the numbers and the collections that they
represent. In other words, until the child has grasped the generic principle
of the natural numbers (2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1, etc.), numbers remain
nothing more than “number-groupings”, not separable from the concrete
nature of the items present, and they can only be recognised by the
principle of pairing (for instance, on seeing two sets of objects lined up next
to each other).
Oddly enough, when a child has acquired the use of speech and learned
to name the first few numbers, he or she often has great difficulty in
symbolising the number three. Children often count from one to two and
then miss three, jumping straight to four. Although the child can recognise,
visually and intuitively, the concrete quantities from one to four, at this
5
NUMBERS AND THE PRIMITIVE MIND
stage of development he or she is still at the very doorstep of abstract
numbering, which corresponds to one, two, many.
However, once this stage is passed (at between three and four years of
age, according to Piaget), the child quickly becomes able to count properly.
From then on, progress is made by virtue of the fact that the abstract
concept of number progressively takes over from the purely perceptual
aspect of a collection of objects. The road lies open which leads on to the
acquisition of a true grasp of abstract calculation. For this reason, teachers
call this phase the “pre-arithmetical stage" of intellectual development.
The child will first learn to count up to ten, relying heavily on the use of
fingers; then the number series is progressively extended as the capacity for
abstraction increases.
ARITHMETIC AND THE BODY
The importance of the hand, and more generally of the body in children’s
acquisition of arithmetic can hardly be exaggerated. Inadequate access to
or use of this “counting instrument” can cause serious learning difficulties.
In earliest infancy, the child plays with his or her fingers. It constitutes
the first notion of the child’s own body. Then the child touches every-
thing in order to make acquaintance with the world, and this also is
done primarily with the hands. One day, a well-intentioned teacher
who wanted arithmetic to be “mental”, forbade finger-counting in his
class. Without realising it, the teacher had denied the children the use
of their bodies, and forbidden the association of mathematics with
their bodies. I’ve seen many children profoundly relieved to be able to
use their hands again: their bodies were at last accepted [ . . . ] Spatio-
temporal disabilities can likewise make learning mathematics very
difficult. Inadequate grasp of the notions of “higher than” and “lower
than” affect the concepts of number, and all operations and relations
between them. The unit digits are written to the right, and the
hundred digits are written to the left, so a child who cannot tell left
from right cannot write numbers properly or begin an operation at all
easily. Number skills and the whole set of logical operations of arith-
metic can thus be seriously undermined by failure to accept the body.
[L. Weyl-Kailey (1985)]
NUMBERS AND THE PRIMITIVE MIND
A good number of so-called primitive people in the world today seem
similarly unable to grasp number as an abstract concept. Amongst these
populations, number is “felt” and “registered”, but it is perceived as
a quality, rather as we perceive smell, colour, noise, or the presence of a
person or thing outside of ourselves. In other words, “primitive” peoples
are affected only by changes in their visual field, in a direct subject-object
relationship. Their grasp of number is thus limited to what their predispo-
sitions allow them to see in a single visual glance.
However, that does not mean that they have no perception of quantity. It
is just that the plurality of beings and things is measured by them not in a
quantitative but in a qualitative way, without differentiating individual
items. Cardinal reckoning of this sort is never fixed in the abstract, but
always related to concrete sets, varying naturally according to the type of
set considered.
A well-defined and appropriately limited set of things or beings,
provided it is of interest to the primitive observer, will be memorised
with all its characteristics. In the primitive’s mental representation
of it, the exact number of the things or beings involved is implicit:
it resembles a quality by which this set is different from another group
consisting of one or several more or fewer members. Consequently,
when he sets eyes on the set for a second time, the primitive knows
if it is complete or if it is larger or smaller than it was previously.
[L. Levy-Bruhl (1928)]
ONE, TWO . . . MANY
In the first years of the twentieth century, there were several “primitive”
peoples still at this basic stage of numbering: Bushmen (South Africa),
Zulus (South and Central Africa), Pygmies (Central Africa), Botocudos
(Brazil), Fuegians (South America), the Kamilarai and Aranda peoples in
Australia, the natives of the Murray Islands, off Cape York (Australia), the
Vedda (Sri Lanka), and many other “traditional” communities.
According to E. B. Tylor (1871), the Botocudos had only two real terms
for numbers: one for “one”, and the other for “a pair”. With these lexical
items they could manage to express three and four by saying something
like “one and two” and “two and two”. But these people had as much
difficulty conceptualising a number above four as it is for us to imagine
quantities of a trillion billions. For larger numbers, some of the Botocudos
just pointed to their hair, as if to say “there are as many as there are hairs
on my head”.
A. Sommerfelt (1938) similarly reports that the Aranda had only two
number-terms, ninta (one), and tarn (two). Three and four were expressed
as tara-mi-ninta (one and two) and tara-ma-tara (“two and two”), and the
number series of the Aranda stopped there. For larger quantities, imprecise
terms resembling “a lot”, “several” and so on were used.
EXPLAINING THE ORIGINS
6
Likewise G. Hunt (1899) records the Murray islanders’ use of the terms
netat and neis for “one" and “two”, and the expressions neis-netat (two +
one) for "three”, and neis-neis (two + two) for “four”. Higher numbers were
expressed by words like “a crowd of . .
Our final example is that of the Torres Straits islanders for whom urapun
meant “one”, okosa “two”, okosa-urapun (two-one) “three”, and okosa-okosa
(two-two) “four”. According to A. C. Haddon (1890) these were the only
terms used for absolute quantities; other numbers were expressed by the
word ras, meaning “a lot”.
Attempts to teach such communities to count and to do arithmetic in the
Western manner have frequently failed. There are numerous accounts of
natives' lack of memory, concentration and seriousness when confronted
with numbers and sums [see, for example, M. Dobrizhoffer (1902)]. It
generally turned out much easier to teach primitive peoples the arts of
music, painting, and sculpture than to get them to accept the interest and
importance of arithmetic. This was perhaps not just because primitive
peoples felt no need of counting, but also because numbers are amongst the
most abstract concepts that humanity has yet devised. Children take longer
to learn to do sums than to speak or to write. In the history of humanity,
too, numbers have proved to be the hardest of these three skills.
PARITY BEFORE NUMBER
These primitive peoples nonetheless possessed a fundamental arithmetical
rule which if systematically applied would have allowed them to manipu-
late numbers far in excess of four. The rule is what we call the principle
of base 2 (or binary principle). In this kind of numbering, five is “two-
two-one”, six is “two-two-two”, seven is “two-two-two-one”, and so on.
But primitive societies did not develop binary numbering because, as
L. Gerschel (1960) reminds us, they possessed only the most basic degree
of numeracy, that which distinguishes between the singular and the dual.
A. C. Haddon (1890), observing the western Torres Straits islanders,
noted that they had a pronounced tendency to count things in groups of
two or in couples. M. Codrington, in Melanesian Languages, noticed the
same thing in many Oceanic populations: “The natives of Duke of York’s
Island count in couples, and give the pairings different names depending
how many of them there are; whereas in Polynesia, numbers are used
although it is understood that they refer to so many pairs of things, not
to so many things.” Curr, as quoted by T. Dantzig (1930), confirms that
Australian aborigines also counted in this way, to the extent that “if two
pins are removed from a set of seven the aborigines rarely notice it, but they
see straight away if only one is removed”.
These primitive peoples obviously had a stronger sense of parity than of
number. To express the numbers three and four, numbers they did not
grasp as abstracts but which common sense allowed them to see in a single
glance, they had recourse only to concepts of one and pair. And so for them
groups like “two-one” or “two-two” were themselves pairs, not (as for us)
the abstract integers (or “whole numbers”) “three” and “four”. So it is easy
to see why they never developed the binary system to get as far as five and
six, since these would have required three digits, one more than the pair
which was their concept of the highest abstract number.
THE LIMITS OF PERCEPTION
The limited arithmetic of “primitive” societies does not mean that their
members were unintelligent, nor that their innate abilities were or are
lesser than ours. It would be a grave error to think that we could do better
than a Torres Straits islander at recognising number if all we had to use
were our natural faculties of perception.
In practice, when we want to distinguish a quantity we have recourse to
our memories and/or to acquired techniques such as comparison, splitting,
mental grouping, or, best of all, actual counting. For that reason it is rather
difficult to get to our natural sense of number. There is an exercise that we
can try, all the same. Looking at Fig. 1.1, which contains sets of objects in
line, try to estimate the quantity of each set of objects in a single visual
glance (that is to say, without counting). What is the best that we can do?
Fig. i.i.
7
THE LIMITS OF PERCEPTION
Everyone can see the sets of one, of two, and of three objects in the
figure, and most people can see the set of four. But that's about the limit of
our natural ability to numerate. Beyond four, quantities are vague, and our
eyes alone cannot tell us how many things there are. Are there fifteen or
twenty plates in that pile? Thirteen or fourteen cars parked along the
street? Eleven or twelve bushes in that garden, ten or fifteen steps on this
staircase, nine, eight or six windows in the facade of that house? The correct
answers cannot be just seen. We have to count to find out!
The eye is simply not a sufficiently precise measuring tool: its natural
number-ability virtually never exceeds four.
There are many traces of the “limit of four" in different languages and
cultures. There are several Oceanic languages, for example, which distin-
guish between nouns in the singular, the dual, the triple, the quadruple,
and the plural (as if in English we were to say one bird, two birdo, three birdi,
four birdu, many birds).
In Latin, the names of the first four numbers ( unus , duos, tres, quatuor)
decline at least in part like other nouns and adjectives, but from five
(quinque), Latin numerical terms are invariable. Similarly, Romans gave
“ordinary” names to the first four of their sons (names like Marcus, Servius,
Appius, etc.), but the fifth and subsequent sons were named only by
a numeral: Quintus (the fifth), Sixtus (the sixth), Septimus (the seventh),
and so on. In the original Roman calendar (the so-called “calendar of
Romulus”), only the first four months had names (Martius, Aprilis, Maius,
Junius), the fifth to tenth being referred to by their order-number: Quintilis,
Sextilis, September, October, November, December.*
Perhaps the most obvious confirmation of the basic psychological rule
of the “limit of four” can be found in the almost universal counting-device
called (in England) the “five-barred gate”. It is used by innkeepers keeping
a tally or “slate” of drinks ordered, by card-players totting up scores, by
prisoners keeping count of their days in jail, even by examiners working out
the mark-distribution of a cohort of students:
1
I
6
m 1
11 tttt I
2
II
7
m 11
12 HH HH II
3
III
8
m hi
B m m hi
4
IIII
9
HH IIII
14 «H Hft IIII
5
HH
10
jlH mi
15 «H fflffll
Fig. 1.2. The five-barred gate
* The original ten-month Roman calendar had 304 days and began with Martius. It was subsequently
lengthened by the addition of two further months, Januarius and Februarius (our January and February).
Julius Caesar further reformed the calendar, taking the start of the year back to 1 January and giving it 365
days in all. Later, the month of Quintilis was renamed Julius (our July) in honour of Caesar, and Sextilis
became Augustus in honour of the emperor of that name.
Most human societies the world has known have used this kind of
number-notation at some stage in their development and all have tried to
find ways of coping with the unavoidable fact that beyond four (IIII)
nobody can “read” intuitively a sequence of five strokes (I1III) or more.
ARAMAIC (Egypt)
Elephantine script: 5th to 3rd centuries BCE
t
a
in
—
\tn
—
win
in in
MUM
WHItlf
Mina/
1
2
3
4
5
6
7
8
9
Fig. 1.3.
ARAMAIC (Mesopotamia)
Khatra script: First decades of CE
l
11
III !
1111
1
>
i>
ll>
lll>
IIII >
1
2
3 1
4
5 l
j
6
7
8
9
Fig. 1.4.
ARAMAIC (Syria)
Palmyrenean script: First decades of CE
f
H
If!
m
y
/y
<r
///
<■
my
my
<r
1
2
3
4
5
6
7
8
9
Fig. 1.5.
CRETAN CIVILISATION
Hieroglyphic script: first half of second millennium BCE
}
>»
m
m
a
*»»»»
0
MW
W9
im
1
2
3
4
5
6
7
8
9
Fig. 1.6.
CRETAN CIVILISATION
Linear script: 1700-1200 BCE
0
SI
III
IIII
-
II
III
III
1000
119
1010
DUO
0 0 011
0010
1
II
01
101
1001
Mllfl
0
1
u
01
110
ION
1
2
3
4
5
6
7
8
9
Fig. 1.7.
EXPLAINING THE ORIGINS
8
EGYPT
Hieroglyphic script: third to first millennium BCE
0
SI
111
IIII
111 SI
101 910
m
991
9919
9095
90S
01
991
009
9019
m
mi
1
2
3
4
5
6
7
8
9
Fig. 1.8.
ELAM
“Proto-Elamite” script: Iran, first half of third millennium BCE
0
on
OOP
rr
OP
ROC
Vvv
BOO
D 00
MSI
ODBC
POOP
I) COBB
1)0000
<■ —
* —
<r
1
2
3
4
5
6
7
8
9
ETRUSCAN CIVILISATION
Italy, 6th to 4th centuries BCE
n
B
A
IA
IIIA
<■
Jill A
<■
D
1
5
6
8
9
Fig. t.io.
GREECE
Epidaurus and Argos, 5th to 2nd centuries BCE
Taurian Chersonesus, Chalcidy, Troezen, 5th to 2nd centuries BCE
c
c<
«<
(«<
r*
n
>
r«
>
r<«
— .>
r««
»
n*
n<
>
n«
■>
n«(
>
n«((
»
1
2
3
4
5
6
7
8
9
*71. initial of pente. five Fig. 1 . 12 .
GREECE
Thebes, Karistos, 5th to 1st centuries BCE
1
11
in
mi
n*
ri
Til
rni
run
1
2
4
5
6
7
8
9
INDUS CIVILISATION
2300-1750 BCE
1
u
III
HU
tint
mm
ilium
—
n
11
in
n
ill
HI
a?
an
un
MU
un
in
ni
in
1
2
3
4
5
6
7
8
9
Fig. 1.14.
HITTITE CIVILISATION
Hieroglyphic: Anatolia, 1500-800 BCE
—
0
II
ill
tin
inn
llltll
1091191
IIIIIIIO
mini
n
S3
111
11
911
IBS
V
■
mil
0111
—
in
111
111
1
2
3
4
5
6
7
8
9
Fig. 1.15.
LYCIAN CIVILISATION
Asia Minor, first half of first millennium BCE
l
11
111
1111
LA
— *
ZJI
—
Zlll
>
^im
— »
1
2
3
4
5
6
7
8
9
Fig. 1.16.
LYDIAN CIVILISATION
Asia Minor, 6th to 4th centuries BCE
1
n
in
1 in
<- —
11 in
<r
III III
1 HI HI
<r
11 in 111
<- —
III III III
1
2
3
4
5
6
7
8
9
Fig. 1.17.
MAYAN CIVILISATION
Pre-Columbian Central America, 3rd to 14th centuries CE
•
• •
• • • |
•
• •
• • •
• • • •
1
2
3
4
5
6
7
8
9
Fig. 1.13.
Fig. 1.18.
9
MESOPOTAMIA
Archaic Sumerian, beginning of third millennium BCE
B
V
V
C
V
V
ev
BP
BP
BP
f
pr
BP
BP
BB
DB
BO
0
BB
r r
OB
BB
BB
BB
BB
or
B
BB
BOD
BBB0
PPB
re
ere
pee
ere
ere
0
BABB
BOBB
per
ore
BOB
1
2
3
4
5
6
7
8
9
MESOPOTAMIA
Sumerian cuneiform, 2850-2000 BCE
Fig. 1.19.
T
TT
TYT
w
w
Iff
EE
ffif
w
i
2
3
4
Tyr
5
6
7
8
9
MESOPOTAMIA Fig. 1.20.
Assyro-Babylonian cuneiform, second to first millennium BCE
r t r
1 2
TIT y
3 4
¥
5
w
6
7
f
8
f
9
CIVILISATIONS OF MAIN & SABA (SHEBA)
Southern Arabia, 5th to 1st centuries BCE
Fig. i.2i.
1 II
1 2
III Mil
3 4
y
5
w
6 *
7
yiH
"Y*
yi'»
PHOENICIAN CIVILISATION
From 6th century BCE
Fig. 1.22.
1 11
1 2
III 1 110
<r
3 4
19 901
<■
5
III III
6
o in mil
<9
7
01 Ul III
<r
8
Ill III Ilf
9
Hieroglyphic script, Armenia, 13th to 9th centuries BCE
Fig. 1.24.
THE LIMITS OF PERCEPTION
To recapitulate: at the start of this story, people began by counting the
first nine numbers by placing in sequence the corresponding number of
strokes, circles, dots or other similar signs representing “one”, more or less
as follows:
i u hi mi mu min rami mum innnn
12345 6 7 8 9
Fig. 1.25.
But because series of identical signs are not easy to read quickly for
numbers above four, the system was rapidly abandoned. Some civilisations
(such as those found in Egypt, Sumer, Elam, Crete, Urartu, and Greece) got
round the difficulty by grouping the signs for numbers from five to nine to
9 according to a principle that we might call dyadic representation:
I
II
III
IIII
III
III
IIII
IIII
mil
II
III
III
IIII
mi
1
2
3
4
5
6
7
8
9
(3 + 2)
(3 + 3)
(4 + 3)
(4 + 4)
(5 + 4)
Fig. 1.26.
Other civilisations, such as the Assyro-Babylonian, the Phoenician, the
Egyptian-Aramaean and the Lydian, solved the problem by recourse to
a rule of three:
I II III III III III III III III
I II III III III III
I II III
12345678 9
(3 + 1) (3 + 2) (3 + 3) (3 + 3 + 1)(3 + 3 + 2)(3 + 3 + 3)
Fig. 1.27.
And yet others, like the Greeks, the Manaeans and Sabaeans, the
Lycians, Mayans, Etruscans and Romans, came up with an idea (probably
based on finger-counting) for a special sign for the number five, proceed-
ing thereafter on a rule of five or quinary system (6 = 5 + 1, 7 = 5 + 2, and
so on).
There really can be no debate about it now: natural human ability to
perceive number does not exceed four!
So the basic root of arithmetic as we know it today is a very rudimentary
numerical capacity indeed, a capacity barely greater than that of some
animals. There’s no doubt that the human mind could no more accede by
innate aptitude alone to the abstraction of counting than could crows or
goldfinches. But human societies have enlarged the potential of these very
limited abilities by inventing a number of mental procedures of enormous
EXPLAINING THE ORIGINS
10
fertility, procedures which opened up a pathway into the universe of
numbers and mathematics , . .
DEAD RECKONING
Since we can discriminate unreflectingly between concrete quantities only
up to four, we cannot have recourse only to our natural sense of number
to get to any quantity greater than four. We must perforce bring into play
the device of abstract counting, the characteristic quality of “civilised"
humanity.
But is it therefore the case that, in the absence of this mental device for
counting (in the way we now understand the term), the human mind is
so enfeebled that it cannot engage in any kind of numeration at all?
It is certainly true that without the abstractions that we call “one”, “two”,
“three”, and so on it is not easy to carry out mental operations. But it does
not follow at all that a mind without numbers of our kind is incapable
of devising specific tools for manipulating quantities in concrete sets.
There are very good reasons for thinking that for many centuries people
were able to reach several numbers without possessing anything like
number-concepts.
There are many ethnographic records and reports from various parts of
Africa, Oceania and the Americas showing that numerous contemporary
“primitive” populations have numerical techniques that allow them to carry
out some “operations”, at least to some extent.
These techniques, which, in comparison to our own, could be called
“concrete”, enable such peoples to reach the same results as we would, by
using mediating objects or model collections of many different kinds (pebbles,
shells, bones, hard fruit, dried animal dung, sticks, the use of notched
bones or sticks, etc.). The techniques are much less powerful and often
more complicated than our own, but they are perfectly serviceable for
establishing (for example) whether as many head of cattle have returned
from grazing as went out of the cowshed. You do not need to be able to
count by numbers to get the right answer for problems of that kind.
ELEMENTARY ARITHMETIC
It all started with the device known as “one-for-one correspondence”.
This allows even the simplest of minds to compare two collections of beings
or things, of the same kind or not, without calling on an ability to count
in numbers. It is a device which is both the prehistory of arithmetic, and
the dominant mode of operation in all contemporary “hard” sciences.
Here is how it works: You get on a bus and you have before you (apart
from the driver, who is in a privileged position) two sets: a set of seats and
a set of passengers. In one glance you can tell whether the two sets have “the
same number” of elements; and, if the two sets are not equal, you can tell
just as quickly which is the larger of the two. This ready-reckoning of
number without recourse to numeration is more easily explained by the
device of one-for-one correspondence.
If there was no one standing in the bus and there were some empty seats,
you would know that each passenger has a seat, but that each seat does not
necessarily have a passenger: therefore, there are fewer passengers than
seats. In the contrary case - if there are people standing and all the seats are
taken - you know that there are more passengers than seats. The third
possibility is that there is no one standing and all seats are taken: as each
seat corresponds to one passenger, there are as many passengers as seats.
The last situation can be described by saying that there is a mapping (or a
biunivocal correspondence, or, in terms of modern mathematics, a bijection)
between the number of seats and the number of passengers in the bus.
At about fifteen or sixteen months, infants go beyond the stage of simple
observation of their environment and become capable of grasping the
principle of one-for-one correspondence, and in particular the property
of mapping. If we give a baby of this age equal numbers of dolls and little
chairs, the infant will probably try to fit one doll on each seat. This kind of
play is nothing other than mapping the elements of one set (dolls) onto the
elements of a second set (chairs). But if we set out more dolls than chairs (or
more chairs than dolls), after a time the baby will begin to fret: it will have
realised that the mapping isn’t working.
Fig. 1.28. Two sets map if for each element of one set there is a corresponding single element of
the other, and vice versa.
This mental device does not only provide a means for comparing two
groups, but it also allows its user to manipulate several numbers without
knowing how to count or even to name the quantities involved.
11
ELEMENTARY ARITHMETIC
If you work at a cinema box-office you usually have a seating plan of the
auditorium in front of you. There is one “box” on the plan for each seat in
the auditorium, and, each time you sell a ticket, you cross out one of the
boxes on the plan. What you are doing is: mapping the seats in the cinema
onto the boxes on the seating plan, then mapping the boxes on the
plan onto the tickets sold, and finally, mapping the tickets sold onto
the number of people allowed into the auditorium. So even if you are too
lazy to add up the number of tickets you’ve sold, you'll not be in any doubt
about knowing when the show has sold out.
To recite the attributes of Allah or the obligatory laudations after
prayers, Muslims habitually use a string of prayer-beads, each bead corre-
sponding to one divine attribute or to one laudation. The faithful “tell their
beads” by slipping a bead at a time through their fingers as they proceed
through the recitation of eulogies or of the attributes of Allah.
Fig. 1 . 29 . Muslim prayer-beads (subha or sebha in Arabic) used for railing the 99 attributes
of Allah or for supererogatory laudations. This indispensable piece of equipment for pilgrims and
dervishes is made of wooden, mother-of-pearl or ivory beads that can be slipped through the fingers.
It is often made up of three groups of beads, separated by two larger “marker" beads, with an even
larger bead indicating the start. There arc usually a hundred beads on a string (33 + 33 + 33 + 1).
but the number varies.
Buddhists have also used prayer-beads for a very long time, as have
Catholics, for reciting Pater nostcr, Ave Maria, Gloria Patri, etc. As these
litanies must be recited several times in a quite precise order and number,
Christian rosaries usually consist of a necklace threaded with five times ten
small beads, each group separated by a slightly larger bead, together with
a chain bearing one large then three small beads, then one large bead
and a cross. That is how the litanies can be recited without counting but
without omission - each small bead on the ring corresponds to one Ave
Maria, with a Gloria Patri added on the last bead of each set of ten, and
a Pater noster is said for each large bead, and so on.
The device of one-for-one correspondence has thus allowed these
religions to devise a system which ensures that the faithful do not lose
count of their litanies despite the considerable amount of repetition
required. The device can thus be of use to the most “civilised” of societies;
and for the completely “uncivilised” it is even more valuable.
Let us take someone with no arithmetical knowledge at all and send him
to the grocery store to get ten loaves of bread, five bottles of cooking oil,
and four bags of potatoes. With no ability to count, how could this person
be trusted to bring back the correct amount of change? But in fact such a
person is perfectly capable of carrying out the errand provided the proper
equipment is available. The appropriate kit is necessarily based on the
principle of one-for-one correspondence. We could make ten purses out of
white cloth, corresponding to the ten loaves, five yellow purses for the
bottles of cooking oil, and four brown purses, for the bags of potatoes.
In each purse we could put the exact price of the corresponding item of
purchase, and all the uneducated shopper needs to know is that a white
purse can be exchanged for a loaf, a yellow one for a bottle of oil and
a brown one for a bag of potatoes.
This is probably how prehistoric humanity did arithmetic for many
millennia, before the first glimmer of arithmetic or of number-concepts
arose.
Imagine a shepherd in charge of a flock of sheep which is brought back
to shelter every night in a cave. There are fifty-five sheep in this flock.
But the shepherd doesn't know that he has fifty-five of them since he does
not know the number “55”: all he knows is that he has “many sheep”. Even
so, he wants to be sure that all his sheep are back in the cave each night.
So he has an idea - the idea of a concrete device which prehistoric
humanity used for many millennia. He sits at the mouth of his cave and
lets the animals in one by one. He takes a flint and an old bone, and cuts
a notch in the bone for every sheep that goes in. So, without realising the
mathematical meaning of it, he has made exactly fifty-five incisions on
the bone by the time the last animal is inside the cave. Henceforth the
shepherd can check whether any sheep in his flock are missing. Every time
he comes back from grazing, he lets the sheep into the cave one by one,
and moves his finger over one indentation in the tally stick for each one.
If there are any marks left on the bone after the last sheep is in the cave,
that means he has lost some sheep. If not, all is in order. And if meanwhile
a new lamb comes along, all he has to do is to make another notch in
the tally bone.
So thanks to the principle of one-for-one correspondence it is possible
to manage to count even in the absence of adequate words, memory or
abstraction.
One-for-one mapping of the elements of one set onto the elements of
EXPLAINING THE ORIGINS
a second set creates an abstract idea, entirely independent of the type
or nature of the things or beings in the one or other set, which expresses
a property common to the two sets. In other words, mapping abolishes
the distinction that holds between two sets by virtue of the type or nature
of the elements that constitute them. This abstract property is precisely
why one-for-one mapping is a significant tool for tasks involving enumera-
tion; but in practice, the methods that can be based on it are only suitable
for relatively small sets.
This is why model collections can be very useful in this domain. Tally
sticks with different numbers of marks on them constitute so to speak a
range of ready-made mappings which can be referred to independently of
the type or nature of the elements that they originally referred to. A stick
of ivory or wood with twenty notches on it can be used to enumerate
twenty men, twenty sheep or twenty goats just as easily as it can be used
for twenty bison, twenty horses, twenty days, twenty pelts, twenty kayaks,
or twenty measures of grain. The only number technique that can be built
on this consists of choosing the most appropriate tally stick from the ready-
mades so as to obtain a one-to-one mapping on the set that you next want
to count.
However, notched sticks are not the only concrete model collections avail-
able for this kind of matching-and-counting. The shepherd of our example
could also have used pebbles for checking that the same number of sheep
come into the cave every evening as went out each morning. All he needs to
do to use this device would be to associate one pebble with each head of
sheep, to put the resulting pile of pebbles in a safe place, and then to count
them out in a reverse procedure on returning from the pasture. If the last
animal in matches the last pebble in the pile, then the shepherd knows
for sure that none of his flock has been lost, and if a lamb has been born
meanwhile, all he needs to do is to add a pebble to the pile.
All over the globe people have used a variety of objects for this purpose:
shells, pearls, hard fruit, knucklebones, sticks, elephant teeth, coconuts,
clay pellets, cocoa beans, even dried dung, organised into heaps or lines
corresponding in number to the tally of the things needing to be checked.
Marks made in sand, and beads and shells, strung on necklaces or made
into rosaries, have also been used for keeping tallies.
Even today, several “primitive” communities use parts of the body for
this purpose. Fingers, toes, the articulations of the arms and legs (elbow,
wrist, knee, ankle . . . ), eyes, nose, mouth, ears, breasts, chest, sternum,
hips and so on are used as the reference elements of one-for-one counting
systems. Much of the evidence comes from the Cambridge Anthropological
Expedition to Oceania at the end of the last century. According to Wyatt
Gill, some Torres Straits islanders “counted visually” (see Fig. 1.30):
12
ENTARY ARITHMETIC
’■a)
EXPLAINING THE ORIGINS
14
They touch first the fingers of their right hand, one by one, then the right
wrist, elbow and shoulder, go on to the sternum, then the left-side
articulations, not forgetting the fingers. This brings them to the number
seventeen. If the total needed is higher, they add the toes, ankle, knee
and hip of the left then the right hand side. That gives 16 more, making
33 in all. For even higher numbers, the islanders have recourse to a
bundle of small sticks. [As quoted in A. C. Haddon (1890)]
Murray islanders also used parts of the body in a conventional order, and
were able to reach 29 in this manner. Other Torres Straits islanders used
similar procedures which enabled them to “count visually” up to 19; the
same customs are found amongst the Papuans and Elema of New Guinea.
NUMBERS, GESTURES, AND WORDS
The question arises: is the mere enumeration of parts of the body in regular
order tantamount to a true arithmetical sequence? Let us try to find the
answer in some of the ethnographic literature relating to Oceania.
The first example is from the Papuan language spoken in what was
British New Guinea. According to the report of the Cambridge Expedition
to the Torres Straits, Sir William MacGregor found that “body-counting”
was prevalent in all the villages below the Musa river. “Starting with the
little finger on the right hand, the series proceeds with the right-hand
fingers, then the right wrist, elbow, shoulder, ear and eye, then on to the left
eye, and so on, down to the little toe on the left foot.” Each of the gestures
to these parts of the body is accompanied, the report continues, by a
ific term in
Papuan, as follows:
NUMBER
NUMBER-GESTURE
GESTURE-WORD
1
right hand little finger
anusi
2
right hand ring finger
doro
3
right hand middle finger
doro
4
right hand index finger
doro
5
right thumb
ubei
6
right wrist
tama
7
right elbow
unubo
8
right shoulder
visa
9
right ear
denoro
10
right eye
diti
11
left eye
diti
12
nose
medo
13
mouth
bee
14
left ear
denoro
NUMBER
NUMBER-GESTURE
GESTURE-WORD
15
left shoulder
visa
16
left elbow
unubo
17
left wrist
tama
18
left thumb
ubei
19
left hand index finger
doro
20
left hand middle finger
doro
21
left hand ring finger
doro
22
left hand little finger
anusi
The words used are simply the names of the parts of the body, and
strictly speaking they are not numerical terms at all. Anusi, for example, is
associated with both 1 and 22, and is used to indicate the little fingers of
both the right and the left hands. In these circumstances how can you know
which number is meant? Similarly the term doro refers to the ring, middle
and index fingers of both hands and “means” either 2 or 3 or 4 or 19 or 20
or 21. Without the accompanying gesture, how could you possibly tell
which of these numbers was meant?
However, there is no ambiguity in the system. What is spoken is the
name of the part of the body, which has its rank-order in a fixed, conven-
tional sequence within which no confusion is possible. So there is no doubt
that the mere enumeration of the parts of the body does not constitute
a true arithmetical sequence unless it is associated with a corresponding
sequence of gestures. Moreover, the mental counting process has no direct
oral expression - you can get to the number required without uttering
a word. A conventional set of “number-gestures” is all that is needed.
In those cases where it is possible to recover the original meanings of the
names given to numbers, it often turns out that they retain traces of body-
counting systems like those we have looked at. Here, for example, are the
number-words used by the Bugilai (former British New Guinea) together
with their etymological meanings:
1
tarangesa
left hand little finger
2
meta kina
next finger
3
guigimeta kina
middle finger
4
topea
index finger
5
rnanda
thumb
6
gaben
wrist
7
trankgimbe
elbow
8
podei
shoulder
9
ngama
left breast
10
dala
right breast
[Source: J. Chalmers (1898)]
15
NUMBERS, GESTURES, AND WORDS
E. C, Hawtrey (1902) also reports that the Lengua people of the Chaco
(Paraguay) use a set of number-names broadly derived from specific
number-gestures. Special words apparently unrelated to body-counting are
used for 1 and 2, but for the other numbers they say something like:
3 "made of one and two”
4 "both sides same”
5 “one hand”
6 “reached other hand, one”
7 “reached other hand, two”
8 “reached other hand, made of one and two”
9 “reached other hand, both sides same”
10 “finished, both hands”
11 “reached foot, one”
12 “reached foot, two”
13 “reached foot, made of one and two”
14 “reached foot, both sides same”
15 “finished, foot”
16 “reached other foot, one”
17 “reached other foot, two”
18 “reached other foot, made of one and two”
19 “reached other foot, both sides same”
20 “finished, feet”
The Zuni have names for numbers which F. H. Cushing (1892) calls
“manual concepts”:
1
topinte
taken to begin
2
kwilli
raised with the previous
3
kha’i
the finger that divides equally
4
awite
all fingers raised bar one
5
dpte
the scored one
6
topali'k’ye
another added to what is counted
already
7
kwillik’ya
two brought together and raised
with the others
8
khailik’ya
three brought together and raised
with the others
9
tenalik’ya
all bar one raised with the others
10
astern ’thila
all the fingers
11
astern 'thila
topaya’thV tona
all the fingers and one more raised
and so on.
All this leads us to suppose that in the remotest past gestures came
before any oral expression of numbers.
CARDINAL RECKONING DEVICES FOR
CONCRETE QUANTITIES
Let us now imagine a group of "primitive” people lacking any conception
of abstract numbers but in possession of perfectly adequate devices for
“reckoning” relatively small sets of concrete objects. They use all sorts of
model collections, but most often they “reckon by eye” in the following
manner: they touch each other’s right-hand fingers, starting with the little
finger, then the right wrist, elbow, shoulder, ear, and eye. Then they touch
each others’ nose, mouth, then the left eye, ear, shoulder, elbow, and wrist,
and on to the little finger of the left hand, getting to 22 so far. If the number
needed is higher, they go on to the breasts, hips, and genitals, then the
knees, ankles and toes on the right then the left sides. This extension allows
19 further integers, or a total of 41.
The group has recently skirmished with a rebellious neighbouring village
and won. The group’s leader decides to demand reparations, and entrusts
one of his men with the task of collecting the ransom. “For each of the
warriors we have lost”, says the chief, “they shall give us as many pearl
necklaces as there are from the little finger on my right hand to my right
eye, as many pelts as there are from the little finger of my right hand to my
mouth, and as many baskets of food as there are from the little finger of
my right hand to my left wrist.” What this means is that the reparation
for each lost soldier is:
10 pearl necklaces
12 pelts
17 baskets of food
In this particular skirmish, the group lost sixteen men. Of course none
amongst the group has a notion of the number “16”, but they have an
infallible method of determining numbers in these situations: on departing
for the fight, each warrior places a pebble on a pile, and on his return each
surviving warrior picks a pebble out of the pile. The number of unclaimed
pebbles corresponds precisely to the number of warriors lost.
One of the leader’s envoys then takes possession of the pile of remaining
pebbles but has them replaced by a matching bundle of sticks, which is
easier to carry. The chief checks the emissaries’ equipment and their
comprehension of the reparations required, and sends them off to parley
with the enemy.
The envoys tell the losing side how much they owe, and proceed to
enumerate the booty in the following manner: one steps forward and says:
EXPLAINING THE ORIGINS
“Bring me a pearl necklace each time I point to a part of my body,” and he
then touches in order the little finger, the ring finger, the middle finger, the
index finger and the thumb of his right hand. So the vanquished bring him
one necklace, then a second, then a third and so on up to the fifth. The
envoy then repeats himself, but pointing to his right wrist, elbow, shoulder,
ear and eye, which gets him five more necklaces. So without having any
concept of the number “10” he obtains precisely ten necklaces.
Another envoy proceeds in identical fashion to obtain the twelve pelts,
and a third takes possession of the seventeen baskets of food that are
demanded.
That is when the fourth envoy conies into the equation, for he possesses
the tally of warriors lost in the battle, in the form of a bundle of sixteen
sticks. He sets one aside, and the three other envoys then repeat their oper-
ations, allowing him to set another stick aside, and so on, until there are no
sticks left in the bundle. That is how they know that they have the full tally,
and so collect up the booty and set off with it to return to their own village.
As can be seen, “primitives” of this kind are not using body-counting in
exactly the same way as we might. Since we know how to count, a conven-
tional order of the parts of the body would constitute a true arithmetical
sequence; each “body-point” would be assimilated in our minds to a
cardinal (rank-order) number, characteristic of a particular quantity of
things or beings. For instance, to indicate the length of a week using this
system, we would not need to remember that it contained as many days
as mapped onto our bodies from the right little finger to the right elbow,
since we could just attach to it the “rank-order number” called “right
elbow”, which would suffice to symbolise the numerical value of any set of
seven elements.
That is because we are equipped with generalising abstractions and in
particular with number-concepts. But “primitive" peoples are not so
equipped: they cannot abstract from the “points" in the numbering
sequence: their grasp of the sequence remains embedded in the specific
nature of the “points” themselves. Their understanding is in effect
restricted to one-for-one mapping; the only "operations” they make are to
add or remove one or more of the elements in the basic series.
Such people do not of course have any abstract concept of the number
“ten”, for instance. But they do know that by touching in order their little
finger, ring finger, middle finger, index finger and thumb on the right hand,
then their right wrist, elbow, shoulder, ear, and eye, they can “tally out” as
many men, animals or objects as there are body-points in the sequence.
And having done so, they remember perfectly well which body-point any
particular tally of things or people reached, and are able to repeat the
operation in order to reach exactly the same tally whenever they want to.
16
17
CARDINAL RECKONING DEVICES
Fig. 1.34.
In other words, this procedure is a simple and convenient means of
establishing ready-made mappings which can then be mapped one-to-one
onto any sets for which a total is required. So when our imaginary tribe
went to collect its ransom, they used only these notions, not any true
number-concepts. They simply mapped three such ready-made sets onto
a set of ten necklaces, a set of twelve pelts, and a set of seventeen baskets
of food for each of the lost warriors.
These body-counting points are thus not thought of by their users as
“numbers”, but rather as the last elements of model sets arrived at after
a regulated (conventional) sequence of body-gestures. This means that
for such people the mere designation of any one of the points is not sufficient
to describe a given number of beings or things unless the term uttered is accom-
panied by the corresponding sequence of gestures. So in discussions concerning
such and such a number, no real “number-term” is uttered: instead, a given
number of body-counting points will be enumerated, alongside the simul-
taneous sequence of gestures. This kind of enumeration therefore fails to
constitute a genuine arithmetical series; participants in the discussion must
also necessarily keep their eyes on the speaker!
All the same, our imaginary tribesmen have unknowingly reached quite
large numbers, even with such limited tools, since they have collected:
16 x 10 = 160 necklaces
16 x 12 = 192 pelts
16 x 17 = 272 baskets of food
or six hundred and twenty-four items in all! (see Fig. 1.34)
There is a simple reason for this: they had thought of associating easily
manipulated material objects with the parts of the body involved in their
counting operations. It is true that they counted out the necklaces, pelts and
food-baskets by their traditional body-counting method, but the determin-
ing element in calculating the ransom to be paid (the number of men lost in
the battle) was “numerated” with the help of pebbles and a bundle of sticks.
Let us now imagine that the villagers are working out how to fix the date
of an important forthcoming religious festival. The shaman who that
morning proclaimed the arrival of the new moon also announced in the
following way, accompanying his words with quite precise gestures of his
hands, that the festival will fall on the thirteenth day of the eighth moon there-
after: “Many suns and many moons will rise and fall before the festival. The
moon that has just risen must first wax and then wane completely. Then it
must wax as many times again as there are from the little finger on my right
hand to the elbow on the same side. Then the sun will rise and set as many
times as there are from the little finger on my right hand to my mouth. That
is when the sun will next rise on the day of our Great Festival.”
EXPLAINING THE ORIGINS
18
This community obviously has a good grasp of the lunar cycle, which is
only to be expected, since, after the rising and the setting of the sun, the
moon’s phases constitute the most obvious regular phenomenon in
the natural environment. As in all empirical calendars, this one is based
on the observation of the first quarter after the end of each cycle. With the
help of model collections inherited from forebears, many generations of
whom must have contributed to their slow development, the community
can in fact “mark time” and compute the date thus expressed without error,
as we shall see.
On hearing the shaman's pronouncement, the chief of the tribe paints a
number of marks on his own body with some fairly durable kind of colour-
ing material, and these marks enable him to record and to recognise the
festival date unambiguously. He first records the series of reappearances
that the moon must make from then until the festival by painting small
circles on his right-hand little finger, ring finger, middle finger, index finger,
thumb, wrist, and elbow. Then he records the number of days that must
pass from the appearance of the last moon by painting a thin line, first of all
on each finger of his right hand, then on his right wrist, elbow, shoulder,
ear, and eye, then on his nose and mouth. To conclude, he puts a thick line
over his left eye, thereby symbolising the dawn of the great day itself.
The following day at sunset, a member of the tribe chosen by the chief
to “count the moons” takes one of the ready-made ivory tally sticks with
thirty incised notches, the sort used whenever it is necessary to reckon the
days of a given moon in their order of succession (see Fig. 1.35). He ties
a piece of string around the first notch. The next evening, he ties a piece of
string around the second notch, and so on every evening until the end
of the moon. When he reaches the penultimate notch, he looks carefully at
the night sky, in the region where the sun has just set, for he knows that the
new moon is soon due to appear.
On that day, however, the first quarter of the new moon is not visible in
the sky. So he looks again the next evening when he has tied the string
around the last notch on the first tally stick; and though the sky is not clear
enough to let him see the new moon, he decides nonetheless that a new
month has begun. That is when he paints a little circle on his right little
finger, indicating that one lunar cycle has passed.
At dusk the following day, our “moon-counter” takes another similar
tally stick and ties a string around the first notch. The day after, he or she
proceeds likewise with the second notch, and so on to the end of the second
month. But at that month’s end the tally man knows he will not need to
scan the heavens to check on the rising of the new moon. For in this tribe,
the knowledge that moon cycles end alternately on the penultimate and
last notches of the tally sticks has been handed down for generations. And
this knowledge is only very slightly inaccurate, since the average length of
a lunar cycle is 29 days and 12 hours.
Fig. 1-35-
I
L
1 day passed
2 days passed
3 days passed
4 days passed
5 days passed
6 days passed
7 days passed
The moon-counter proceeds in this manner through alternating months
of 29 and 30 days until the arrival of the last moon, when he paints a little
circle on his right elbow. There are now as many circles on the counter’s
body as on the chief’s: the counter’s task is over: the “moon tally” has been
reached.
The chief now takes over as the "day-counter”, but for this task tally
sticks are not used, as the body-counting points suffice. The community
will celebrate its festival when the chief has crossed out all the thin lines
from his little finger to his mouth and also the thick line over his left eye,
that is to say on the thirteenth day of the eighth moon (Fig. 1.36)
This reconstitution of a non-numerate counting system conforms to
many of the details observed in Australian aboriginal groups, who are able
to reach relatively high numbers through the (unvocalised) numeration
of parts of the body when the body-points have a fixed conventional order
and are associated with manipulable model collections - knotted string,
bundles of sticks, pebbles, notched bones, and so on.
Valuable evidence of this kind of system was reported by Brooke,
observing the Dayaks of South Borneo. A messenger had the task of inform-
ing a number of defeated rebel villages of the sum of reparations they had
to pay to the Dayaks.
The messenger came along with some dried leaves, which he broke
into pieces. Brooke exchanged them for pieces of paper, which were
more convenient. The messenger laid the pieces on a table and used his
fingers at the same time to count them, up to ten; then he put his foot
on the table, and counted them out as he counted out the pieces of
paper, each of which corresponded to a village, with the name of its
chief, the number of warriors and the sum of the reparation. When he
had used up all his toes, he came back to his hands. At the end of the
list, there were forty-five pieces of paper laid out on the table.* Then
* Each linger is associated with one piece of paper and one village, in this particular system, and each toe
with the set often fingers.
19
CARDINAL RECKONING DEVICES
he asked me to repeat the message, which I did, whilst he ran through
the pieces of paper, his fingers and his toes, as before.
“So there are our letters," he said. “You white folk don't read the way
we do.”
Later that evening he repeated the whole set correctly, and as he put
his finger on each piece of paper in order, he said:
“So, if I remember it tomorrow morning, all will be well; leave the
papers on the table.”
Then he shuffled them together and made them into a heap.
As soon as we got up the next morning, we sat at the table, and he
re-sorted the pieces of paper into the order they were in the previous
day, and repeated all the details of the message with complete
accuracy. For almost a month, as he went from village to village, deep
in the interior, he never forgot the different sums demanded. [Adapted
from Brooke, Ten Years in Sarawak]
All this leads us to hypothesise the following evolution of counting
systems:
First stage
Only the lowest numbers are within human grasp. Numerical ability
remains restricted to what can be evaluated in a single glance. “Number” is
indissociable from the concrete reality of the objects evaluated.* In order
to cope with quantities above four, a number of concrete procedures are
developed. These include finger-counting and other body-counting
systems, all based on one-for-one correspondence, and leading to the devel-
opment of simple, widely-available ready-made mappings. What is
articulated (lexicalised) in the language are these ready-made mappings,
accompanied by the appropriate gestures.
Second stage
By force of repetition and habit, the list of
the names of the body-parts in their
numerative order imperceptibly acquire
abstract connotations, especially the first
five. They slowly lose their power to suggest
the actual parts of the body, becoming
progressively more attached to the corre-
sponding number, and may now be applied
to any set of objects. (L. Levy-Bruhl)
Third stage
A fundamental tool emerges: numerical
nomenclature, or the names of the
numbers.
Fig. 1.37. Detail from a “material model" of a lunar
calendar formerly in use amongst tribal populations in
former Dahomey (West Africa). It consists of a strip of
cloth onto which thirty objects (seeds, kernels, shells,
hard fruit, stories, etc.) have been sewn, each standing
for one of the days of the month. (The fragment above
represents the last seven days). From the Mush' de
I Homme, Paris.
Thus as I.. Lcvy-Iiruhl reports. Fijians and Solomon islanders have collective nouns for tens of arbitrarily
selected items that express neither the number itself nor the objects collected into the set. In Fijian, bold
means “a hundred dugouts", kora “a hundred coconuts", salava ‘‘a thousand coconuts". Natives ofMola say
aka peperiui (“butterfly two dugout") for a pair of dugouts” because of the .appearance of the sails. See also
Codrington, F. Stephen and 1.. 1.. Conanl.
EXPLAINING THE ORIGINS
20
counting: a human faculty
The human mind, evidently, can only grasp integers as abstractions if it has
fully available to it the notion of distinct units as well as the ability to
“synthesise” them. This intellectual faculty (which presupposes above all a
complete mastery of the ability to analyse, to compare and to abstract from
individual differences) rests on an idea which, alongside mapping and
classification, constitutes the starting point of all scientific advance.
This creation of the human mind is called “hierarchy relation” or “order
relation”: it is the principle by which things are ordered according to their
“degree of generality”, from individual, to kind, to type, to species, and so on.
Decisive progress towards the art of abstract calculation that we now
use could only be made once it was clearly understood that the integers
could be classified into a hierarchised system of numerical units whose terms
were related as kinds within types, types within species, and so on.
Such an organisation of numerical concepts in an invariable sequence
is related to the generic principle of “recurrence” to which Aristotle
referred ( Metaphysics 1057, a) when he said that an integer was a “multi-
plicity measurable by the one”. The idea is really that integers are
“collections” of abstract units obtained successively by the adjunction of
further units.
Any clement in the regular sequence of the integers (other than 1) is obtained
by adding 1 to the integer immediately preceding in the "natural" sequence that is
so constituted (see Fig. 1.38). As the German philosopher Schopenhauer put
it, any natural integer presupposes its preceding numbers as the cause of its
existence: for our minds cannot conceive of a number as an abstraction
unless it subsumes all preceding numbers in the sequence. This is what we
called the ability to “synthesise” distinct units. Without that ability,
number-concepts remain very cloudy notions indeed.
But once they have been put into a natural sequence, the set of integers
permits another faculty to come into play: numeration. To numerate the
items in a group is to assign to each a symbol (that is to say, a word, a
gesture or a graphic mark) corresponding to a number taken from the
natural sequence of integers, beginning with 1 and proceeding in order
until the exhaustion of that set (Fig. 1.40). The symbol or name given to
each of the elements within the set is the name of its order number within
the collection of things, which becomes thereby a sequence or procession
of things. The order number of the last element within the ordered group
is precisely equivalent to the number of elements in the set. Obviously
the number obtained is entirely independent of the order in which the
elements are numerated - whichever of the elements you begin with, you
always end up with the same total.
1
1
1 +1
2
1+1 + 1
3
1 + 1 + 1+1
4
1 + 1 + 1 + 1+1
5
• • • • •
• • •
1 +n... + 1
n
n
1 + 1 + • • • + 1 +1
n
n +1
• • • • •
• • •
Fig. i . 3 8 . The generation of integers by the so-called procedure of recurrence
Fig. 1.39. Numeration of a “cloud" of dots
21
counting: a human faculty
For example, let us take a box containing “several" billiard balls. We take
out one at random and give it the “number" 1 (for it is the first one to come
out of the box). We take another, again completely at random, and give
it the “number" 2. We continue in this manner until there are no billiard
balls left in the box. When we take out the last of the balls, we give it
a specific number from the natural sequence of the integers. If its number
is 20, we say that there are “twenty” balls in the box. Numeration has
allowed us to transform a vague notion (that there are “several” billiard
balls) into exact knowledge.
In like manner, let us consider a set of “scattered" points, in other words
dots in a “disordered set” (Fig. 1.39). To find out how many dots there are,
Fig. 1.40. Numeration allowing us to advance from concrete plurality to abstract number
all we have to do is to connect them by a “zigzag” line passing through each
dot once and no dot twice. The points then constitute what is commonly
called a chain. We then give each point in the chain an order-number, start-
ing from one of the ends of the chain we have just made. The last number,
given therefore to the last point in the chain, provides us with the total
number of dots in the set.
So with the notions of succession and numeration we can advance from
the muddled, vague and heterogeneous apperception of concrete plurality
to the abstract and homogenous idea of “absolute quantity”.
So the human mind can only “count” the elements in a set if it is in
possession of all three of the following abilities:
1. the ability to assign a “rank-order” to each element in a procession;
2. the ability to insert into each unit of the procession the memory of
all those that have gone past before;
3. the ability to convert a sequence into a “stationary” vision.
The concept of number, which at first sight seemed quite elementary,
thus turns out to be much more complicated than that. To underline this
point I should like to repeat one of P. Bourdin’s anecdotes, as quoted in R.
Balmes (1965):
I once knew someone who heard the bells ring four as he was trying to
go to sleep and who counted them out in his head, one, one, one, one.
Struck by the absurdity of counting in this way, he sat up and shouted:
“The clock has gone mad, it’s struck one o’clock four times over!”
THE TWO SIDES OF THE INTEGERS
The concept of number has two complementary aspects: cardinal
numbering, which relies only on the principle of mapping, and ordinal
numeration, which requires both the technique of pairing and the idea
of succession.
Here is a simple way of grasping the diff erence. January has 31 days. The
number 31 represents the total number of days in the month, and is thus in
this expression a cardinal number. However, in expressions such as “31
January 1996”, the number 31 is not being used in its cardinal aspect
(despite the terminology of grammar books) because here it means some-
thing like “the thirty-first day” of the month of January, specifying not
a total, but a rank-order of a specific (in this case, the last) element in a set
containing 31 elements. It is therefore unambiguously an ordinal number.
We have learned to pass with such facility from cardinal to ordinal
number that the two aspects appear to us as one. To determine the
plurality of a collection, i.e. its cardinal number, we do not bother any
more to find a model collection with which we can match it - we count
EXPLAINING THF. ORIGINS
22
it. And to the fact that we have learned to identify the two aspects of
number is due our progress in mathematics. For whereas in practice
we are really interested in the cardinal number, this latter is incapable
of creating an arithmetic. The operations of arithmetic are based
on the tacit assumption that we can always pass from any number to its
successor, and this is the essence of the ordinal concept.
And so matching by itself is incapable of creating an art of
reckoning. Without our ability to arrange things in ordered succession
little progress could have been made. Correspondence and succession,
the two principles which permeate all mathematics - nay, all realms of
exact thought - are woven into the very fabric of our number-system.
[T. Dantzig (1930)]
TEN FINGERS TO COUNT BY
Humankind slowly acquired all the necessary intellectual equipment
thanks to the ten fingers on its hands. It is surely no coincidence if children
still learn to count with their fingers - and adults too often have recourse to
them to clarify their meaning.
Traces of the anthropomorphic origin of counting systems can be found
in many languages. In the Ali language (Central Africa), for example, “five”
and “ten” are respectively moro and mbouna : moro is actually the word for
“hand” and mbouna is a contraction of moro (“five”) and bouna, meaning
“two” (thus “ten” = “two hands”).
CARDINAL ASPECT ORDINAL ASPECT
ft is therefore very probable that the Indo-European, Semitic and
Mongolian words for the first ten numbers derive from expressions related
to finger-counting. But this is an unverifiable hypothesis, since the original
meanings of the names of the numbers have been lost.
In any case, the human hand is an extremely serviceable tool and
constitutes a kind of “natural instrument” well suited for acquiring the first
ten numbers and for elementary arithmetic.
Because there are ten fingers and because each can be moved indepen-
dently of the others, the hand provides the simplest “model collection” that
people have always had - so to speak - to hand.
The asymmetric disposition of the fingers puts the hand in harmony
with the normal limitation of the human ability to recognise number
visually (a limit set at four). As the thumb is set at some distance from
the index finger it is easy to treat it as being “in opposition” to the elemen-
tary set of four, and makes the first five numbers an entirely natural
sequence. Five thus imposes itself as a basic unit of counting, alongside
the other natural grouping, ten. And because each of the fingers is actually
different from the others, the human hand can be seen as a true succession
of abstract units, obtained by the progressive adjunction of one to the
preceding units.
In brief, one can say that the hand makes the two complementary
aspects of integers entirely intuitive. It serves as an instrument permitting
natural movement between cardinal and ordinal numbering. If you need to
show that a set contains three, four, seven or ten elements, you raise or
bend simultaneously three, four, seven or ten fingers, using your hand as
cardinal mapping. If you want to count out the same things, then you bend
or raise three, four, seven or ten fingers in succession, using the hand as an
ordinal counting tool (Fig. 1.41).
The human hand can thus be seen as the simplest and most natural
counting machine. And that is why it has played such a significant role
in the evolution of our numbering system.
Fig. 1.41.
23
NUMBERS AND THEIR SYMBOLS
CHAPTER 2
BASE NUMBERS
AND THE BIRTH OF NUMBER-SYSTEMS
NUMBERS AND THEIR SYMBOLS
Once they had grasped abstract numbers and learned the subtle distinction
between cardinal and ordinal aspects, our ancestors came to have a different
attitude towards traditional “numbering tools” such as pebbles, shells,
sticks, strings of beads, or points of the body. Gradually these simple
mapping devices became genuine numerical symbols, which are much
better suited to the tasks of assimilating, remembering, distinguishing and
combining numbers.
Another great step forward was the creation of names for the numbers.
This allowed for much greater precision in speech and opened the path
towards real familiarity with the universe of abstract numbers.
Prior to the emergence of number-names, all that could be referred to in
speech were the “concrete maps” which had no obvious connection
amongst themselves. Numbers were referred to by intuitive terms, often
directly appealing to the natural environment. For instance, 1 might have
been “sun”, “moon”, or “penis”; for 2, you might have found “eyes",
“breasts”, or “wings of a bird”; “clover” or “crowd” for 3; “legs of a beast” for
4; and so on. Subsequently some kind of structure emerged from body-
counting. At the start, perhaps, you had something like this: “the one
to start with” for 1; “raised with the preceding finger” for 2; “the finger in
the middle” for 3; “all fingers bar one” for 4; “hand” for 5; and so on.
Then a kind of anatomical mapping occurred, so that “little finger" = 1,
“ring finger” = 2, “middle finger” = 3, “index finger” = 4, “thumb" = 5, and
so on. However, the need to distinguish between the number-symbol
and the name of the object or image being used to symbolise the number
led people to make an ever greater distinction between the two names, so
that eventually the connection between them was entirely lost. As people
progressively learned to rely more and more on language, the sounds
superseded the images for which they stood, and the originally concrete
models took on the abstract form of number-words. The idea of a natural
sequence of numbers thus became ever clearer; and the very varied set of
initial counting maps or model collections turned into a real system of
number-names. Habit and memory gave a concrete form to these abstract
ideas, and, as T. Dantzig says (p. 8), that is how “mere words became
measures of plurality”.
WRITTEN SYMBOLS
Unmotivated Figures
(without visual motivation)
oo
-J
O
CQ
2
>*
OO
w
Q£
O
u-
\
Use of
concrete
objects
(pebbles,
shells, sticks,
clay shapes,
etc.)
Notched
wood
or bone
Use of
knotted string
Intuitive
finger-
counting
Conventional
hand-gesture
E
Ll
e
Alphabetic
letters with
numerical
values:
fifth letter
of the
alphabet
Ll
F
Alphabetical
letters with
numerical
values:
Ll
n
initial letter of
relevant
number-name
BOD
DO
Motivated
PC
CO
0
Figures
(related to
direct visual
tuition)
• ••
• •
— i
H
m
Z
oo
CD
o
Hand
Thumb
Five
Cinq
Motivated
number-names
Unmotivated
number-names
1. Greek letter “epsilon"
2. Hebrew letter “he”
3. Initial letter of the
word for “five”
4. Greek letter “pi”, for
“pente”, meaning 5
ORAL SYMBOLS
Fig. 2 . 1 .
Of course, concrete symbols and spoken expressions were not the only
devices that humanity possessed for mastering numbers. There was also
writing, even if that did arise much later on. Writing involves figures, that
is to say, graphic signs, of whatever kind (carved, drawn, painted, or scored
on clay or stone; iconic signs, letters of the alphabet, conventional signs,
and so on). We should note that figures are not numbers. “Unit”, “pair of”,
“triad” are “numbers”, whilst 1, 2, 3 are “figures”, that is to say, conven-
tional graphic signs that represent number-concepts. A figure is just one
of the “dresses” that a number can have: you can change the way a number
is written without changing the number-concept at all.
These were very important developments, for they allowed “operations”
on things to be replaced by the corresponding operation on number-
symbols. For numbers do not come from things, but from the laws of the
BASE NUMBERS
24
human mind as it works on things. Even if numbers seem latent in the
natural world, they certainly did not spring forth from it by themselves.
2
3
one-one
one-one-one
4
one-one-one-one
THE DISCOVERY OF BASE NUMBERS
There were two fundamental principles available for constructing number-
symbols: one that we might call a cardinal system, in which you adopt a
standard sign for the unit and repeat it as many times as there are units in
the number; and another that we could call an ordinal system, in which
each number has its own distinctive symbol.
In virtue of the first principle, the numbers 2 to 4 can be represented by
repeating the name of the number 1 two, three or four times, or by laying out
in a line, or on top of each other, the appropriate number of “unit signs" in
pebbles, fingers, notches, lines, or dots (see Fig. 2.2).
The second principle gives rise to representations for the first four
numbers (in words, objects, gestures or signs) that are each different from
the others (see Fig. 2.3).
Either of these principles is an adequate basis for acquiring a grasp of
ever larger sets - but the application of both principles quickly runs into
difficulty. To represent larger numbers, you can’t simply use more and
more pebbles, sticks, notches, or knotted string; and the number of fingers
and other counting points on the body is not infinitely extensible. Nor is
it practicable to repeat the same word any number of times, or to create
unique symbols for any number of numbers. (Just think how many
different symbols you would need to say how many cents there are in
a ten-dollar bill!)
To make any progress, people had first to solve a really tricky problem:
What in practice is the smallest set of symbols in which the largest numbers
can in theory be represented? The solution found is a remarkable example
of human ingenuity.
The solution is to give one particular set (for example, the set of ten, the
set of twelve, the set of twenty or the set of sixty) a special role and to
classify the regular sequence of numbers in a hierarchical relationship
to the chosen (“base”) set. In other words, you agree to set up a ladder
and to organise the numbers and their symbols on ascending steps of the
ladder. On the first step you call them “first-order units”, on the second
step, “units of the second order”, on the third step, “units of the third
order”, and so on. And that is all there was to the invention of a number-
system that saves vast amounts of effort in terms of memorisation and
writing-out. The system is called “the rule of position" (or “place-value
system"), and its discovery marked the birth of numbering systems where
the “base” is the number of units in the set that constitutes the unit of the
Fig. 2.2. "Cardinal” representations of the first four numbers
next order. The place-value system can be applied to material “relays”, to
words in a language, or to graphic marks - producing respectively concrete,
oral, and written numbering systems.
WHY BASE lO?
Not so long ago shepherds in certain parts of West Africa had a very
practical way of checking the number of sheep in their flocks. They would
make the animals pass by one by one. As the first one went through the
gate, the shepherd threaded a shell onto a white strap: as the second went
through, he threaded another shell, and so on up to the ninth. When the
25
WHY BASE 10?
tenth went through, he took the shells off the white strap and put one on
a blue strap, which served for counting in tens. Then he began again
threading shells onto the white strap until the twentieth sheep went
through, when he put a second shell on the blue strap. When there were ten
shells on the blue strap, meaning that one hundred sheep had now been
counted, he undid the blue strap and threaded a shell onto a red strap,
which was the “hundreds” counting device. And so he continued until the
whole flock had been counted. If there were for example two hundred and
fifty-eight head in the flock, the shepherd would have eight shells on the
white strap, five on the blue strap and two on the red strap. There’s nothing
“primitive” about this method, which is in effect the one that we use now,
though with different symbols for the numbers and orders of magnitude.
The basic idea of the system is the primacy of grouping (and of the
rhythm of the symbols in their regular sequence) in “packets” of tens,
hundreds (tens of tens), thousands (tens of tens of tens), and so on. In the
shepherd’s concrete technique, each shell on the white strap counts as a
simple unit, each shell on the blue strap counts for ten, and each shell
on the red strap counts for a hundred. This is what is called the principle
of base 10. The shepherd’s device is an example of a concrete decimal
number-system.
Obviously, instead of using threaded shells and leather straps, we could
apply the same system to words or to graphic signs, producing oral or
written decimal numeration. Our current number system is just such, using
the following graphic signs, often referred to as Arabic numerals:
1234567890
The first nine symbols represent the simple units, or units of the first
decimal order (or “first magnitude”). They are subject to the rule of
position, or place-value, since their value depends on the place or position
that they occupy in a written numerical expression (a 3, for instance, counts
for three units, three tens, or three hundreds depending on its position in
a three-digit numerical expression). The tenth symbol above represents
what we call “zero", and it serves to indicate the absence of any unit of
a particular decimal order, or order of magnitude. It also has the meaning
of “nought” - for example, the number you obtain when you subtract
a number from itself.
The base of ten, which is the first number that can be represented by two
figures, is written as 10, a notation which means “one ten and no units”.
The numbers from 11 to 99 are represented by combinations of two of
the figures according to the rule of position:
11 “one ten, one unit”
12 “one ten, two units”
20 “two tens, no units”
21 “two tens, one unit"
30 “three tens, no units”
40 “four tens, no units”
50 “five tens, no units”
The hundred, equal to the square of the base, is written: 100, meaning “one
hundred, no tens, no units”, and is the smallest number that can be written
with three figures.
Numbers from 101 to 999 are represented by combinations of three of
the basic figures:
101 “one hundred, no tens, one unit”
358 “three hundreds, five tens, eight units”
There then comes the thousand, equal to the cube of the base, which is
written : 1,000 (“one thousand, no hundreds, no tens, no units”), and is the
smallest number that can be written with four figures. The following step
on the ladder is the ten thousand, the base to the power of four, which is
written 10,000 (“one ten thousand, no thousands, no hundreds, no tens, no
units”) and is the smallest number that can be written with five figures; and
so on.
In oral (spoken) numeration constructed in the same way things proceed
in very similar general manner, but with one difference that is inherent
to the nature of language: all the numbers less than or equal to ten and also
the several powers of ten (100, 1,000, 10,000, etc.) have individual names
entirely unrelated to each other, whereas all other numbers are expressed by
words made up of combinations of the various number-names.
In English, if we restrict ourselves for a moment to cardinal numbers, the
system would proceed in theory as follows. For the first ten numbers we
have individual names:
one two three four five six seven eight nine ten
1234567 8 9 10
The first nine are “units of the first decimal order of magnitude” and the
tenth constitutes the “base” of the system (and by definition is therefore
the sign for the “unit of the second decimal order of magnitude”). To
name the numbers ffom 11 to 19, the units are grouped in “packets” of ten
and we proceed (in theory) by simple addition:
11 one-ten (= 1 + 10)
12 two-ten (= 2 + 10)
13 three-ten (= 3 + 10)
14 four-ten (= 4 + 10)
BASF NUMBERS
26
15 five-ten (= 5 + 10)
16 six-ten (= 6 + 10)
17 seven-ten (=7 + 10)
18 eight-ten (= 8 + 10)
19 nine-ten (= 9 + 10)
Multiples of the base, from 20 to 90, are the “tens”, or units of the second
decimal order, and they are expressed by multiplication:
20
two -tens
(= 2 x 10)
30
three-tens
(= 3 x 10)
40
four- tens
(= 4 x 10)
50
five-tens
(= 5 x 10)
60
six-tens
(= 6 x 10)
70
seven-tens
(= 7 x 10)
80
eight-tens
(= 8 x 10)
90
nine-tens
(= 9 x 10)
If the number of tens is itself equal to or higher than ten, then the tens are
also grouped in packets of ten, constituting the “units of the third decimal
order”, as follows:
100
hundred
(= io 2 )
200
two hundreds
(= 2 x 100)
300
three hundreds
(= 3 x 100)
400
four hundreds
(= 4 x 100)
The hundreds are themselves then grouped into packets of ten, constituting
“units of the fourth decimal order”, or thousands:
1,000
one thousand
(=10 3 )
2,000
two thousands
(2 x 1,000)
3,000
three thousands
(3 x 1,000)
Then come the ten thousands, which used to be called myriads, correspond-
ing to the “units of the fifth decimal order”:
10.000 a myriad (=10 4 )
20.000 two myriads (= 2 X 10,000)
30.000 three myriads (= 3 x 10,000)
Using only these words of the language, the names of all the other numbers
are obtained by creating expressions that rely simultaneously on multipli-
cation and addition in strict descending order of the powers of the base 10:
53,781
five-myriads three-thousands seven-hundreds eight-tens one
(=5x10,000 +3x1,000 +7x 100 +8x 10 + 1)
Such then are the general rules for the formation of the names of the
cardinal numbers in the “base 10” system of the English language.
It must have taken a very long time for people to develop such an
effective way of naming numbers, as it obviously presupposes great
powers of abstraction. However, what we have laid out is evidently a
purely theoretical naming system, which no language follows with absolute
strictness and regularity. Particular oral traditions and the rules of
individual languages produce a wide variety of irregularities: here are some
characteristic examples from around the world.
Numbers in Tibetan
[For sources and further details, see M. Lalou (1950), S. C. Das (1915);
H. Bruce Hannah (1912). Information kindly supplied by Florence and
Helene Bequignon]
Tibetan has an individual name for each of the first ten numbers:
geig gnyis gsum bzhi Inga drug bdun brgyad dgu bcu
123456789 10
For numbers from 11 to 19, Tibetan uses addition:
11
beu-geig
(= 10 + 1)
12
bcu-gnyis
(= 10 + 2)
13
bcu-gsum
(= 10 + 3)
14
bcu-bzhi
(= 10 + 4)
15
bcu-lnga
(= 10 + 5)
16
bcu-drug
(= 10 + 6)
17
bcu-bdun
(= 10 + 7)
18
hcu-brg)iad
(= 10 + 8)
19
bcu-dgu
(= 10 + 9)
And for the tens, multiplication is applied:
20
gnyis-bcu
“two-tens”
(=2
X
10)
30
gsum-bcu
“three-tens”
(=3
X
10)
40
bzhi-bcu
“four-tens”
(=4
X
10)
50
Inga-bcu
“five-tens”
(=5
X
10)
60
drug-bat
“six-tens”
(=6
X
10)
70
bdun-bcu
“seven-tens”
(=7
X
10)
80
brgyad-bcu
“eight-tens”
(=8
X
10)
90
dgu-bcu
“nine-tens”
(=9
X
10)
27
For a hundred (=10 2 ) there is the word brgya, and the corresponding
multiples are obtained by the same principle of multiplication:
200 gnyis-brgya “two-hundreds” (= 2 x 100)
300 gsum-brgya “three-hundreds” (-3x 100)
400 bzhi-brgya “four-hundreds” (= 4 x 100)
500 Inga-brgya “five-hundreds” (= 5 x 100)
600 drug-brgya “six-hundreds” (= 6 x 100)
700 bdun-brg)'a “seven-hundreds” (= 7 x 100)
800 brgyad-brgya “eight-hundreds" (= 8 x 100)
900 dgu-brgya “nine-hundreds” (= 9 x 100)
There are similarly individual words for “thousand”, “ten thousand” and so
on, producing a very simple naming system for all intermediate numbers:
21: gnyis-bcu rtsa gag “two-tens and one”
(= 2 x 10 + 1)
560: Inga-brgya rtsa drug-bcu “five-hundreds and six-tens”
(= 5 x 100 + 6 x 10)
Numbers in Mongolian
[Source: L. Hambis (1945)]
Numbering in Mongolian is similarly decimal, but with some variations on
the regular system we have seen in Tibetan. It has the following names for
the first ten numbers:
nigdn qoyar g urban dorban tabun jirgu’an dolo’an naiman yisiinarban
123 456 7 89 10
and proceeds in a perfectly normal way for the numbers from eleven
to nineteen:
11 arhan nigdn (“ten-one”)
12 arban qoyar (“ten-two”)
However, the tens are formed rather differently. Instead of using analytic
combinations of the “two-tens", “three-tens" type, Mongolian has specific
words formed from the names of the corresponding units, subjected to a
kind of “declension” or alteration of the ending of the word:
20 qorin (from qoyar = 2)
30 g ucin (from g urban = 3)
40 docin (from dorban = 4)
50 tabin (from tabun = 5)
WHY BASH 10?
60 jirin (from jirgu’an = 6)
70 dalan (from dolo’an = 7)
80 nay an (from naiman = 8)
90 jarin (from yisiin = 9)
From one hundred, however, numbers are formed in a regular way based
on multiplication and addition, as explained above:
100 ja’un (“hundred”)
200 qoyar ja’un (“two-hundreds”)
300 g urban ja’un (“three-hundreds”)
400 dorban ja’un (“four-hundreds”)
1.000 minggan (“thousand”)
2.000 qoyar minggan (“two-thousands”)
3.000 g urban minggan (“three-thousands”)
10.000 tiirndn (“myriad”)
20.000 qoyar tiimdn (“two-myriads”)
20,541 qoyar tiimdn tabun ja’un docin nigdn
“two myriads five-hundreds forty one”
(= 2 x 10,000 + 5 x 100 +40 + 1)
Ancient Turkish numbers
[Source: A. K. von Gabain (1950)]
This section describes the numerals in spoken Turkish of the eighth century
CE as deduced from Turkish inscriptions found in Mongolia. The system
has some remarkable features.
The first nine numbers are as follows:
bir iki iic tiirt bes alti yeti sdkiz tokuz
1234567 8 9
For the tens, the following set of names are used:
10 on
20 yegirmi
30 otuz
40 kirk
50 dllig
60 altrriis
70 yet mis
BASE NUMBERS
28
80 sakiz on
90 tokuz on
The tens from 20 to 50 do not seem to have any etymological relation with
the corresponding units. However, altmis (= 60) and yetmis(= 70) derive
respectively from alt'i (= 6) and yeti (= 7) by the addition of the ending
mis (or mis). The words for 80 and 90 also derive from the names of 8 and
9, but by analytical combination with the word for 10, so that they mean
something like “eight tens” and “nine tens”.
The word for 50, however, is very probably derived from the ancient
method of finger-counting, since dllig is clearly related to dl (or alig), the
Turkish word for “hand”. (Turkish finger-counting is still done in
the following way: using one thumb, you touch in order on the other
hand the tip of the little finger, the ring finger, the middle finger and the
index finger, which gets you to 4; for 5, you raise the thumb of the “counted”
hand; then you bend back the thumb and raise in order the index finger,
the middle finger, the ring finger, the little finger, and finally the thumb
again, so that for 10 you have all the fingers of the “counted” hand stretched
out. This technique represents the trace of an even older system in which
the series was extended by raising one finger of the other hand for each ten
counted, so that one hand with all fingers stretched out meant 10, and the
other hand with all fingers stretched out meant 50.)
The system then gives the special name of yiiz to the number 100, and
proceeds by multiplication for the names of the corresponding multiples
of a hundred:
100
yiiz
200
iki yiiz
“two-hundreds”
(= 2 x 100)
300
iic yiiz
“three-hundreds”
(= 3 x 100)
400
tort yiiz
“four-hundreds”
(= 4 X 100)
500
bes yiiz
“five-hundreds”
(= 5 x 100)
600
altiyiiz
“six-hundreds”
(= 6 X 100)
700
yeti yiiz
“seven-hundreds”
(= 7 x 100)
800
sakiz yiiz
“eight-hundreds”
(= 8 x 100)
900
tokuz yiiz
“nine-hundreds”
(= 9 x 100)
The word for a thousand is bing (which in some Turkic dialects also means
“a very large amount”), and the multiples of a thousand are similarly
expressed by analytical combinations of the same type:
1.000 bing
2.000 iki bing “two-thousands” (= 2 x 1,000)
3.000 iic bing “three-thousands” (= 3 x 1,000)
4,000
tort bing
“four-thousands” (=4x1,000)
5,000
bes bing
“five-thousands” (= 5 x 1,000)
6,000
alt'i bing
“six-thousands” (= 6 x 1,000)
7,000
yeti bing
“seven-thousands” (= 7 x 1,000)
8,000
sakiz bing
“eight-thousands” (= 8 x 1,000)
9,000
tokuz bing
“nine-thousands” (= 9 x 1,000)
What is unusual about the ancient Turkish system is the way the
numbers from 11 to 99 are expressed. In this range, what is given is first
the unit, and then, not the multiple of ten already counted, but the
multiple not yet reached, by
a kind of “prospective account”. This gives,
for example:
11
biryegirmi
literally: “one, twenty”
12
ikiyegirmi
literally: “two, twenty”
13
iicyegirmi
literally: “three, twenty”
21
bir otuz
literally: “one, thirty”
22
iki otuz
literally: “two, thirty”
53
iic altmis
literally: “three, sixty”
65
bes yet mis
literally: “five, seventy”
78
sakiz sakiz on
literally: “eight, eighty”
99
tokuz yiiz
literally: “nine, one hundred”
What is involved is neither a multiplicative nor a subtractive principle
but something like an ordinal device, as follows:
11
“the
first unit before twenty”
12
“the
second unit before twenty”
21
“the
first unit before thirty”
23
“the
third unit before thirty”
53
“the
third unit before sixty”
87
“the
seventh unit before ninety”
99
“the
ninth unit before a hundred”
This way of counting is reminiscent of the way time is expressed in
contemporary German, where, for “a quarter past nine” you say viertel
zehn, meaning “a quarter of ten" (= “the first quarter before ten”), or for
“half past eight” you say halb neun, meaning “half nine” (= “the first half
before nine”).
However, around the tenth century CE, under Chinese influence,
which was very strong in the eastern Turkish-speaking areas, this
rather special way of counting was “rationalised” (by the Uyghurs, first
of all, who always have been close to Chinese civilisation). Using the
Turkic stem artuk, meaning “overtaken by”, the following expressions
were created:
29
WHY BASE 10?
11 on artuk'i bir (“ten overtaken by one”)
23 yegirmi artuk'i iic (“twenty overtaken by three”)
53 allig artuk'i iic (“fifty overtaken by three”)
87 sakiz on artuk'i yeti (“eighty overtaken by seven”)
Whence come the simplified versions still in use today:
11 on bir (= 10 + 1)
23 yegirmi iic (= 20 + 3)
53 allig iic (= 50 + 3)
87 sakiz on yeti (=80 + 7)
11
eka-dasa
“one-ten”
(= 1 + 10)
12
dva-dasa
"two-ten”
(=2 + 10)
13
tri- dasa
“three-ten”
(= 3 + 10)
14
catvari-dasa
“four-ten”
(= 4 + 10)
15
panca-dasa
“five-ten”
(= 5 + 10)
16
sat-dasa
“six-ten”
(= 6 + 10)
17
sapta-dasa
"seven-ten”
(=7 + 10)
18
asta-dasa
“eight-ten”
(= 8 + 10)
19
ndva-dasa
“nine-ten”
(= 9 + 10)
For the following multiples of 10, Sanskrit has names with particular features:
Sanskrit numbering
The numbering system of Sanskrit, the classical language of northern
India, is of great importance for several related reasons. First of all, the
most ancient written texts that we have of an Indo-European language
are the Vedas, written in Sanskrit, from around the fifth century BCE, but
with traces going as far back as the second millennium BCE. (All modern
European languages with the notable exceptions of Finnish, Hungarian,
Basque, and Turkish belong to the Indo-European group: see below).
Secondly, Sanskrit, as the sacred language of Brahmanism (Hinduism), was
used throughout India and Southeast Asia as a language of literary and
scholarly expression, and (rather like Latin in mediaeval Europe) provided
a means of communication between scholars belonging to communities
and lands speaking widely different languages. The numbering system of
Sanskrit, as a part of a written language of great sophistication and
precision, played a fundamental role in the development of the sciences
in India, and notably in the evolution of a place-value system.
The first ten numbers in Sanskrit are as follows:
1 eka
2 dvau, dva, dve, dvi
3 trayas, tisras, tri
4 catvaras, catasras, catvari, catur
5 panca
6 sat
7 sapta
8 astau, asta
9 nava
10 dasa
Numbers from 11 to 19 are then formed by juxtaposing the number of
units and the number 10:
20 vimsati
30 trimsati
40 catvarimsati
50 pahcasat
60 sasti
70 sapti
80 asiti
90 navati
Broadly speaking, the names of the tens from 20 upwards are formed from
a word derived from the name of the corresponding unit plus a form for the
word for 10 in the plural.
One hundred is satam or sata, and for multiples of 100 the regular
formula is used:
100 satam, sata
200 dvisata (= 2 x 100)
300 trisata (= 3 x 100)
400 catursata (= 4 x 100)
500 pancasata (= 5 x 100)
For 1,000, the word sahasram or sahasra is used, in analytical combi-
nation with the names of the units, tens and hundreds to form multiples
of the thousands, the ten thousands, and hundred thousands:
1,000
2.000
sahasra
dvisahdsra
(= 2 X 1,000)
3,000
trisahasra
(= 3 X 1,000)
10,000
dasasahasra
(= 10 x 1,000)
20,000
vimsatsahasra
(= 20 x 1,000)
30,000
trimsatsahasra
(= 30 x 1,000)
BASE NUMBERS
30
100.000 satasahasra (= 100 x 1,000)
200.000 dvisatasahasra (= 200 x 1,000)
300.000 trisatasahasra (= 300 x 1,000)
This gives the following expressions for intermediate numbers:
4,769:
nava
sasti
saptasata
ca
catursahasra
(“nine
sixty
seven-hundreds
and
four-thousands”)
(=9
+ 60
+ 7 x 100
+
4 x 1,000)
Sanskrit thus
has a
decimal numbering system,
like ours, but with
combinations done “in reverse", that is to say starting with the units and
then in ascending order of the powers of 10.
WHAT IS INDO-EUROPEAN?
“Indo-European” is the name of a huge family of languages spoken
nowadays in most of the European land-mass, in much of western Asia, and
in the Americas. There has been much speculation about the geographical
origins of the peoples who first spoke the language which has split into the
many present-day branches of the Indo-European family. Some theories
hold that the Indo-Europeans originally came from central Asia (the Pamir
mountains, Turkestan); others maintain that they came from the flat lands
of northern Germany, between the Elbe and the Vistula, and the Russian
steppes, from the Danube to the Ural mountains. The question remains
unresolved. All the same, some things are generally agreed. The Indo-
European languages derive from dialects of a common “stem” spoken by
a wide diversity of tribes who had numerous things in common. The Indo-
Europeans were arable farmers, hunters, and breeders of livestock; they
were patriarchal and had social ranks or castes of priests, farmers, and
warriors; and a religion that involved the cult of ancestors and the worship
of the stars. However, we know very little about the origins of these peoples,
who acquired writing only in relatively recent times.
The Indo-European tribes began to split into different branches in the
second millennium BCE at the latest, and over the following thousand
years the following tribes or branches appear in early historical records:
Aryans, in India, and Kassites, Hittites, and Lydians in Asia Minor; the
Achaeans, Dorians, Minoans, and Hellenes, in Greece; then the Celts in
central Europe, and the Italics in the Italian peninsula. Further migrations
from the East occurred towards the end of the Roman Empire in the fourth
to sixth centuries CE, bringing the Germanic tribes into western Europe.
The Indo-European language family is thus spread over a very wide area
and is traditionally classified in the following branches, for each of which
the earliest written traces date from different periods, but none from before
the second millennium BCE:
• The Indo- Aryan branch: Vedic, classical Sanskrit, and their numerous
modern descendants, of which there are five main groups:
- the western group, including Sindhi, Gujurati, Landa, Mahratta,
and Rajasthani
- the central group, including Punjabi, Pahari, and Hindi
- the eastern group, including Bengali, Bihari, and Oriya
- the southern group (Singhalese)
- so-called “Romany” or gypsy languages
• The Iranian branch, including ancient Persian (spoken at the time
of Darius and Xerxes), Avestan (the language of Zoroaster), Median,
Scythian, as well as several mediaeval and modern languages spoken in
the area of Iran (Sogdian, Pahlavi, Caspian and Kurdish dialects,
Ossetian (spoken in the Caucasus), Afghan, and Baluchi)
• A branch including the Anatolian language of the Hittite Empire as
well as Lycian and Lydian
• The Tokharian branch. This language (with its two dialects, Agnaean
and Kutchian) was spoken by an Indo-European population settled in
Chinese Turkestan between the fifth and tenth centuries CE, but became
extinct in the Middle Ages. As an ancient language related to Hittite as
well as to Western branches of the Indo-European family (Greek, Latin,
Celtic, Germanic), it is of great importance for historical linguists and is
often used in tracing the etymologies of common Indo-European words
• The Armenian branch, with two dialects, western (spoken in
Turkey) and eastern (spoken in Armenia)
• The Hellenic branch, which includes ancient dialects such as
Dorian, Achaean, Creto-Minoan, as well as Homeric (classical) Greek,
Koine (the spoken language of ancient Greece), and Modern Greek
• The Italic branch, which includes ancient languages such as
Oscan, Umbrian and Latin, and all the modern Romance languages
(Italian, Spanish, Portuguese, Provencal, Catalan, French, Romanian,
Sardinian, Dalmatian, Rhaeto-Romansch, etc.)
• The Celtic branch, which has two main groups:
-“continental” Celtic dialects, including the extinct language of
the Gauls
-“island” Celtic, itself possessing two distinct subgroups, the
Brythonic (Breton, Welsh, and Cornish) and the Gaelic (Erse,
Manx, and Scots Gaelic)
• The Germanic branch, which has three main groups:
- Eastern Germanic, of which the main representative is Gothic
31
I N D O - F. U R O P K A N NUMBER-SYSTEMS
- the Nordic languages (Old Icelandic, Old Norse, Swedish, Danish)
- Western Germanic languages, including Old High German and
its mediaeval and modern descendants (German), Low German,
Dutch, Friesian, Old Saxon, Anglo-Saxon, and its mediaeval and
modern descendants (Old English, Middle English, contempo-
rary British and American English)
• The Slavic branch, of which there are again three main groups:
- Eastern Slavic languages (Russian, Ukrainian, and Belorussian)
- Southern Slavic languages (Slovenian, Serbo-Croatian, Bulgarian)
-Western Slavic languages (Czech, Slovak, Polish, Lekhitian,
Sorbian, etc.)
• The Baltic branch, comprising Baltic, Latvian, Lithuanian, and Old
Prussian
• Albanian, a distinct branch of the Indo-European family, with no
“close relatives” and two dialects, Gheg and Tosk
• The Thraco -Phrygian branch, with traces found in the Balkans
(Thracian, Macedonian) and in Asia Minor (Phrygian)
• And finally a few minor dialects with no close relatives, such as
Venetian and Illyrian.
INDO-EUROPEAN NUMBER-SYSTEMS
Sanskrit is thus a particular case of a very large “family” of languages (the
Indo-European family) all of whose members use decimal numbering
systems. The general rule that all these systems have in common is that the
numbers from 1 to 9 and each of the powers of 10 have individual names,
all other numbers being expressed as analytical combinations of these
names.
Nonetheless, some of these languages have additional number-names
that seem to have no etymological connection with the basic set of names:
for example “eleven” and “twelve” in English, like the German e/fand zwolf
have no obvious connection to the words for “ten” (zehn) and “one” (eins) or
ten” and “two” (zwei) respectively, whereas all the following numbers are
formed in regular fashion:
ENGLISH GERMAN
13
thirteen
(= three+ten)
dreizehn
(= drei+zehn )
14
fourteen
(= four+ten)
vierzehn
(= vier+zehn )
15
fifteen
(= five+ten)
junjzchn
( =funf+zehn )
16
sixteen
(= six+ten)
sechszehn
(= sechs+zehn )
17
seventeen
(= seven+ten)
siebzehn
(= sieben+zehn)
18
eighteen
(= eight+ten)
achtzehn
(= acht+zehn)
19
nineteen
(= nine+ten)
neunzehn
(= neun+zehn)
The “additional” number-names in the range 11-19 in the Romance
languages, on the other hand, are obvious contractions of the analytical
Latin names (with the units in first position) from which they are all derived:
LATIN
ITALIAN
FRENCH
S PA N I S
11
undecim
(“one-ten”)
undid
onze
once
12
duodecim
("two-ten”)
dodici
douze
doce
13
tredecim
(“three-ten”)
tredici
treize
trece
14 quattuordecim
(“four-ten”)
quattordici
quatorze
catorce
15
quindccim
(“five-ten”)
quindici
quinze
quince
16
sedecim
(“six-ten”)
sedici
sdze
17
septendecim
(“seven-ten”)
18
octodecim
(“eight-ten”)
19 undeviginti (“one from twenty”)
The remaining numbers before 20 are constructed analytically: dix-scpt,
dix-huit, da-neuf (French), dieci-sette, dieci-otto (Italian), etc.
In the Germanic languages, the tens are constructed in regular fashion
using an ending clearly derived from the word for "ten” on the stem of the
word for the corresponding unit : in English, twenty = “two - ty”, thirty =
“three - ty”, and so on. and in German, zwanzig = "zwei - zig, dreissig =
drei + sig, and so on. In order to avoid confusion between the “teens” and
the “tens” in Latin, multiples of 10, which similarly have the unit-name in
first position, use the ending “-ginta”, giving the following contractions
in the Romance languages derived from it:
LATIN
ITALIAN
FRENCH
SPANISH
30
triginta
trenta
trente
treinta
40
quadraginta
quaranta
quarante
cuarenta
50
quinquaginta
cinquanta
cinquante
cincuenta
60
sexaginta
sessanta
soixante
sessanta
70
septuaginta
settanta
septante*
setenta
80
octoginta
ottanta
octante*
ochenta
90
nonaginta
novanta
nonante *
noventa
The French numerals marked with an asterisk are the “regular” versions
found only in Belgium and French-speaking Switzerland; “standard” French
uses irregular expressions for 70 (soixantc-dix, “sixty-ten”), 80 ( quatre -
vingts, “four-twenty”), and 90 ( quatre-vingt-dix , “four-twenty-ten”). In
addition, of course, we have omitted from the table above the Latin and
Romance names for the number 20, which seems to be a problem at first
sight. In Latin it is viginti, a word with no relation to the words for “ten”
(decern) or for “two” ( duo ); and its Romance derivatives, with the exception
of Romanian, follow the irregularity (i venti in Italian, vingt in French, veinte
in Spanish). So where does the "Romance twenty” come from?
BASF NUMBERS
Roots
The richness of the descendance of the original Indo-European language
means that, by comparison and deduction, it is possible to reconstruct
the form that many basic words must have had in the “root” or “stem”
language, even though no written trace remains of it. Indo-European root
words, being hypothetical, are therefore always written with an asterisk.
The original number-set is believed to have been this:
1 *oi-no, *oi-ko, *oi-wo
2 *dwd, *dwu, *dwoi
3 *tri (and derivative forms: *treyes, *tisores)
4 *kwetwores, *kwetesres, *kwetwor
5 *penkwe, *kwenkwe
6 *seks, *sweks
7 *septm
8 *oktd, *oktu
9 *newn
10 *dekm
This helps us to see that despite their apparent difference, the words for
“one” in Sanskrit, Avestan, and Czech, for example (respectively eka, aeva
and jeden) are all derived from the same “root” or prototype, as are the
Latin unus, German eins and Swedish en.
All trace has been lost of the concrete meanings that these Indo-European
number-names might have had originally. However, Indo-European lan-
guages do bear the visible marks of that long-distant time when, in the
absence of any number-concept higher than two, the word for all
other numbers meant nothing more than “many".
The first piece of evidence of this ancient number-limit is the grammatical
distinction made in several Indo-European languages between the singular,
the dual, and the plural. In classical Greek, for example, ho lukos means “the
3 2
wolf”, hoi lukoi means “the wolves”, but for “two wolves” a special ending,
the mark of the “dual”, is used: to luko.
Another piece of the puzzle is provided by the various special meanings
and uses of words closely associated with the name of the number 3. Anglo-
Saxon thria (which becomes “three” in modern English) is related to the
word throp, meaning a pile or heap; and words like throng are similarly
derived from a common Germanic root having the sense of “many”. In the
Romance languages there are even more evident connections between
the words for “three” and words expressing plurality or intensity: the
Latin word tres (three) has the same root as the preposition and prefix
trans- (with meanings related to “up until”, “through”, “beyond”), and in
French, derivations from this common stem produce words like tres
(“very”), “trop" (too much), and even troupe (“troop”). It can be deduced
from these and many other instances that in the original Indo-European
stem language, the name of the number “three” (tri) was also the word for
plurality, multiplicity, crowds, piles, heaps, and for the beyond, for what
was beyond reckoning.
The number-systems of the Indo-European languages, which are all
strictly decimal, have remained amazingly stable over many millennia, even
whilst most other features of the languages concerned have changed
beyond recognition and beyond mutual comprehension. Even the apparent
irregularities within the system are for the most part explicable within the
logic of the original decimal structure - for example, the problem
mentioned above of the “Romance twenty”. French vingt, Spanish veinte,
etc. derive from Latin viginti, which is itself fairly self-evidently a derivative
of the Sanskrit vimsati. And Sanskrit “twenty” is not irregular at all, being
a contraction of a strictly decimal dvi-dasati (“two-tens”) => visati =>
vimsati. Similar derivations can be found in other branches of the Indo-
European family of languages. In Avestan, 20 is visaiti, formed from hat,
“two”, and dasa (= 10); and in Tokharian A, where wu = 2 and sak = 10,
wi-saki (- 2 x 10) became wiki, “twenty”.
33
INDO-EUROPEAN NUMBER-SYSTEMS
THE NAMES OF THE NUMBER 1
Indo-European
*oi-no, *oi-ko.
prototypes:
*oi-wo
SANSKRIT
eka
AVESTAN
aeva
GREEK
hen
EARLY LATIN
oinos, oinom
LATIN
unus, unum
ITALIAN
uno
SPANISH
uno
FRENCH
un
PORTUGUESE
um
ROMANIAN
uno
OLD ERSE
oen
MODERN IRISH
oin
BRETON
eun
SCOTS GAELIC
un
WELSH
un
GOTHIC
ain (-s)
DUTCH
een
OLD ICELANDIC
einn
SWEDISH
en
DANISH
en
OLD SAXON
en
ANGLO-SAXON
an
ENGLISH
one
OLD HIGH GERMAN
ein, eins
GERMAN
ein
CHURCH SLAVONIC
inu
RUSSIAN
odin
CZECH
jedert
POLISH
jeden
LITHUANIAN
vienas
BALTIC
vienes
Fig. 2.4A.
THE NAMES OF THE NUMBER 2
Indo-European *dwd, *dwu,
prototypes: *dwoi
SANSKRIT
dvau, dva, dvi
AVESTAN
bae
HITTITE
ta
TOKHARIAN A
wu, we
ARMENIAN
erku
GREEK
duo
LATIN
duo, duae
SPANISH
dos
FRENCH
deux
ROMANIAN
doi
OLD ERSE
dau, do
MODERN IRISH
da
BRETON
diou
SCOTS GAELIC
dow
WELSH
dwy, dau
GOTHIC
twai, twa
DUTCH
twee
OLD ICELANDIC
tveir
SWEDISH
tva
DANISH
to
OLD SAXON
twene
ANGLO-SAXON
twegen
ENGLISH
two
OLD HIGH GERMAN
zwene
GERMAN
zwei
CHURCH SLAVONIC
duva, duve
RUSSIAN
dva
POLISH
dwa
LITHUANIAN
du, dvi
ALBANIAN
dy, dyj
Fig. 2.4B.
THE NAMES OF THE NUMBER 3
Indo-European
*treyes,
prototypes:
*tisores, *tri
SANSKRIT
trayas, tisras,
tri
AVESTAN
thrayo, tisro,
tri
HITTITE
tri
TOKHARIAN B
trai
ARMENIAN
erekh
GREEK
Ireis
OSCAN
tris
LATIN
tres, tria
ITALIAN
tre
SPANISH
Ires
FRENCH
trois
ROMANIAN
trei
OLD ERSE
teoir, tri
WELSH
tri, tair
GOTHIC
threis, thrija
DUTCH
drie
OLD ICELANDIC
thrir
SWEDISH
tre
OLD SAXON
thria
ANGLO-SAXON
thri
ENGLISH
three
OLD HIGH GERMAN
dri
GERMAN
drei
CHURCH SLAVONIC
trije, tri
RUSSIAN
tri
POLISH
trzy
LITHUANIAN
trys
ALBANIAN
tre, tri
Fig. 2.4c.
THE NAMES OF THE NUMBER 4
Indo-European
prototypes:
*kwetwores,
*kwetesres,
*kwetwor
SANSKRIT
catvaras,
catasras,
catvari, catur
AVESTAN
cathwaro
TOKHARIAN A
st war
TOKHARIAN B
stwer
ARMENIAN
corkh
ANCIENT GREEK
tettares,
tessares,
tetores
OSCAN
pettiur, petora
LATIN
quattuor
ITALIAN
quattro
SPANISH
cuatro
FRENCH
quatre
ROMANIAN
patru
OLD ERSE
cethir, cethoir
BRETON
pevar
WELSH
pedwar
SCOTS GAELIC
peswar
GOTHIC
fidwor
OLD ICELANDIC
jjorer
SWEDISH
jym
OLD SAXON
fiuwar
ANGLO-SAXON
foevter
ENGLISH
four
OLD HIGH GERMAN
vier
GERMAN
vier
CHURCH SLAVONIC
cetyre
RUSSIAN
cetyre
CZECH
ctyri
POLISH
cilery
LITHUANIAN
keturi
BALTIC
keturi
Fig. 2.4D.
BASF. NUMBERS
3 4
THE NAMES OF THE NUMBER 5 THE NAMES OF THE NUMBER 6 THE NAMES OF THE NUMBER 7 THE NAMES OF THE NUMBER 8
Indo-European
*penkwe.
prototypes:
*kwenkwe
Indo-European
*seks, *sweks
prototypes:
Indo-European
*septm
prototype:
Indo-European
*okto, *oktu
prototypes:
SANSKRIT
sapta
AVESTAN
hapta
HITTITE
sipta
TOKHARIAN A
spat
ARMENIAN
ewhtn
GREEK
hepta
LATIN
septem
SPANISH
siete
FRENCH
sept
ROMANIAN
shapte
OLD ERSE
secht
MODERN IRISH
secht
WELSH
saith
BRETON
seiz
GOTHIC
sibun
DUTCH
zeven
OLD ICELANDIC
siau
SWEDISH
sju
OLD SAXON
sibun
ENGLISH
seven
OLD HIGH GERMAN
siben
GERMAN
sieben
CHURCH SLAVONIC
sedmJ
RUSSIAN
sem
POLISH
siedem
LITHUANIAN
septyni
SANSKRIT
ast’d, astau
AVESTAN
asta
TOKHARIAN B
okt
ARMENIAN
uth
GREEK
okto
LATIN
odd
SPANISH
ocho
FRENCH
huit
ROMANIAN
opt
OLD ERSE
ocht
MODERN IRISH
ocht
WELSH
wyth
BRETON
eiz
GOTHIC
ahtau
DUTCH
acht
OLD ICELANDIC
alta
SWEDISH
dtta
OLD SAXON
ahto
ANGLO-SAXON
eahta
ENGLISH
eight
OLD HIGH GERMAN
ahto
GERMAN
acht
CHURCH SLAVONIC
osmi
RUSSIAN
vosem ’
POLISH
osiem
LITHUANIAN
astuoni
SANSKRIT
sat
AVESTAN
xsvas
TOKHARIAN A
sak
ARMENIAN
vec
ANCIENT GREEK
weks
MODERN GREEK
hex
LATIN
sex
ITALIAN
sei
SPANISH
seis
FRENCH
six
ROMANIAN
shase
OLD ERSE
se
MODERN IRISH
se
WELSH
chwech
BRETON
c'houec’h
GOTHIC
saihs
DUTCH
zes
OLD ICELANDIC
sex
SWEDISH
sex
OLD SAXON
sehs
ANGLO-SAXON
six
ENGLISH
six
OLD HIGH GERMAN
sehs
GERMAN
sechs
CHURCH SLAVONIC
sesti
RUSSIAN
chest’
POLISH
szesc
LITHUANIAN
sesi
ALBANIAN
giashte
SANSKRIT
pahca
AVESTAN
panca
HITTITE
panta
TOKHARIAN A
pah
TOKHARIAN B
pis
ARMENIAN
king
GREEK
pente
LATIN
quinque
SPANISH
cinco
FRENCH
cinq
ROMANIAN
cinci
OLD ERSE
coic
MODERN IRISH
coic
WELSH
pump
BRETON
pemp
GOTHIC
fimf
DUTCH
viif
OLD ICELANDIC
fimm
SWEDISH
fern
OLD SAXON
fif
ANGLO-SAXON
flf
ENGLISH
five
OLD HIGH GERMAN
finf
GERMAN
fiinf
CHURCH SLAVONIC
peti
RUSSIAN
piat'
POLISH
piec
LITHUANIAN
penki
ALBANIAN
pgse
35
INDO-EUROPEAN NUMBER-SYSTEMS
THE NAMES OF THE NUMBER 9 THE NAMES OF THE NUMBER 10
Indo-European prototype: *newn
Indo-European prototype: *dekm
SANSKRIT
nava
AVF.STAN
nava
TOKHARIAN A
hu
TOKHARIAN B
hu
ARMENIAN
inn
GREEK
en-nea
LATIN
novem
ITALIAN
nove
SPANISH
nueve
FRENCH
neuf
ROMANIAN
noue
PORTUGUESE
noue
OLD ERSE
noin
MODERN IRISH
noi
WELSH
naw
BRETON
nao
GOTHIC
nium
DUTCH
negon
OLD ICELANDIC
nio
SWEDISH
nio
OLD SAXON
nigun
ANGLO-SAXON
nigon
ENGLISH
nine
OLD HIGH GERMAN
niun
GERMAN
tieun
CZECH
devet
RUSSIAN
deviat'
POLISH
dziewiec
LITHUANIAN
devyni
ALBANIAN
nende
SANSKRIT
dasa
AVESTAN
dasa
TOKHARIAN A
sdk
TOKHARIAN B
sak
ARMENIAN
tasn
GREEK
deka
LATIN
decern
ITALIAN
died
SPANISH
diez
FRENCH
dix
ROMANIAN
zece
PORTUGUESE
dez
OLD ERSE
deich
MODERN IRISH
deich
WELSH
deg
BRETON
dek
GOTHIC
taikun
DUTCH
tien
OLD ICELANDIC
tio
SWEDISH
tio
OLD SAXON
techan
ANGLO-SAXON
tyn
ENGLISH
ten
OLD HIGH GERMAN
zehan
GERMAN
zehn
CZECH
deset
RUSSIAN
desiat’
POLISH
dziesicc
LITHUANIAN
desimt
ALBANIAN
diete
LATIN
ITALIAN
FRENCH
SPANISH
ROMANIAN
1
unus
uno
un
uno
uno
2
duo
due
deux
dos
doi
3
tres
tre
trois
tres
trei
4
quattuor
quattro
quatre
cuatro
patru
5
quinque
cinque
cinq
cinco
cinci
6
sex
sei
six
seis
shase
7
septem
selte
sept
siete
shapte
8
octo
otto
huit
ocho
opt
9
novem
nove
neuf
nueve
noue
10
decern
died
dix
diez
zece
11
undecim
undid
onze
once
un spree zece
12
duodecim
dodici
douze
doce
doi spree zece
20
viginti
venti
vingt
veinte
doua-zeci
30
triginta
trenta
trente
treinta
trei-zeci
40
quadraginta
quaranta
quarante
cuarenta
patru-zeci
50
quinquaginta
cinquanta
cinquante
cincuenta
cinci-zeci
60
sexaginta
sessanta
soixanle
sesenta
shase-zeci
70
septuaginta
settanta
soixante-dix
setenta
shapte-zeci
80
octoginta
ottanta
quatre-vingts
ochenta
opt-zeci
90
nonagirtla
novanta
quatre-vingt-dix
noventa
noua-zeci
100
centum
cento
cent
ciento
osuta
1,000
mille
mille
mille
mil
0 mie
GOTHIC
OLD HIGH
GERMAN
GERMAN
ANGLO-SAXON
ENGLISH
1
ains
ein
eins
an
one
2
twa
zwene
zwei
twegen
two
3
preis
dri
drei
pri
three
4
fidwoor
vier
vier
feower
four
5
fimf
finf
fimf
fif
five
6
saihs
sehs
sechs
six
six
7
si bun
siben
siebcn
seofou
seven
8
ahtau
ahte
acht
eahta
eight
9
niun
niun
neun
nigon
nine
10
taihun
zehan
zehn
tyn
ten
11
ain-lif
einlif
ef
endleofan
eleven
12
twa-lif
zwelif
zwblf
twelf
twelve
20
twai-tigjus
zwein-zug
zwanzig
twenlig
twenty
30
threo-tigjus
driz-zug
dreifiig
thritig
thirty
40
fidwor-tigjus
fior-zug
vierzig
feowertig
forty
50
fimf-tigjus
M-zug
funfzig
fifi'g
fifty
60
saihs-tigjus
sehs-zug
sechzig
sixtig
sixty
70
sibunt-ehund
sibun-zo
siebzig
hund-seofontig
seventy
80
ahtaut-ehund
ahto-zo
achtzig
hund-eahtatig
eighty
90
niunt-ehund
niun-zo
neunzig
hund-nigontig
ninety
100
talhun-taihund
zehan-zo
hundert
hund-teontig
hundred
1,000
thusundi
dusent
tausend
thusund
thousand
Fig. 2. 41-
Fig. 2.4J.
Fig. 2.5. The decimal nature of Indo-European number-names
BASF N U M B F, K S
OTHER SOLUTIONS TO THE PROBLEM
OF THE BASE
Not all civilisations came up with the same solution to the problem of the
base. In other words, base 10 is not the only way of constructing a number-
system.
There are many examples of numeration built on a base of 5. For
example: Api, a language spoken in the New Hebrides (Oceania), gives
individual names to the first five numbers only:
1 tai
2 lua
3 tolu
4 vari
5 luna (literally, “the hand”)
and then uses compounds for the numbers from 6 to 10:
6 otai (literally, “the new one")
7 olua (literally, “the new two”)
8 otolu (literally, “the new three”)
9 ovari (literally, “the new four”)
10 lualuna (literally, “two hands”)
The name of 10 then functions as a new base unit:
11
lualuna tai
(=2x5 + 1)
12
lualuna lua
(=2x5 + 2)
13
lualuna tolu
(=2x5 + 3)
14
lualuna vari
(=2x5 + 4)
15
toluluna
(=3X5)
16
toluluna tai
(=3x5 + 1)
17
toluluna lua
(=3x5 + 2)
and so on. [Source: T. Dantzig (1930), p. 18]
Languages that use base 5 or have traces of it in their number-systems
include Carib and Arawak (N. America): Guarani (S. America); Api and
Houailou (Oceania); Fulah, Wolof, Serere (Africa), as well as some other
African languages: Dan (in the Mande group), Bete (in the Kroo group),
and Kulango (one of the Voltaic languages); and in Asia, Khmer. [See:
M. Malherbe (1995); F. A. Pott (1847)].
Other civilisations preferred base 20 - the “vigesimal base” - by which
things are counted in packets or groups of twenty. Amongst them we
find the Tamanas of the Orinoco (Venezuela), the Eskimos or Inuits
(Greenland), the Ainus in Japan and the Zapotecs and Maya of Mexico.
3 6
The Mayan calendar consisted of “months” of 20 days, and laid out cycles
of 20 years, 400 years (= 20 2 ) 8,000 years (= 20 3 ), 160,000 years (= 20 4 ),
3,200,000 years (= 20 5 ), and even 64,000,000 years (= 20 6 ).
Like all the civilisations of pre-Columbian Central America, the Aztecs
and Mixtecs measured time and counted things in the same way, as shown
in numerous documents seized by the conquistadors. The goods collected
by Aztec administrators from subjugated tribes were all quantified in
vigesimal terms, as Jacques Soustelle explains:
For instance, Toluca was supposed to provide twice a year 400 loads of
cotton cloth, 400 loads of decorated ixtle cloaks, 1,200 (3 X 20 2 ) loads
of white ixtle cloth . . . Quahuacan gave four yearly tributes of 3,600
(9 x 20 2 ) beams and planks, two yearly tributes of 800 (2 x 20 2 ) loads
of cotton cloth and the same number of loads of ixtle cloth . . .
Quauhnahuac supplied the Imperial Exchequer with twice-yearly deliv-
eries of 3,200 (8 x 20 2 ) loads of cotton cloaks, 400 loads of loin-cloths,
400 loads of women’s clothing, 2,000 (5 x 20 2 ) ceramic vases, 8,000
(20 3 ) sheaves of “paper” . . .
[From the Codex Mendoza]
This is how the Aztec language gives form to a quinary-vigesimal base:
1
ce
H
matlactli-on-ce (10 + 1)
2
ome
12
matlactli’ on-ome (10 + 2)
3
yey
13
matlactli-on-yey (10 + 3)
4
naui
14
matlactli-on-naui (10 + 4)
5
chica or macuilli
15
caxtulli
6
chica-ce (5 + 1)
16
caxtulli-on-ce (15 + 1)
7
chica-ome (5 + 2)
17
caxtulli-on-ome (15 + 2)
8
chica-ey (5 + 3)
18
caxtulli-on-yey (15 + 3)
9
chica-naui (5 + 4)
19
caxtulli-on-naui (15 + 4)
10
matlactli
20
cem-poualli (1 x 20, “a score”)
30
cem-poualli- on- matlactli
(20 + 10)
40
ome-poualli
(2 X 20)
50
ome-poualU-on-matlactli
(2 x 20 + 10)
100
macuiTpoualli
(5 x 20)
200
matlactli-poualli
(10 x 20)
300
caxtullipoualli
(15 x 20)
400
cen-tzuntli
(1 x 400, “one four-hundreder")
800
ome-tzuntli
(2 x 400)
1,200
yey-tzuntli
(3 X 400)
8,000
cen-xiquipilli
(1 x 8,000, “one eight-thousander”)
Fig. 2.6.
There are many populations outside of America and Europe (for
instance, the Malinke of Upper Senegal and Guinea, the Banda of Central
Africa, the Yebu and Yoruba people of Upper Senegal and Nigeria, etc.) who
37
OTHER SOLUTIONS
continue to count in this fashion. Yebu numeration is as follows, according
to C. Zaslavsky (1973):
1
otu
2
abuo
3
ato
4
ano
5
iso
6
isii
7
asaa
8
asato
9
toolu
10
iri
20
ohu
30
ohu na iri
(=
20 +10)
40
ohu abuo
(=
20x2)
50
ohu abuo na iri
(=
20 x 2 +10)
60
ohu ato
(=
20x3)
100
ohu iso
(=
20 x 5)
200
ohu iri
(=
: 20 x 10)
400
nnu
(=
20 2 )
8,000
nnu khuru ohu
(=
20 3 = “400 meets 20”)
160,000
nnu khuru nnu
(=
20 4 = “400 meets 400”)
The Yoruba, however, proceed in a quite special way, using additive and
subtractive methods alternately [Zaslavsky (1973)]:
1
ookan
2
eeji
3
eeta
4
eerin
5
aarun
6
eeta
7
eeje
8
eejo
9
eesan
10
eewaa
11
ookan laa
(= 1 + 10: laa from le ewa, “added to 10”)
12
eeji laa
(= 2 + 10)
13
eeta laa
(= 3 + 10)
14
eerin laa
(= 4 + 10)
15
eedogun
(=20-5; from aarun
din ogun, “5 taken from 20”)
16
erin din logun
(= 20 - 4)
17
eeta din logun
(= 20 - 3)
18
eeji din logun
(= 20 - 2)
19
ookan din logun
(=20-1)
20
ogun
21
ookan le loogun
(= 20 + 1)
25
eedoogbon
(= 30-5)
30
ogbon
35
aarun din logoji
(= (20 x 2) - 5)
40
logoji
(= 20 x 2)
50
aadota
(= (20 x 3) -10)
60
ogota
(= 20 x 3)
100
ogorun
(= 20 x 5)
400
irinwo
2,000
egbewa
(= (20 x 10) x 10)
4,000
egbaaji
(= 2,000 x 2)
20,000
egbaawaa
(= 2,000 x 10)
40,000
egbaawaa lonan meji
(= (2,000 x 10) x 2)
1,000,000
egbeegberun (literally:
“1,000x1,000”)
The source of this bizarre vigesimal system lies in the Yorubas’
traditional use of cowrie shells as money: the shells are always gathered
in “packets” of 5, 20, 200 and so on.
According to Mann (JAI, 16), Yoruba number-names have two mean-
ings - the number itself, and also the things that the Yoruba count most
of all, namely cowries. “Other objects are always reckoned against
an equivalent number of cowries . . .” he explains. In other words, Yoruba
numbering retains within it the ancient tradition of purely cardinal numer-
ation based on matching sets.
Various other languages around the world retain obvious traces of
a 20-based (vigesimal) number-system. For example, Khmer (spoken in
Cambodia) has some combinations based on an obsolete word for 20,
and, according to F. A. Pott (1847), used to have a special word (slik) for 400
(= 20 x 20). Such features are of course also to be found in European
languages, and nowhere more clearly than in the English word score. “Four
score and seven years ago ...” is the famous opening sentence of Abraham
Lincoln’s Gettysburg Address. Since to score also means to scratch, mark,
or incise (wood, stone or paper), we can see the very ancient origin of
B A S U NUMBERS
its use for the number 20: a score was originally a counting stick “scored”
with twenty notches.
French also has many traces of vigesimal counting. The number 80 is
“four-twenties” ( quatre-vingts ) in modern French, and until the seventeenth
century other multiples of twenty were in regular use. Six-vingts (6 x 20 =
120) can be found in Moliere’s Le Bourgeois Gentilhomme (Act III, scene iv);
the seventeenth-century corps of the sergeants of the city of Paris, who
numbered 220 in all, was known as the Corps des Onze-Vingts (11 x 20), and
the hospital, originally built by Louis XI to house 300 blind veteran
soldiers, was and still is called the Hopital des Quinze-Vingts (15 x 20 = 300).
Danish also has a curious vigesimal feature. The numbers 60 and 80
are expressed as “three times twenty" ( tresindstyve ) and “four times twenty”
( firsindstyve ); 50, 70 and 90, moreover, are halvtresindstyve, halvfirsindstyve,
and halvfemsindstyve, literally “half three times twenty”, “half four times
twenty”, and “half five times twenty”, respectively. The prefix “half” means
that only half of the last of the multiples of 20 should be counted. This
accords with the kind of “prospective account” that we observed in ancient
Turkish numeration (see above, p. 000):
50 = 3 x 20 minus half of the third twenty = 3 x 20 - 10
70 = 4 x 20 minus half of the third twenty = 4 x 20 - 10
90 = 5 x 20 minus half of the third twenty = 5 x 20 - 10
Even clearer evidence of vigesimal reckoning is found in Celtic languages
(Breton, Welsh, Irish). In modern Irish, for example, despite the fact that
100 and 1,000 have their own names by virtue of the decimality that is
common to all Indo-European languages, the tens from 20 to 50 are
expressed as follows:
20 fiche (“twenty”)
30 deich ar fiche (“ten and twenty”)
40 da fiche (“two-twenty”)
50 deich ar da fiche (“ten and two-twenty”)
We can only presume that the Indo-European peoples who settled long
ago in regions stretching from Scandinavia to the north of Spain, including
the British Isles and parts of what is now France, found earlier inhabitants
whose number-system used base 20, which they adopted for the common-
est numbers up to 99, integrating these particular vigesimal expressions
into their own Indo-European decimal system. Since all trace of the
languages of the pre-Indo-European inhabitants of Western Europe has
disappeared, this explanation, though plausible, is only speculation, but it
is supported, if not confirmed, by the use of base 20 in the numbering
system of the Basques, one of the few non-Indo-European languages spoken
38
in Western Europe and whose presence is not accounted for by any
recorded invasion or conquest.
Irish
WELSH
BRETON
1
oin
un
cun
2
da
dau
diou
3
tri
tri
tri
4
celhir
petwar
pevar
5
coic
pimp
pemp
6
se
eh we
chouech
7
scdlt
scilh
seiz
8
ocht
wyth
eiz
9
nbi
naw
nao
10
deich
dec, deg
dek
11
oin deec
1 + 10
un ar dec
1 + 10
unnek
1 + 10
12
da deec
2 + 10
dou ar dec
2 + 10
daou-zek
2 + 10
13
tri deec
3 + 10
tri ar dec
3 + 10
tri-zek
3 + 10
14
cethir deec
4 + 10
petwar ac
dec
4 + 10
pevar-zek
4 + 10
15
coic deec
5 + 10
hymthec
5 + 10
pem-zek
5 + 10
16
se deec
6 + 10
un ar
hymthec
1 + 15
choue-zek
6 + 10
17
seek deec
7 + 10
dou ar
hymthec
2 + 15
seit-zek
7 + 10
18
ocht deec
8 + 10
tri ar
hymthec 1
3 + 15
eiz-zek 2
8 + 10
19
noi deec
9 + 10
pedwar ar
hymthec
4 + 15
naou-zek
9 + 10
20
fiche
20
ugeint
20
ugent
20
30
deich ar
dec ar
fiche
10+20
ugeint
10 + 20
tregont
40
da fiche
2x20
de-ugeint
2x20
daou- ugent
2x20
50
deich ar
dccar
dafichc
10 + (2x20)
de-ugeint
10 + (2 X 20)
hanter-kant
half-100
60
tri fiche
3x20
tri- ugeint
3x20
tri-ugent
3x20
70
dech ar
decar
dek ha
tri fiche
10 + (3 x 20)
tri- ugeint
10 + (3 X 20)
tri-ugent 10 + (3 X 20)
80
ceithri
pedwar-
pevar-
fiche
4x20
ugeint
4x20
ugent
4x20
90
deich ar
dec ar
dek ha
ceithri
pedwar-
pevar-
fiche
10 + (4 X 20)
ugeint
10 + (4 x 20)
ugent 10 + (4 x 20)
100
cet
cant
kant
1.000
mile
mil
mil
Alternatively, deu tiaw (=
2X9)
Alternatively, tri-ouech (-3X6)
Fig. 2.7. Celtic number-names
39
THE COMMONEST BASE IN HISTORY
Basque numbers are as follows:
1
bat
16
hamasei =10 + 6
2
bi, biga, bida
17
hamazazpi =10 + 7
3
hiru, hirur
18
hamazortzi =10 + 8
4
lau, laur
19
hemeretzi =10 + 9
5
host, bortz
20
hogei = 20
6
sei
30
hogeitabat = 10 + 20
7
zazpi
40
berrogei =2 x 20
8
zortzi
50
berrogei-tamar = (2 x 20) + 10
9
bederatzi
60
hirurogei = 3 x 20
10
hamar
70
hirurogei-tamar = (3 x 20) + 10
11
hamaika
irregular
80
laurogei = 4 x 20
12
hamabi =
10 + 2
90
laurogei-tamar = ( 4 x 20) + 10
13
hamahiru =
10 + 3
100
ehun
14
hamalau =
10 + 4
1,000
mila
15
hamabost =
10 + 5
The mystery of Basque remains entire. As can be seen, it is a decimal
system for numbers up to 19, then a vigesimal system for numbers from 20
to 99, and it then reverts to a decimal system for larger numbers. It may be
that, like the Indo-European examples given above (Danish, French, and
Celtic), it was originally a decimal system which was then “contaminated" by
contact with populations using base 20; or, on the contrary, Basque may have
been originally vigesimal, and subsequently “reformed” by contact with
Indo-European decimal systems. The latter seems to be supported by the
obviously Indo-European root of the words for 100 (not unlike “hundred”)
and 1,000 (almost identical to Romance words for “thousand”); but neither
hypothesis about the origins of Basque numbering can be proven.
ASSYRIANS:
Mesopotamia, from the start of the second
millennium BCE to c. 500 BCE
BAMOUNS :
Cameroon
BAOULE :
Ivory Coast
BERBERS :
Fair-skinned people settled in North Africa
since at least Classical times
SHAN :
Indo-China, from second century CE
CHINESE :
from the origins
EGYPTIANS :
from the origins
ELAMITES:
Khuzestan, southwestern Iran, from fourth
century BCE
ETRUSCANS :
probably from Asia Minor, settled in Tuscany
from the late seventh century BCE
GOURMANCHES :
Upper Volta
GREEKS :
from the Homeric period
HEBREWS :
before and after the Exile
HITTITES:
Anatolia, from second millennium BCE
INCAS :
Peru, Ecuador, Bolivia, twelfth to sixteenth
centuries CE
INDIA :
All civilisations of northern and southern
India
INDUS CIVILISATION:
River Indus area, c. 2200 BCE
LYCIANS :
MALAYSIANS
Asia Minor, first half of first millennium BCE
THE COMMONEST BASE IN HISTORY: lO
Base 20, although quite widespread, has never been predominant in the
history of numeration. Base 10, on the other hand, has always been by far
the commonest means of establishing the rule of position. Here is a (non-
exhaustive) alphabetical listing of the languages and peoples who have used
or still use a numbering system built on base 10:
amorites: Northwestern Mesopotamia, founders of Babylon
c. 1900 BCE, and of the first Babylonian dynasty
ARABS: before and after the birth of Islam
Aramaeans: Syria and northern Mesopotamia, second half of
second millennium BCE
Malagasy : Madagascar
MANCHUS
minoans : Crete, second millennium BCE
MONGOLIANS
Nubians : Northeast Africa, since Pharaonic times
PERSIANS
PHOENICIANS
ROMANS
TIBETANS
ugaritic people : Syria, second millennium BCE
Urartians : Armenia, seventh century BCE
BASE NUMBERS
40
In the world today, base 10 is used by a multitude of languages,
including:
Albanian; the Altaic languages (Turkish, Mongolian, Manchu);
Armenian; Bamoun (Cameroon); Baoule (Ivory Coast); Batak; Chinese;
the Dravidian languages (Tamil, Malayalam, Telugu); the Germanic
languages (German, Dutch, Norwegian, Danish, Swedish, Icelandic,
English); Gourmanche (Upper Volta); Greek; Indo-Aryan languages
(Sindhi, Gujurati, Mahratta, Hindi, Punjabi, Bengali, Oriya,
Singhalese); Indonesian; Iranian languages (Persian, Pahlavi, Kurdish,
Afghan); Japanese; Javanese; Korean; Malagasy; Malay; Mon-Khmer
languages (Cambodian, Kha); Nubian (Sudan); Polynesian languages
(Hawaiian, Samoan, Tahitian, Marquesan); the Romance languages
(French, Spanish, Italian, Portuguese, Romanian, Catalan, Provencal,
Dalmatian); Semitic languages (Hebrew, Arabic, Amharic, Berber); the
Slavic languages (Russian, Slovene, Serbo-Croat, Polish, Czech,
Slovak); Thai languages (Laotian, Thai, Vietnamese); Tibeto-Burmese
languages (Tibetan, Burmese, Himalayan dialects); Uralian (Finno-
Ugrian) languages (Finnish, Hungarian).
These lists show, if it needed to be shown, just how successful base 10 has
been and ever remains.
ADVANTAGES AND DRAWBACKS OF BASE lO
The ethnic, geographical, and historical spread of base 10 is enormous, and
we can say that it has become a virtually universal counting system. Is that
because of its inherent practical or mathematical properties? Certainly not!
To be sure, base 10 has a distinct advantage over larger counting units
such as 60, 30, or even 20: its magnitude is easily managed by the human
mind, since the number of distinct names or symbols that it requires is
quite limited, and as a result addition and multiplication tables using base
10 can be learned by rote without too much difficulty. It is far, far harder to
learn the sixty distinct symbols of a base 60 system, even if large numbers
can then be written with far fewer symbols; and the multiplication tables
for even very simple Babylonian arithmetic require considerable feats of
memorisation (sixty tables, each with sixty lines.)
At the other extreme, small bases such as 2 and 3 produce very small
multiplication and addition tables to learn by heart; but they require very
lengthy strings to express even relatively small numbers in speech or
writing, a difficulty that base 10 avoids.
Let us look at a concrete alternative system, an English oral numbering
system using base 2. Initially such a system would have only two number-
names: “one” to express the unit, and “two” (let us call it “twosome”) to
express the base.
1 2
one twosome
It would then acquire special names for each of the powers of the base: let
us say “foursome” for 2 2 , “eightsome” for 2 3 , “sixteensome” for 2 4 , and so
on. Analytical combinations would therefore produce a set of number-
names
; something like this:
1
one
10
eightsome twosome
2
twosome
11
eightsome twosome-one
3
twosome-one
12
eightsome foursome
4
foursome
13
eightsome foursome-one
5
foursome one
14
eightsome foursome twosome
6
foursome twosome
15
eightsome foursome twosome-one
7
foursome twosome-one
16
sixteensome
8
eightsome
17
sixteensome-one
9
eightsome one
and so on.
If our written number-system, using the rule of position, were
constructed on base 2, then we would need only two digits, 0 and 1. The
number two (“twosome”), which constitutes the base of the system, would
be written 10, just like the present base “ten”, but meaning “one twosome
and no units"; three would be written 11 (“one twosome and one unit”),
and so on:
l
would be written
i
2
would be written
10
3
would be written
11
= 1x2 + 1
4
100
= 1x2 2 +0x2 + 0x1
5
101
= 1x2 2 +0x2 + 1x1
6
110
= 1x2 2 +1x2 + 0x1
7
111
= 1x2 2 +1x2 + 1x1
8
1000
= 1x2 3 + Ox2 2 + Ox2 + Ox1
9
1001
= 1x2 3 +0x2 2 + 0x2 + 1x1
10
1010
= 1x2 3 +0x2 2 +1x2 + 0x1
11
1011
= 1x2 3 +0x2 2 +1x2 + 1x1
12
1100
= 1x2 3 +1x2 2 +0x2 + 0x1
13
1101
= 1x2 3 +1x2 2 + 0x2 + 1x1
14
1110
= 1x2 3 +1x2 2 +1x2 + 0x1
15
1111
= 1x2 3 +1x2 2 +1x2 + 1x1
16
10000=
lx 2 4 + 0 x 2 3 + 0 x 2 2 + 0 x 2 + 0
17
10001=
1x2 4 + 0 x 2 3 + 0 x 2 2 + 0 x 2 + 1
Fig. 2 . 8.
41
ADVANTAGES AND DRAWBACKS OF BASE 10
Now, whilst we now require only four digits to express the number two
thousand four hundred and forty-eight (2,448) in a base 10 number system,
a base 2 or binary system (which is in fact the system used by computers)
requires no fewer than twelve digits:
100110010000
(= 1x2 u + Ox2 10 +Ox2 9 + 1 x 2 8 + 1 x2 7 +0 x2 6 + 0 x 2 5 + lx
2 4 + 0x2 3 +0x2 2 + 0x2+0)
Using these kinds of expressions would produce real practical problems in
daily life: cheques would need to be the size of a sheet of A3 paper in order
to be used to pay the deposit on a new house, for example; and it would
take quite a few minutes just to say how much you think a second-hand
Ferrari might be worth.
Nonetheless, there are several other numbers that could serve as base
just as well as 10, and in some senses would be preferable to it.
There is nothing impossible or impracticable about changing the “steps
on the ladder” and counting to a different base. Bases such as 7, 11, 12,
or even 13 would provide orders of magnitude that would be just as
satisfactory as base 10 in terms of the human capacity for memorisation. As
for arithmetical operations, they could be carried out just as well in these
other bases, and in exactly the same way as we do in our present decimal
system. However, we would have to lose our mental habit of giving a special
status to 10 and the powers of 10, since the corresponding names and
symbols would be just as useless in a 12-based system as they would in
one based on 11.
If we were to decide one day on a complete reform of the number-system,
and to entrust the task of designing the new system to a panel of experts,
we would probably see a great battle engaged, as is often the case, between
the “pragmatists” and the “theoreticians". “What we need nowadays is
a system that is mathematically satisfactory,” one of them would assert.
“The best systems are those with a base that has the largest number of
divisors,” the pragmatist would propose. “And of all such bases, 12 seems
to me to be by far the most suitable, given the limits of human memory.
I don’t need to remind you how serviceable base 12 was found to be by
traders in former times - nor that we still have plenty of traces of the
business systems of yore, such as the dozen and the gross (12 x 12), and that
we still count eggs, oysters, screws and suchlike in that way. Base 10 can
only be divided by 2 and 5; but 12 has 2, 3, 4, and 6 as factors, and that’s
precisely why a duodecimal system would be really effective. Just think how
useful it would be to arithmeticians and traders, who would much more
easily be able to compute halves, thirds, quarters, and even sixths of every
quantity or sum. Such fractions are so natural and so common that they
crop up all the time even without our noticing. And that’s not the whole
story! Just think how handy it would be for calculations of time: the
number of months in the year would be equal to the base of the system; a
day would be twice the base in terms of hours; an hour would be five times
the base in minutes; and a minute the same number of seconds. It would be
enormously helpful as well for geometry, since arcs and angles would
be measured in degrees equal to five times the base in minutes, and minutes
would be the same number of seconds. The full circle would be thirty times
the base 12, and a straight line just fifteen times the base. Astronomers too
would find it more than handy . . .’’
“But those are not the most important considerations in our day and
age,” the theoretician would argue. “IVe no historical example to support
what I’m going to propose, but enough time has passed for my ideas to
stand up on their own. The main purpose of a written number system - I’m
sure everyone will agree - is to allow its users to represent all numbers
simply and unambiguously. And I do mean all numbers - integers, frac-
tions, rational and irrational numbers, the whole lot. So what we are
looking for is a numbering system with a base that has no factor other than
itself, in other words, a number system having a prime number as its base.
The only example I’ll give is base 11. This would be much more useful
than base 10 or 12, since under base 11 most fractions are irreducible:
they would therefore have one and only one possible representation in
a system with base 11. For instance: the number which in our present
decimal system is written 0.68 corresponds in fact to several other
fractions - 68/100, 34/50, and 17/25. Admittedly, these expressions all
refer to the same fraction, but there is an ambiguity all the same in repre-
senting it in so many different ways. Such ambiguities would vanish
completely in a system using base 11 or 7 (or indeed, any system with
a prime number as its base), since the irreducibility of fractions would
mean that any number had one and only one representation. Just think of
the mathematical advantages that would flow from such a reform . .
So, since it has only two factors and is not a prime number, base 10
would have no supporters on such a committee of experts!
Base 12 really has had serious supporters, even in recent times. British
readers may recall the rearguard defence of the old currency - 12 pence (d)
to the shilling, 20 shillings to the pound sterling - at the time it was
abandoned in 1971: the benefits of teaching children to multiply and divide
by 2, 3, 4, and 6 (for the smaller-value coins of 3d and 6d) and by 8 (for
the “half-crown”, worth 2s 6d) were vigorously asserted, and many older
people in Britain continue to maintain that youngsters brought up on
decimal coinage no longer “know how to count”. In France, a civil servant
by the name of Essig proposed a duodecimal system for weights and
BASF. NUMBERS
42
measures in 1955, but failed to persuade the nation that first universalised
the metric system to all forms of measurement.
It seems quite unrealistic to imagine that we could turn the clock back
now and modify the base number of both spoken and written number-
systems. The habit of counting in tens and powers of 10 is so deeply
ingrained in our traditions and minds as to be well nigh indestructible. The
best thing to do was to reform the bizarre divisions of older systems of
weights and measures and to replace them with a unified system founded
on the all-powerful base of 10. That is precisely what was done in France in
the Revolutionary period: the Convention (a form of parliament) created
the metric system and imposed it on the nation by the Laws of 18 Germinal
Year III, in the revolutionary calendar (8 April 1795) and 19 Frimaire Year
VIII (19 December 1799).
A BRIEF HISTORY OF THE METRIC SYSTEM
Until the late eighteenth century, European systems of weights and
measures were diverse, complicated, and varied considerably from one area
to another. Standards were fixed with utter whimsicality by local rulers,
and quite arbitrary objects were used to represent lengths, volumes, etc.
From the late seventeenth century onwards, as the experimental sciences
advanced and the general properties of the physical world became better
understood, scholars strove to devise stable and coherent measuring
systems based on permanent, universal and unmodifiable standards. The
growth of trade throughout the eighteenth century also created a need for
common measurements at least within each country, and a uniform system
of weights and measures. Thus the metric system emerged towards the
end of the eighteenth century. It is a fully consistent and coherent measure-
ment system using base 10 (and therefore fully compatible with the
place-value system of written numbering that the Arabs had brought to
Europe in the Middle Ages, having themselves learned it from the Indians),
which the French Revolution offered “to all ages and to all peoples, for their
greater benefit”. It produced astounding progress in applied areas, since it
is perfectly adapted to numerical calculation and is extremely simple to
operate in fields of every kind.
Around 1660: In order to harmonise measurement of time and length
and also so as to compare the various standards used for measuring
length around the world, the Royal Society of London proposed to establish
as the unit of length the length of a pendulum that beats once per second.
The idea was taken up by Abbe Jean Picard in La Mesure de la Terre (“The
Measurement of the Earth”) in 1671, by Christian Huygens in 1673, and by
La Condamine in France, John Miller in England, and Jefferson in America.
1670: Abbe Gabriel Mouton suggested using the sexagesimal minute
of the meridian (= 1/1000 of the nautical mile) as the unit of length. But
this unit, of roughly 1.85 metres, was too long to be any practical use.
1672: Richer discovered that the length of a pendulum that beats once
per second is less at Cayenne (near the Equator) than in Paris. The conse-
quence of this discovery was that, because of the variation in length of
the pendulum caused by the variation in gravity at different points on the
globe, the choice of the location of the standard pendulum would be polit-
ically very tricky. As a result the idea of using the one-second pendulum as
a unit of length was eventually abandoned.
1758: In Observations sur les principes metaphysiques de la geomctrie
(“Observations on the Metaphysical Principles of Geometry”), Louis Dupuy
suggested unifying measurements of length and weight by fixing the unit of
weight as that of a volume of water defined by units of length.
1790: 8 May: Talleyrand proposed, and the Assemblee constituante (Con-
stituent Assembly) approved the creation of a stable, simple and uniform
system of weights and measures. The task of defining the system was
entrusted to a committee of the Academy of Sciences, with a membership
consisting of Lagrange, Laplace, and Monge (astronomical and calendrical
measurements), Borda (physical and navigational measurement), and
Lavoisier (chemistry). The base unit initially chosen was the length of the
pendulum beating once per second.
1791: 26 March: The committee decided to abandon the pendulum as
the base unit and persuaded the Constituent Assembly to choose as the unit
of length the ten-millionth part of one quarter of the earth’s meridian,
which can be measured exactly as a fraction of the distance from the pole to
the Equator. At Borda’s suggestion, this unit would be called the metre
(Greek for “measure”).
What the committee then had to do was to produce conventional
equivalencies between the various units chosen so that all of them (except
units of time) could be derived from the metre. So, for measuring surface
area, the unit chosen was the are, a square with a side of 10 metres; for
measuring weight, the kilogram was defined as the weight of a unit of
volume (1 litre) of pure water at the temperature of melting ice, corrected
for the effects of latitude and air pressure. All that now had to be done to
set up the entire metric system was to make the key measurement, the
distance from the pole to the Equator - a measurement that was all
the more interesting at that time as Isaac Newton had speculated that the
globe was an ellipsoid with flattened ends (contradicting Descartes, who
believed it was a sphere with elongated or pointed ends).
1792: The “meridian expedition" began. A line was drawn from Dunkirk
to Barcelona and measured out by triangulation points located thanks to
43
A BRIEF HISTORY OF THE METRIC SYSTEM
Borda’s goniometer, with some base stretches measured out with greater
precision on the ground. Under the direction of Mechain and Delambre,
one team was in charge of triangulation, one was responsible for the
standard length in platinum, and one for drafting the users’ manuals of
the new system. Physicists such as Coulomb, Hauy, Hassenfrantz, and
Borda, and the mathematicians Monge, Lagrange, and Laplace were
amongst the many scientists who collaborated on this project which was
not fully completed until 1799.
1793: 1 August: The French government promulgated a decree requiring
all measures of money, length, area, volume, and weight to be expressed in
decimal terms : all the units of measure would henceforth be hierarchised
according to the powers of 10. As it overturned all the measures in current
use (most of them using base 12), the decimalisation decree required new
words to be invented, but also created the opportunity for much greater
coherence and accuracy in counting and calculation.
1795: 7 April: Law of 18 Germinal, Year III, which organised the metric
system, gave the first definition of the metre as a fraction of the terrestrial
meridian, and fixed the present nomenclature of the units (decimetre,
centimetre, millimetre; are, deciare, centiare, hectare; gram, decigram,
centigram, kilogram; franc, centime; etc.)
1795: 9 June: Lenoir fabricated the first legal metric standard, on the
basis of the calculation made by La Caille of the distance between the pole
and the Equator at 5,129,070 toises de Paris (in 1799, Delambre and
Mechain obtained a different, but actually less accurate figure of 5,130,740
toises de Paris).
1795: 25 June: Establishment of the Bureau des Longitudes (Longitude
Office) in Paris.
1799: First meeting in Paris of an international conference to discuss
universal adoption of the metric system. The system was considered “too
revolutionary” to persuade other nations to “think metric” at that time.
1799: 22 June: The definitive standard metre and kilogram, made of
platinum, were deposited in the French National Archives.
1799: 10 December: Law of 10 Frimaire, Year VIII, which confirmed
the legal status of the definitive standards, gave the second definition of
the metre (the length of the platinum standard in the National Archives,
namely 3 feet and 11.296 “lines” of the toise de Paris), and in theory made
the use of the metric system obligatory. (In fact, old habits of using pre-
metric units of measurement persisted for many years and were tolerated.)
1840: 1 January: With the growing spread of primary education in
France, the law was amended to make the use of the metric system
genuinely obligatory on all.
1875: Establishment of the International Bureau of Weights and
Measures at Sevres (near Paris). The Bureau created the new international
standard metre, made of iridoplatinum.
1876: 22 April: The new international standard metre was deposited in
the Pavilion de Breteuil, at Sevres, which was then ceded by the nation to
the International Weights and Measures Committee and granted the status
of “international territory”.
1899: The General Conference on Weights and Measures met and
provided the third definition of the metre. The length of the meridian was
abandoned as a basis of calculation. Henceforth, the metre was defined as
the distance at 0°C of the axis of the three median lines scored on the inter-
national standard iridoplatinum metre.
1950s: The invention of the laser allowed significant advances in optics,
atomic physics, and measurement sciences. Moreover, quartz and atomic
clocks resulted in the discovery of variations in the length of the day, and
put an end to the definition of units of time in terms of the earth’s rotation
on its axis.
1960: 14 October: Fourth definition of the metre as an optical standard
(one hundred times more accurate than the metre of 1899): the metre now
becomes equal to 1,650,763.73 wave-lengths of orange radiation in
a void of krypton 86 (krypton 86 being one of the isotopes of natural
krypton).
1983: 20 October: The XVIIth General Conference on Weights and
Measures gives the fifth definition of the metre, based on the speed of light
in space (299,792,458 metres per second): a metre is henceforth the
distance travelled by light in space in 1/299,792,458 of a second. As for
the second, it is defined as the duration of 9,192,631,770 periods of radia-
tion corresponding to the transition between the two superfine levels of the
fundamental state of an atom of caesium 133. At the same conference,
definitions of the five other basic units (kilogram, amp, kelvin, mole, and
candela) were also adopted, as well as the standards that constitute the
current International Standards system (IS).*
THE ORIGIN OF BASE 10
Well, then: where does base 10 come from?
In the second century CE, Nicomachus of Gerasa, a neo-Pythagorean from
Judaea, wrote an Arithmetical Introduction which, in its many translations,
influenced Western mathematical thinking throughout the Middle Ages.
For Nicomachus, the number 10 was a “perfect” number, the number of the
divinity, who used it in his creation, notably for human toes and fingers,
* For information contained in this section on the metric system I am indebted to Jean Dhombres, President
of the French Association for the History of Science.
HASH NUMB K It S
44
and inspired all peoples to base their counting systems on it. For many
centuries, indeed, numbers were thought to have mystical properties; in
Pythagorean thinking, 10 was held to be “the first-born of the numbers, the
mother of them all, the one that never wavers and gives the key to all
things".
Such attitudes to numbers, which had their place in a world-view which
was itself mystical through and through, now seem as circular and self-
defeating as the observation that God had the wisdom to cause rivers to
flow through the middle of towns.
In fact, the almost universal preference for base 10 comes from nothing
more obscure than the fact that we learn to count on our fingers, and that
we happen to have ten of them. We would use base 10 even if we had no
language, or were bound to a vow of total silence: for just like the North
African shepherd and his shells and straps discussed on p. 24-25 above, we
could use our raised fingers to count out the first ten in silence, a colleague
could then raise one finger to keep count of the tens, and so on to 99, when
(for numbers of 100 and more) the fingers of a third colleague would be
needed. Fig. 2.9 shows the position of the three silent colleagues’ hands at
number 627.
Helper No. 3
Helper No. 2
Helper No. 1
Left
Right
Left Right
Left Right
to
%
\
'I
w.t J
600
20
7
Fig. 2.9.
The obvious practicality of such a non-linguistic counting system using
only our own bodies shows that the idea of grouping numbers into packets of
ten and powers of ten is based on the “accident of nature” that is the physiol-
ogy of the human hand. Since that physiology is universal, base 10 necessarily
occupies a dominant, not to say inexpugnable position in counting systems.
If nature had given us six fingers, then the majority of counting systems
would have used base 12. If on the other hand evolution had brought us
down to four fingers on each hand (as it has for the frog), then we would
doubtless have long-standing habits and traditions of counting on base 8.
THE ORIGINS OF THE OTHER BASES
The reason for the adoption of vigesimal (base 20) systems in some cultures
can be seen by the basic idea of Aztec numbering as laid out in Fig. 2.6
above. In the language of the Aztecs
• the names of the first five numbers can be associated with the
fingers of one hand;
• the following five numbers can be associated with the fingers of the
other hand;
• the next five numbers can be associated with the toes of one foot;
• and the last five numbers can be associated with the toes on the
other foot.
And so 20 is reached with the last toe of the second foot (see Fig. 2.10).
This is no coincidence. It is simply that some communities, because they
realised that by leaning forward a little they could count toes as well as
fingers, ended up using base 20.
One remarkable fact is that both the Inuit (Greenland) and the Tamanas
(in the Orinoco basin) used the same expression for the number 53, literally
meaning: “of the third man, three on the first foot".
According to C. Zaslavsky (1973), the Banda people in Central Africa
express the number 20 by saying something like “a hanged man”: presum-
ably because when you hang a man you can see straight away all his fingers
and toes. In some Mayan dialects, the expression hun uinic, which means 20,
also means “one man". The Malinke (Senegal) express 20 and 40 by saying
respectively “a whole man” and “a bed” - in other words, two bodies in a bed!
In the light of all this there can be no doubt at all that the origin of
vigesimal systems lies in the habit of counting on ten fingers and ten
toes . . .
The origin of base 5 is similarly anthropomorphic. Quinary reckoning
is founded on learning to count using the fingers of one hand only.
The following finger-counting technique, which is found in various parts
of Oceania and is also currently used by many Bombay traders for various
specific purposes, is a good example of how a primitive one-hand counting
system can give rise to more elaborate numbering. You use the five fingers
of the left hand to count the first five units. Then, once this number is
reached, you extend the thumb of the right hand, and go on counting to 10
with the fingers of the left hand; then you extend the index finger of the
right hand and count again on the left hand from 11 to 15; and so on, up to
25. The series can be extended to 30 since the fingers of the left hand are
usable six times over in all.
However, this obviously fails to resolve the basic mystery: why did base
5 - which must be considered the most natural base by far, since it is
45
THE ORIGINS OF OTHER BASES
Fig. 2.11.
virtually dictated by the basic features of the human body and must be
self-evident from the very moment of learning to count - why did base 5
not become adopted as the universal human counting tool? Why, in other
words, was the apparently inevitable construction of quinary counting
generally avoided? Why did so many cultures go up to 10, to 20 or, in the
case of the Sumerians, whom we will discuss again, as far as 60? Even
more mysterious are those cultures which possessed a concept of number
and knew how to count, but went back down to 4 for their numerical base.
L. L. Conant (1923) tackled the whole problem in detail without claim-
ing to have found the final answer. The anthropologist Levy-Bruhl, on the
other hand, thought it was a false problem. In his view, we should not
suppose that people ever invented number-systems in order to carry out
arithmetical operations or devised systems that were intended to be best
suited to operations that, prior to the devising of the system, could not be
imagined. "Numbering systems, like languages, from which they can hardly
be distinguished, are in the first place social phenomena, closely dependent
on collective mentalities,” he claimed, “The mentality of any society is
completely bound up with its internal functioning and its institutions.”
To conclude this chapter we shall return with Levy-Bruhl to very
BASF NUMBFRS
46
primitive counting systems which do not yet clearly distinguish between
the cardinal and ordinal aspects of number. In the kind of “body-counting"
explained above and demonstrated in Fig. 1.30, 1.31 and 1.32, there are
no “privileged” points or numbers, and therefore no concept of a base
at all. Using Petitot’s dictionary of the language of Dene-Dindjie Indians
(Canada), Levy-Bruhl explains how things are counted out in a system
with no base:
You hold out your left hand (always the left hand) with the palm
turned towards your face, and bend your little finger, saying
for 1 : the end is bent
or on the end
Then you bend your ring finger and say:
for 2: it's bent again
Then you bend your middle finger and say
for 3: the middle one is bent
Then you bend your index finger, leaving the thumb stretched out,
and say:
for 4: there’s only that left
Then you open out your whole hand and say:
for 5: it’s OK on my hand
or on a hand
or my hand
Then you fold back three fingers together on your left hand, keeping
the thumb and index stretched out, and touch the left thumb with the
right thumb, saying:
for 6: there’s three on each side
or: three by three
Then you bend down four fingers on your left hand and touch your left
thumb (still stretched out) with the thumb and index finger of your
right hand, and say:
for 7: on one side there are four
or there are still three bent
or three on each side and one in the middle
Then you stretch out three fingers of your right hand and touch the
outstretched thumb of your left hand, creating two groups of four
fingers (bent and extended), and say:
for 8: four on four
or four on each side
Then you show the little finger of your right hand, the only one now
bent, and say:
for 9: there's still one down
or one still short
or the little finger's lying low
Then you start the gestures over again, saying “one full plus one", or “one
counted plus one”, “one counted plus two”, “one counted plus three”, and so
on.
Levy-Bruhl argues that in this system, which does not prevent the
Dene-Dindjie from counting properly, there is no concept of a quinary base:
6 is not “a second one”, 7 not “a new two”, as we find in so many other
numbering systems. On the contrary, he says, 6 here is “three and three” -
which shows that finishing the count on one hand is in no way a “marker”
or a “privileged number” in this system. The periodicity of numbers is not
derived from the physical manner of counting, does not come from the
series of movements made to indicate the sequence of the numbers.
In this view, numbering systems relate much more directly to the
“mental world” of the culture or civilisation, which may be mythical rather
than practical, attributing more significance to the four cardinal points of
the compass, or to the four legs of an animal, than to the five fingers
of the hand. We do not have to try and guess why this base rather than
another was “chosen” by a given people for their numbering system, even if
they do effectively use the five fingers of their hand for counting things
out. Where a numbering system has a base, the base was never “chosen”,
Levy-Bruhl asserts. It is a mistake to think of “the human mind” construct-
ing a number system in order to count: on the contrary, people began to
count, slowly and with great difficulty, long before they acquired the
concept of number.
However, it is clear that the adoption of base 5 is related to the way we
count on the fingers of our hands. But why did those cultures that adopted
base 5 not extend it, like so many others, to the base 10 that corresponds
to the fingers of both hands? Dantzig has speculated that it may have to do
with the conditions of life in warrior societies, in which men rarely go about
unarmed. If they want to count, they tuck their weapon under their left arm
and count on the left hand, using the right hand as a check-off. The right
hand remains free to seize the weapon if needed. This may explain why the
left hand is almost universally used by right-handed people for counting,
and vice versa [T. Dantzig (1930), p. 13].
However this may be, base numbers arise for many reasons, many of
which have nothing at all to do with their suitability for counting or for
arithmetical operations; and they may indeed have arisen long before any
kind of abstract arithmetic was invented.
47
EARLY WAYS OF COUNTING ON FINGERS
CHAPTER 3
THE EARLIEST CALCULATING
MACHINE - THE HAND
That uniquely flexible and useful tool, the human hand, has also been the
tool most widely used at all times as an aid to counting and calculation.
Greek writers from Aristophanes to Plutarch mention it, and Cicero tells
us that its use was as common in Rome: tuns digitns novi - “1 well know your
skill at calculating on your fingers” ( Epistutae ad Atticum, V, 21, 13); Seneca
says much the same: “Greed was my teacher of arithmetic: I learned to make
my fingers the servants of my desires” ( Epistles , LXXXVII); and, later,
Tertullian said: “Meanwhile, I have to sit surrounded by piles of papers,
bending and unbending my fingers to keep track of numbers.” The famous
orator Quintilian stressed the importance of calculating on the fingers,
especially in the context of pleading at law: "Skill with numbers is needed not
only by the Orator, but also by the pleader at the Bar. An Advocate who
stumbles over a multiplication, or who merely exhibits hesitation or clumsi-
ness in calculating on his fingers, immediately gives a bad impression of his
abilities;” and the digital techniques he referred to, which were in common
use by the inhabitants of Rome, required very considerable dexterity (see
Fig. 3.13). Pliny the Elder, in his Natural History (XVI), described how King
Numa offered up to the God Janus (the god of the Year, of Age, and of Time)
a statue whose fingers displayed the number of days in a year. Such practices
were by no means confined to the Greeks and Romans. Archaeologists,
historians, ethnologists, and philologists have come upon them at all times
and in all regions of the world, in Polynesia, Oceania, Africa, Europe,
Ancient Mesopotamia, Egypt under the Pharaohs, the Islamic world, China,
India, the Americas before Columbus, and the Western world in the Middle
Ages. We can conclude, therefore, that the human hand is the original
“calculating machine”. In the following we shall show how, once people had
grasped the principle of the base, over the ages they developed the arith-
metical potential of their fingers to an amazing degree. Indeed, certain
details of this are evidence of contacts and influences between different
peoples, which could never have been inferred in any other way.
EARLY WAYS OF COUNTING ON FINGERS
The simplest method of counting on the fingers consists of associating an
integer with each finger, in a natural order. This may be done in many ways.
One may start with the fingers all bent closed, and count by successively
straightening them; or with the fingers open, and successively close them.
One may count from the left thumb along the hands to the right little
finger, or from the little finger of the left hand through to the thumb of
the right, or from the index finger to the little finger and finally the thumb
(see Fig. 3.3). The last method was especially used in North Africa. It seems
likely that at the time of Mohammed the Arabs used this method. One
of the Hadiths tells how the Prophet showed his disciples that a month
could have 29 days, showing “his open hands three times, but with one
finger bent the third time”. Also, a Muslim believer always raises the
index finger when asserting the unity of Allah and expressing his faith in
Islam, in performing the prayer of Shahadah (“witness”).
Fig. 3.1. Finger-counting among the Aztecs (Pre-Columbian Mexico). Detail of a mural by Diego
Rivera. National Museum of Mexico
THK EARLIEST CALCULATING MACHINE
T HE HAND
48
Fig. 3 . 2 . Boethius (480-524 CE), the philosopher and mathematician, counting on his fingers.
From a painting by Justus of Ghent (15th C.). See P. Dedron and J. hard (1959).
A STRANGE WAY OF BARGAINING
There is a similar method, of very ancient origin, which persisted late in
the East and was common in Asia in the first half of the twentieth century. It
is a special way of finger-counting used by oriental dealers and their clients
in negotiating their terms. Their very curious procedure was described by
the celebrated Danish traveller Carsten Niebuhr in the eighteenth century,
as follows:
I have somewhere read, I think, that the Orientals have a special way of
settling a deal in the presence of onlookers, which ensures that none
of these becomes aware of the agreed price, and they still regularly
make use of it. I dreaded having someone buy something on my behalf
in this way, for it allows the agent to deceive the person for whom he
< CQ U Q
is acting, even when he is watching. The two parties indicate what
price is asked, and what they are willing to pay, by touching fingers
or knuckles. In doing so, they conceal the hand in a corner of their
dress, not in order to conceal the mystery of their art, but simply in
order to hide their dealings from onlookers. . . . ( Beschreibung von
Arabien, 1772)
To indicate the number 1, one of the negotiators takes hold of the index
finger of the other; to indicate 2, 3 or 4, he takes hold of index and middle
fingers, index, middle and fourth, or all four fingers. To indicate 5, he grasps
the whole hand. For 6, he twice grasps the fingers for 3 (2 x 3), for 7, the
fingers for 4 then the fingers for 3 (4 + 3), for 8, he twice grasps the fingers for
49
COUNTING ALONG THE FINGERS
4 (2 x 4), and for 9, he grasps the whole hand, and then the fingers for 4
(5 + 4). For 10, 100, 1,000 or 10,000 he again takes hold of the index finger (as
for 1); for 20, 200, 2,000 or 20,000 the index and middle fingers (as for 2),
and so on (see Fig. 3.4). This does not, in fact, lead to confusion because the
two negotiators will have agreed beforehand on roughly what the price will
be (whether about 40 dinars, or 400, for example). Niebuhr does not tell that
he himself saw such a deal take place, but J. G. Lemoine found traces of
the method in Bahrain, a place famous for its pearl fishery, when he made
a study of this topic at the beginning of this century. He gathered infor-
mation from pearl dealers in Paris, who had often visited Bahrain and had
occasion to employ this procedure in dealing with the Bahrainis. He states:
The two dealers, seated face to face, bring their right hands together
and, with the left hand, hold a cloth over them so that their right hands
are concealed. The negotiation, with all its “discussions”, takes place
without a word being spoken, and their faces remain totally impassive.
Those who have observed this find it extremely interesting, for the
slightest visible sign could be taken to the disadvantage of one or other
of the dealers.
Similar methods of negotiation have been reported from the borders of
the Red Sea, from Syria, Iraq and Arabia, from India and Bengal, China and
Mongolia, and - from the opposite end of the world - Algeria. P. J. Dols
(1918), reporting on “Chinese life in Gansu province", describes how
dealings were still being conducted in China and Mongolia in the early
twentieth century.
The buyer puts his hands into the sleeves of the seller. While talking,
he takes hold of the seller’s index finger, thereby indicating that he is
offering 10, 100 or 1,000 francs. “No!” says the other. The buyer then
takes the index and middle fingers together. “Done!” says the seller.
The deal has been struck, and the object is sold for 20, or for 200,
francs. Three fingers together means 30 (or 300 or 3,000), four fingers
40 (or 400 or 4,000). When the buyer takes the whole hand of the
seller, it is 50 (or 500 or 5,000). Thumb and little finger signify 60 (note
the difference from the Middle Eastern system described above).
Placing the thumb in the vendor’s palm means 70, thumb and index
together 80. When the buyer, using his thumb and index finger
together, touches the index finger of the seller, this indicates 90.
COUNTING ALONG THE FINGERS
There is more to fingers than a single digit: they have a knuckle, two
joints, and three bones (but one joint and two bones for the thumbs).
Amongst many Asiatic peoples, this more detailed anatomy has been
1
.10
too
1,000
2
20
200
2,000
3
30
300
3,000
4
40
400
4,000
Fig. 3.4. Method of counting on the fingers, once used in
bargaining between oriental dealers
exploited for counting. In southern China, Indo-China and India, for
example, people have counted one for each joint, including the knuckle,
working from base to tip of finger (or in reverse) and from little finger to
thumb, pointing with a finger of the other hand. Thus each hand can count
up to 14, and both hands up to 28 (see Fig. 3.5). A Chinaman from Canton
once told me of a singular application of this method. Since a woman’s
monthly cycle lasts 28 days, his mother used to tie a thread around each joint
as above for each day of her cycle, to detect early or late menstruation. The
Venerable Bede (673-735 CE, a monk in the Monastery of Saints Peter and
Paul at Wearmouth and Jarrow and author of the influential De ratione
temporum, “Of the Division of Time”), applied similar counting methods for
his calculations of time. To count the twenty-eight years of the solar cycle,
beginning with a leap year, he started from the tip of the little finger and
counted across the four fingers, winding back and forth and working down
to the base of the fingers to count up to 12, then moving to the other hand to
count up to 24, finally using the two thumbs to count up to 28 (see Fig. 3.6).
THE EARLIEST CALCULATING MACHINE
THE HAND
50
Fig. 3 . 5 . The method used in China, Indo-
China, and India, using the fourteen
finger-joints of each hand
Fig. 3 . 6 . The Venerable Bede ’s method of
counting the twenty-eight years of the solar cycle
on the knuckles (7th C.). Leap years are marked
with asterisks
Fig. 3 . 7 . Bede’s method of counting the nine-
teen years of the lunar cycle, using the knuckles
and fingernails of the left hand
Fig. 3 . 8 . The Indian and Bengali method,
using the knuckles of fingers and thumb, and the
ball of the thumb
He used a similar method to count the nineteen years of the lunar cycle,
counting up to 14 on the knuckles of the left hand, going on up to 19 by
also counting the fingernails (see Fig. 3.7). The objective was to determine
the date of Easter, the subject of complicated controversies in the early
Church. In particular there was dispute between the British and Irish
Churches, on the one hand, and the Roman Church on the other, regarding
which lunar cycle to adopt for the date of Easter. Bede’s calculations
brought together the solar year and the solar and lunar cycles of the Julian
calendar, and its leap years.
A different method of counting on the knuckles was long used in
northeast India, and is still found in the regions of Calcutta and Dacca.
It was reported by seventeenth- and eighteenth-century travellers, espe-
cially the Frenchman J. B. Tavernier (1712). According to N. Halhed (1778),
the Bengalis counted along the knuckles from base to tip, starting with the
little finger and ending with the thumb, using the ball of the thumb as well
and thus counting up to 15 (Fig. 3.8).
This method of counting on the knuckles has given rise to the practice,
common among Indian traders, of fixing a price by offering the hand under
cover of a cloth; they then touch knuckles to raise or lower their proposed
prices (Halhed).
There are 15 days in the Hindu month, the same number as can be
counted on a hand; and, according to Lemoine, this is no coincidence. The
Hindu year (360 days) consists of 12 seasons ( Nitus ) each of two “months”
( Masas ). One month of 15 days corresponds to one phase of the moon, and
the following month to another phase. The first, waxing, phase is called
Rahu, and the second, waning, phase is called Ketu. In this connection we
may refer to the legend which tells how, before the raising up of the oceans,
these two “faces” of the moon formed a single being, subsequently cut in
two by Mohini (Vishnu). Such a system for counting on the hands is also
found throughout the Islamic world, but mainly for religious purposes in
this case. Muslims use it when reciting the 99 incomparable attributes of
Allah or for counting in the litany of subha (which consists of the 33-fold
repetition of each of the three “formulas”), which is recited following the
obligatory prayer. To do either of these conveniently, a count of 33 must be
achieved. This is done by counting the knuckles, from base to tip, of each
finger and the thumb (including the ball of the thumb), first on the left
hand and then on the right. In this way a count of 30 is attained, which is
brought up to 33 by further counting on the tips of the little, ring, and
middle fingers of the right hand.
Fig. 3 . 9 . The Muslim method of counting up to 33, used for reciting the 99 (= 3 times 33) attributes
of Allah, and for the 33-fold repetitions of the subha
Nowadays, Muslims commonly adopt a rosary of prayer-beads for this
purpose, but the method just described may still be adopted if the beads are
not to hand. However, the hand-counting method is extremely ancient and
undoubtedly pre-dates the use of the beads. Indeed, it finds mention in the
oral tradition, in which the Prophet is described as admonishing women
51
THE GAME OF MORRA
believers against the use of pearls or pebbles, and encouraging them to use
their fingers to count the Praises of Allah. I. Godziher (1890) finds in this
tradition some disapproval, by the Islamic authorities, of the use of the
rosary subsequent to its emergence in the ninth century CE, which
persisted until the fifteenth century CE. Abu Dawud al Tirmidhi tells it as
follows: “The Prophet of Allah has said to us, the women of Medina: Recite
the tasbih, the tahlil and the taqdis] and count these Praises on your
fingers, for your fingers are for counting.” This parallelism between Far-
Eastern commercial practices and the ancient and widespread customs of
Islamic religious tradition is extremely interesting.
THE GAME OF MORRA
For light relief, let us consider the game of Morra, a simple, ancient and
well-known game usually played between two players. It grew out of finger-
counting. The two players stand face to face, each holding out a closed fist.
Simultaneously, the two players open their fists; each extends as many
fingers as he chooses, and at the same time calls out a number from 1 to 10.
If the number called by a player equals the sum of the numbers of fingers
shown by both players, then that player wins a point. (The players may also
use both hands, in which case the call would be between 1 and 20.) The
game depends not only on chance, but also on the quickness, concentra-
tion, judgment and anticipation of the players. Because the game is so well
defined, and also of apparently ancient origin, it is very interesting for our
purpose to follow its traces back into history, and across various peoples;
and we shall come upon many signs of contacts and influences which will
be important for us. It is still popular in Italy (where it is called morra), and
is also played in southeast France (la mourn), in the Basque region of Spain,
in Portugal, in Morocco and perhaps elsewhere in North Africa. As a child
1 played a form of it myself in Marrakesh, with friends, as a way of choosing
“ft”. We would stand face to face, hands behind our backs. One of the two
would bring forward a hand with a number of fingers extended. The other
would call out a number from 1 to 5, and if this was the same as the number
of fingers then he was “It”; otherwise the first player was “It”. In China and
Mongolia the game is called hua quart (approximately, “fist quarrel”).
According to Joseph Needham, it is a popular entertainment in good
circles. P. Perny (1873) says: “If the guests know each other well, their host
will propose qing hua quart (‘let us have a fist quarrel’). One of the guests is
appointed umpire. For reasons of politeness, the host and one of the guests
commence, but the host will soon give way to someone else. The one who
loses pays the ‘forfeit’ of having to drink a cup of tea.” The game of Morra
was very popular, in Renaissance times, in France and Italy, amongst valets,
pages and other servants to while away their idle hours. “The pages would
play Morra at a flick of the fingers” (Rabelais, Pantagruel Book IV, Ch. 4);
“Sauntering along the path like the servants sent to get wine, wasting
their time playing at Morra” (Malherbe, Lettres Vol. II p. 10). Fifteen
hundred years earlier, the Roman plebs took great delight in the game,
which they called micatio (Fig. 3.10). Cicero’s phrase for a man one could
trust was: “You could play micatio with him in the dark.” He says it was a
common turn of phrase, which indicated the prevalence of the game in the
popular culture.
The game also served in the settling of disputes, legal or mercantile,
when no other means prevailed, much as in “drawing the short straw”, and
was even forbidden by law in public markets (G. Lafaye, 1890). Vases and
other Ancient Greek relics depict the game (Fig. 3.11). According to legend
it was Helen who invented the game, to amuse her lover Paris.
Much earlier, the Egyptians had a similar game, as shown in the two
funerary paintings reproduced in Fig. 3.12. The top is from a tomb at Beni
Fig. 3.11. The game of Mona in Greek times. (Left) Painted vase in the Lambert Collection, Paris.
(Right) Painted vase, Munich Museum. (DAGR, pp. 1889-90)
IHK F.AR1. IK ST CALCULATING MAC II INK
T HH HAND
52
Fig. 3.12. Two Egyptian funeral paintings showing the game o/Morra. (Above) Tomb no. 9 of
Beni-Hassan (Middle Kingdom). See Newberry. ASF. vol. 2 (1893). plate 7. (Below) Theban tomb
no. 36 (Aba's tomb. XXVIth dynasty). See Wilkinson (1837), vol. 2. p. 55 (Fig. 307). See also photo
no. 9037 by Schott at the Gottingen Institute of Egyptology.
Hassan dating from the Middle Kingdom (21st- 17th centuries BCE), and
it shows two scenes. In the first scene, one man holds his hand towards
the eyes of the other, hiding the fingers with his other hand. The other
man holds his closed fist towards the first. The lower scene depicts similar
gestures, but directed towards the hand. According to J. Yoyotte (in
G. Posener, 1970), the hieroglyphic inscriptions on these paintings mean:
Left legend : Holding the Ip towards the forehead; Right legend: Holding the
Ip towards the hand. The Egyptian word Ip means “count" or “calculate”, so
these paintings must refer to a game like Morra. The lower painting, from
Thebes, is from the time of King Psammetichus I (seventh century BCE)
and was (according to Leclant) copied from an original from the Middle
Kingdom. This too shows two pairs of men, showing each other various
combinations of open and closed fingers.
We may therefore conclude that the game of Morra, in one form or
another, goes back at least to the Middle Kingdom of Pharaonic Egypt. In
the world of Islam, Morra is called mukharaja (“making it stick out”). At
the start of the present century it was played in its classical form in remote
areas of Arabia, Syria and Iraq. Mukharaja was above all, however, a divina-
tion ritual amongs the Muslims and was therefore forbidden to the faithful
(fortune-telling is proscribed by both Bible and Koran); so it was a much
more serious matter than a mere game. An Arabian fortune-telling manual
shows circular maps of the universe (Za ’ irjat al alam), divided into sectors
corresponding to the stars, where each star has a number. There are also
columns of numbers which give possible “answers” to questions which
might be asked. The mukharaja was then used to establish a relationship
between the two sets of numbers.
COUNTING AND SIGN-LANGUAGE
There is a much more elaborate way of counting with the hand which, from
ancient times until the present day, has been used by the Latins and can
also be found in the Middle East where, apparently, it may go back even
further in time. It is rather like the sign language used by the deaf and
dumb. Using one or both hands at need, counting up to 9,999 is possible by
this method. From two different descriptions we can reconstruct it in its
entirety. These are given in parallel to each other in Fig. 3.13.
The first was written in Latin in the seventh century by the English
monk Bede (“The Venerable”) in his De ratione temporum, in the chapter
De computo vel loquela digilorum (“Counting and talking with the fingers”).
The other is to be found in the sixteenth-century Persian dictionary Farhangi
Djihangiri. There is a most striking coincidence between these two descrip-
tions written nine centuries apart and in such widely separated places.
With one hand (the left in the West and the right in the East), the
little finger, fourth and middle fingers represented units, and either
the thumb or the index finger (or both) was used for tens. With the other
hand, hundreds and thousands were represented in the same way as
the units and tens.
Both accounts also describe how to show numbers from 10,000 upwards.
In the Eastern description: “for 10,000 bring the whole top joint of the
thumb in contact with the top joint of the index finger and part of its
second joint, so that the thumbnail is beside the nail of the index finger and
the tip of the thumb is beside the tip of the index finger.” For his part, Bede
says: “For 10,000 place your left hand, palm outwards, on your breast, with
the fingers extended backwards and towards your neck.” Therefore the two
descriptions diverge at this point.
Let us however follow Bede a little further.
For 20,000 spread your left hand wide over your breast. For 30,000 the
left hand should be placed towards the right and palm downwards,
with the thumb towards the breastbone. For 50,000 similarly place the
left hand at the navel. For 60,000 bring your left hand to your left
thigh, inclining it downwards. For 70,000 bring your left hand to the
53
COUNTING AND SIGN- LANGUAGE
WESTERN DESCRIPTION
From the Latin of the Venerable Bede,
seventh century
EASTERN DESCRIPTION
From a sixteenth-century Persian dictionary
A. U
NITS
\k
w
3
When you say “one”, bend
your left little finger so as
to touch the central fold of
your palm
For “two”, bend your next
finger to touch the same spot
When you say “three", bend
your third finger in the same
way
When you say “four”, raise
up your little finger from
its place
Saying “five”, raise your
second finger in the same
way
When you say “six”, you also
raise your third finger, but
you must keep your ring
finger in the middle of
your palm
For 1, bend down your little
finger
For 2, your ring finger must
join your little finger
For 3, bring your middle
finger to join the other two
For the number 5, also raise
your ring finger
For the number 6, raise the
middle finger, keeping your
ring finger down (so that
its tip is in the centre
of the palm)
For 4, raise the little finger
(the other fingers should stay (1 jj
where they were before) *
A. UNITS (continued)
Saying “seven”, raise all your
other fingers except the little
finger, which should be bent
onto the edge of the palm
For 7, the ring finger is also
raised, but the little finger is
lowered so that its tip points
towards the wrist
8
To say “eight”, do the same
with the ring finger
For 8, do the same with the
ring finger
Saying “nine”, you place the
middle finger also in the
same place
For 9, do just the same with
the middle finger
9
B. TENS
When you say “ten", place
the nail of the index finger
into the middle joint of
the thumb
For 10, the nail of the right
index finger is placed on the
first joint (counting from
the tip) of the thumb, so
that the space between the
fingers is like a circle
For “twenty” put the tip of
For 20, place the middle- VI
the thumb between the
finger side of the lower joint Jl/illj
| \ index and the middle fingers
of the index finger over the jS j V|
V i
face of the thumbnail, ( \
so that it appears that the l . J
20
tip of the thumb is gripped 2 q
between the index and
middle fingers. But the middle finger
must not take part in this gesture, for
by varying the position of this one also
you may obtain other numbers. The
number 20 is expressed solely by the
contact between the thumbnail and the
lower joint of the index finger
Fig. 3.13.
T H F. FARM E S T CALCULATING M A CHI N E
T HE HAND
54
B. TF.NS (continued)
30
For “thirty”, touch thumb
and index in a gentle kiss
For 30, hold the thumb
straight and touch the tip
of the thumbnail with
the index finger, so that
together they resemble the
arc of a circle with its chord
(if you need to bend the
thumb somewhat, the number will be
equally well indicated and no confusion
should result)
For “forty”, place the inside
of the thumb against the
side or the back of the index
finger, keeping both of them
straight
For 40, place the inside of
the tip of the thumb on the
back of the index finger, so
that there is no space
between the thumb and
the edge of the palm
flh For “fifty”, bend the thumb
J across the palm of the hand.
For 50, hold the index finger
straight up, hut bend the
, Jh
with the top joint bent over,
thumb and place it in the
pf
j like the Greek letter
palm of the hand, in front of
V'V
the index finger 1
V J
50
50
For “sixty”, with the thumb
bent as for fifty, the index
finger is brought down
to cover the face of the
thumbnail
For 60, bend the thumb and
place the second phalanx of
the index Finger on the face
of the thumbnail
60 i
For “seventy”, with the index
finger as before, that is
closely covering the thumb-
nail, raise the thumbnail
across the middle joint of the
index finger
For 70, raise the thumb and
place the underside of the
first joint of the index finger
on the tip of the thumbnail
so that the face of the thumb-
nail remains uncovered
70
B. TENS (continued)
80
For “eighty”, with the index
raised as above, and the
thumb straight, place the
thumbnail within the bent
middle phalanx of the index
finger
For 80, hold the thumb
straight and place the tip of
the index finger on the curve
of its top joint. (Note the
discrepancy between the two
accounts here)
For “ninety”, press the nail of
the index finger against the
root of the thumb
For 90, put the nail of the
index finger over the joint
of the second phalanx of the
thumb (just as, for 10, you
place it over the joint of the
first phalanx)
90
C. HUNDREDS AND THOUSANDS
When you say “a hundred”, on your right
hand do as for ten on the left hand; "two
hundred” on the right hand is like twenty
on the left; “three hundred” on the right
like thirty on the left; and so on up to
“nine hundred”
When you say “a thousand”, with your
right hand you do as for one with the left;
“two thousand”, on the right is like two
on the left; “three thousand” on the right
like three on the left, and so on up to
“nine thousand”
Once you have mastered these eighteen
numbers, the nine combinations of the
little, ring and middle fingers as well as
the nine combinations of the thumb and
the index finger then you can readily
understand that what serves on the right
hand to show the units from 1 to 9 will on
the left hand show from 1,000 to 9,000;
and that what on the right hand shows the
tens, on the left hand shows the hundreds
from 100 to 900
u/ U.
1,000 2,000 3,000 4,000 5,000
6,000 7,000 8,000 9,000
6,000 7,000 8,000 9,000
Fig. 3.13. (continued)
55
FINGER-COUNTING THROUGHOUT HISTORY
same place, but palm outwards. For 80,000 grasp your thigh with your
hand. For 90,000 grasp your loins with the left hand, the thumb
towards the genitals.
Bede continues by describing how, by using the same signs on the right-
hand side of the body, and with the right hand, the numbers from 100,000
to 900,000 may be represented. Finally he explains that one million may be
indicated by crossing the two hands, with the fingers intertwined.
FINGER-COUNTING THROUGHOUT HISTORY
The method described above is extremely ancient. It is likely that it goes
back to the most extreme antiquity, and it remained prominent until recent
times in both the Western and Eastern worlds and, in the latter, persisted
until recent times. In the Egypt of the Pharaohs it was in use from the
Old Kingdom (2800-2300 BCE), as it would seem from a number of funeral
paintings of the period. For example, Fig. 3.14 shows, from right to left,
three men displaying numbers on their fingers according to the method
just described. The first figure seems to be indicating 10 or 100, the fourth
6 or 6,000 and the sixth 7 or 7,000. According to traditions which have been
repeated by various authors, Egypt clearly appears to have been the source
of this system.
Fig. 3.14. Finger-counting shown on a Egyptian monument of the Old Kingdom (Fifth Dynasty,
26th century BCE). Mastaba D2 at Saqqara. See Borchardt (1937), no. 1534A, plate 48.
C. Pellat (1977) quotes two Arab manuscripts. One of these is at the
University of Tunis (no. 6403) and the other is in the library of
the Waqfs in Baghdad ( Majami ‘ 7071/9). The counting system in question
is, in the first manuscript, attributed to “the Copts of Egypt”; the title of the
second clearly suggests that it is of Egyptian origin. ( Treatise on the Coptic
manner of counting with the hands).
A qasida (poem in praise of a potential patron) attributed to Mawsili al
Hanbali describes “the sign language of the Copts of Egypt, which expresses
numbers by arranging the fingers in special ways”. Ibn al Maghribi states,
See! I follow in the steps of every learned man. The spirit moves me to
write something of this art and to compose a Ragaz, to be called The Table
of Memory, which shall include the art of counting of the Copts.” Finally,
Juan Perez de Moya ( Alcala de Henares, 1573) comes to the following
conclusion: “No one knows who invented this method of counting, but
since the Egyptians loved to be sparing of words (as Theodoret has said), it
must be from them that it has come.”
There is also evidence for its use in ancient Greece. Plutarch ( Lives of
Famous Men ) has it that Orontes, son-in-law of Artaxerxes King of Persia,
said: “Just as the fingers of one who counts are sometimes worth ten
thousand and sometimes merely one, so also the favourites of the King may
count for everything, or for nothing.”
The method was also used by the Romans, as we know in the first
instance from “number-tiles” discovered in archaeological excavations from
several parts of the Empire, above all from Egypt, which date mostly from
the beginning of the Christian era (Fig. 3.15). These are small counters or
tokens, in bone or ivory, each representing a certain sum of money. The
Roman tax collectors gave these as “receipts”. On one side there was a
representation of one of the numbers according to the sign system
described above, and on the other side was the corresponding Roman
numeral. (It would seem, however, that these numbers never went above
15 in these counters from the Roman Empire).
Fig. 3.15. Roman numbered tokens (tesserae,) from the first century CE. The token on the left
shows on one face the number 9 according to a particular method of finger-counting; on the reverse
face, the same value is shown in Roman numerals. British Museum. The token on the right shows a
man making the sign for 15, according to the same system, on the fingers of his left hand.
Bibliotheque nationale (Paris). Tessera no. 316. See Frohner (1884).
We also know about this from the writings of numerous Latin authors.
Juvenal (c. 55-135 CE) speaks thus of Nestor, King of Pylos, who lived, it is
said, for more than a hundred years: “Fortunate Nestor who, having
attained one hundred years of age, henceforth shall count his years on
his right hand!” This tells us that the Romans counted tens and units on
the left hand, and hundreds and thousands on the right hand. Apuleus
(c. 125-170 CE) describes in his Apologia how, having married a rich
widow, a certain Aemilia Pudentilla, he was accused of resorting to magic
means to win her heart. He defended himself before the Pro-Consul
THE EARLIEST CALCULATING MACHINE
THE HAND
56
Claudius Maximus in the presence of his chief accuser Emilianus. Emilianus
had ungallantly declared that Aemilia was sixty years old, whereas she was
really only forty. Here is how Apuleus challenges Emilianus.
How dare you, Emilianus, increase her true number of years by one
half again? If you said thirty for ten, we might think that you had ill-
expressed it on your fingers, holding them out straight instead of
curved (Fig. 3.16). But forty, now that is easily shown: it is the open
hand! So when you increase it by half again this is not a mistake, unless
you allow her to be thirty years old and have doubled the consular
years by virtue of the two consuls.
Fig. 3.16.
And we may cite Saint Jerome, Latin philologist of the time of Saint
Augustine:
One hundred, sixty, and thirty are the fruits of the same seed in the
same earth. Thirty is for marriage, since the joining of the two fingers
as in a tender kiss represents the husband and the wife. Sixty depicts
the widow in sadness and tribulation. And the sign for one hundred
(pay close attention, gentle reader), copied from the left to right with
the same fingers, shows the crown of virginity (Fig. 3.17).
Fig. 3-17-
Again, the patriarch Saint Cyril of Alexandria (376-444) gives us the
oldest known description of this system ( Liber de computo, Chapter
CXXXVIII: De Flexibus digitorum, III, 135). The description exactly matches a
passage in a sixth-century Spanish encyclopaedia, Liber etymologiarum,
which was the outcome of an enormous compilation instituted by Bishop
Isidor of Seville (570-636). The Venerable Bede, in his turn, drew inspira-
tion from it in the seventh century for his chapter De computo vel loquela
digitorum.
One of the many reasons why this system remained popular was its
secret, even mysterious, aspect. J. G. Lemoine (1932) says: “What a splendid
method for a spy to use, from the enemy camp, to inform his general at a
distance of the numerical force of the enemy, by a simple, apparently casual,
gesture or pose.” Bede also gives an example of such silent communication:
“A kind of manual speech [manualis loquela ] can be expressed by the system
which 1 have explained, as a mental exercise or as an amusement.” Having
established a correspondence between the Latin letters and the integers,
he says: “To say Caute age (‘look out!’) to a friend amongst doubtful or
dangerous people, show him (the following finger gestures)” (Fig. 3.18).
Fig. 3.18.
Following the fall of the Roman Empire, the same manual counting
remained extraordinarily in vogue until the end of the Middle Ages (Fig. 3.19
to 3.21), and played a most important part in mediaeval education. The finger
counting described in Bede’s De computo vel loquela digitorum (cited above)
was extensively used in the teaching of the Trivium of grammar, rhetoric and
logic during the undergraduate years leading to the B.A. degree, which, with
the Quadrivium (literally “crossroads”, the meeting of the Four Ways of arith-
metic, geometry, astronomy, and music) studied in the following years
leading to the M.A. degree, made up the Seven Liberal Arts of the
scholarly curriculum, from the sixth to the fifteenth centuries. Barely four
hundred years ago, a textbook of arithmetic was not considered complete
without detailed explanations of this system (Fig. 3.22). Only when written
arithmetic became widespread, with the adoption of the use of Arabic
numerals, did the practice of arithmetic on hands and fingers finally decline.
57
Fig. 3.19. The system describee! in Fig. 3.13 illustrated in a manuscript by the Spanish
theologian Rabano Maura (780-856). Codex Alcobacense 394, folio 152 V National
Library of Lisbon. See Burnam (1912-1925), vol 1, plate XIV.
Fig. 3.20. The same system again, in a Spanish manuscript of 1130. Detail of a codex
from Catalonia (probably from Santa Maria de Ripoll). National Library of Madrid,
Codex matritensis A 19, folio 3 V. See Burnam (1912-1925), vol. 3, plate XUI1.
F I N G F. R- C O U N T I N G T HROU G II OUT HISTORY
F 1 c; . 3.21. The same system yet again in a mathematical work published in
Vienna in 1494. Extract from the work by Luca Pacioli: Summa de
Arithmetica, Geometrica, proportioni e proportionalila
Fig. 3 - 22 . The same system of signs in a work on arithmetic published in
Germany in 1727: Jacob Leupold, Theatrum Arithmetico-Geometricum
THE EARLIEST CALCULATING MACHINE
THE HAND
58
Fig. 3.23. In the Arab-Persian system of
number gestures, the number 93 is shown by
placing the nail of the index finger right on the
joint of the second phalanx of the thumb (which
represents 90), and then bending the middle,
ring and little fingers (which represents 3); and
this, nearly enough, gives rise to a closed fist.
In the Islamic world, the system was at least as widely spread as in the
West, as recounted by many Arab and Persian writers from the earliest
times. From the beginning of the Hegira, or Mohammedan era (dated from
the flight of Mohammed from Mecca to Medina on 15 July 622 CE), we
find an oblique allusion among these poets when they say that a mean or
ungenerous person’s hand “makes 93” (see the corresponding closed hand,
symbol of avarice, in Fig. 3.23). One of them, Yahya Ibn Nawfal al Yamani
(seventh century) says: “Ninety and three, which a man may show as a fist
closed to strike, is not more niggardly than thy gifts, Oh Yazid.” Another,
Khalil Ibn Ahmad (died 786), grammarian and one of the founders of Arab
poetry, writes: “Your hands were not made for giving, and their greed is
notorious: one of them makes 3,900 (the mirror image of 93) and the other
makes 100 less 7.”
One of the greatest Persian poets, Abu’l Kassim Firdusi, dedicated Shah
Nameh (The Book of Kings) to Sultan Mahmud le Ghaznavide but found
himself poorly rewarded. In a satire on the Sultan’s gross avarice, he wrote:
“The hand of King Mahmud, of noble descent, is nine times nine and three
times four.”
A qasida of the Persian poet Anwari (died 1189 or 1191) praises the
Grand Vizir Nizam al Mulk for his precocity in arithmetic: “At the age
when most children suck their thumbs, you were bending the little finger of
your left hand” (implying that the Vizir could already count to a thousand)
(Fig. 3.13C).
A dictum of the Persian poet Abu’l Majid Sanayi (died 1160) reminds us
that by twice doing the same thing in one’s life, one may take away from
its value: “What counts for 200 on the left hand, on the right hand is worth
no more than 20” (Fig. 3.24). The poet Khaqanl (1106-1200) exclaims:
“If I could count the turns of the wheel of the skies, I would number them
on my left hand!” and: “Thou slayest thy lover with the glaive of thy glances,
so many as thou canst count on thy left hand” (the left hand counts the
hundreds and the thousands).
Another quotation from Anwari: “One night, when the service I
rendered thee did wash the face of my fortune with the water of kindness,
you did give to me that number (50) which thy right thumb forms when it
tries to hide its back under thy hand” (Fig. 3.25).
And some verses of Al Farazdaq (died 728) refer to forming the number
30 by opposing thumb and forefingi
lice (Fig. 3.26).
■, in a description of crushing pubic
Fig. 3.25. Fig. 3.26.
According to Levi della Vida (1920), one of the earliest datable references
from the Islamic world to this numerical system can be found in the
following quotation from Ibn Sa’ad (died c. 850): “Hudaifa Ibn al Yaman,
companion of the Prophet, signalled the murder of Khalif ‘Otman as one
shows the number 10 and sighed: ‘This will leave a void [forming
a round between finger and thumb, Fig. 3.27] in Islam which even a
mountain could hardly fill.’”
A poem attributed to Al Mawsili al Hanbali says: “If you place the thumb
against the forefinger like - listen carefully - someone who takes hold of
an arrow, then it means 60” (Fig. 3.28); and, in verses attributed to Abul
Hassan ‘All: “For 60, bend your forefinger over your thumb, as a bowman
grasps an arrow [Fig. 3.28] and for 70 do like someone who flicks a dinar
to test it” (Fig. 3.29).
Ahmad al Barbir al Tarabulusi (a writer on secular Arab and Persian
texts), talking of what he calls counting by bending the fingers, says: “We
59
FINGER-COUNTING THROUGHOUT HISTORY
know the traditionalists use it, because we find references; and it is the
same with the fuqaha* for these lawyers refer to it in relation to prayer in
connection with the Confession of Faith; 1- they say that, according to the
rule of tradition, he who prays should place his right hand on his thigh
when he squats for the Tashahud, forming the number 53” (Fig. 3.30).
From the poet Khaqani we have; “What struggle is this between Rustem
and Bahrain? What fury and dispute is it that perturbs these two sons of
noble lines? Why, they fight day and night to decide which army shall do a
20 on the other’s 90.”
This may seem obscure to the modern reader, unfamiliar with the finger
signs in question. But look closely at the gestures that correspond to the
numbers (Fig. 3.31): “90” undoubtedly represents the anus (and, by
extension, the backside), as it commonly did in vulgar speech; while “to
do a 20 on someone” is undoubtedly an insulting reference to the sexual
act (apparently expressed as “to make a thumb” in Persian) and therefore
(by extension in this military context) to “get on top of”.
More obscenely, Ahmad al Barbir al Tarabulusi could not resist offering
his pupils the following mnemonic for the gestures representing 30 and 90:
90 20 90 30
Fig. 3.31. Fig. 3.32.
“A poet most elegantly said, of a handsome young man: Khalid set out
with a fortune of 90 dirhams, but had only one third of it left when he
returned!” plainly asserting that Khalid was homosexual (Fig. 3.32), having
started “narrow” (90) but finished “wide” (30).
These many examples amply show how numbers formed by the fingers
served as figures of speech, no doubt much appreciated by the readers of
the time. These ancient origins find etymological echo today, as in digital
computing. There is no longer any question of literally counting on the
fingers, but the Latin words digiti (“fingers") and articuli (“joints”) came to
represent “units” and “tens”, respectively, in the Middle Ages, whence digiti
* Islamic lawyers who concern themselves with every kind of’ social or personal matter, with the order of
worship and with ritual requirements
* Asserting that Allah is One, affirming belief in Mohammed, at the same time raising the index finger and
closing the others
in turn came to mean the signs used to represent the units of the decimal
system. The English word digit, meaning a single decimal numeral, is
derived directly from this. In turn, this became applied to computation,
hence the term digital computing in the sense of “computing by numbers”.
With the development and recent enormous spread of “computers” ( digital
computers), the meaning of “digital computation” has been extended to
include every aspect of the processing of information by machine in which
any entity, numerical or not, and whether or not representing a variable
physical quantity, is given a discrete representation (by which is meant that
distinct representations correspond to different values or entities, there is
a finite - though typically enormous - number of possible distinct repre-
sentations, and different repesentations are encoded as sequences of
symbols taken from a finite set of available symbols). In the modern digital
computer, the primitive symbols are two in number and denoted by “0”
and “1” (the binary system) and realised in the machine in terms of distinct
physical states which are reliably distinguishable.
HOW TO CALCULATE ON YOUR FINGERS
After this glance at the modern state of the art of digital information pro-
cessing, let us see how the ancients coped with their “manual informatics”.
The hand can be used not only for counting, but also for systematically
performing arithmetical calculations. I used to know a peasant from the
Saint-Flour region, in the Auvergne, who could multiply on his fingers, with
no other aid, any two numbers he was given. In so doing, he was following
in a very ancient tradition.
For example, to multiply 8 by 9, he closed on one hand as many fingers as
the excess of 8 over 5, namely 3, keeping the other two fingers extended. On
the other hand, he closed as many fingers as the excess of 9 over 5, namely
4, leaving the fifth finger extended (Fig. 3.33). He would then (mentally)
multiply by 10 the total number (7) of closed fingers (70), multiply together
the numbers of extended fingers on the two hands (2x1 = 2), and finally
add these two results together to get the answer (72). That is to say:
8 x 9 = (3 + 4) x 10 + (2 x 1) = 72
THE EARLIEST CALCULATING MACHINE
THE HAND
Similarly, to multiply 9 by 7, he closed on one hand the excess of 9 over 5,
namely 4, and on the other the excess of 7 over 5, namely 2, in total 6;
leaving extended 1 and 3 respectively, so that by his method the result is
obtained as
9 x 7 = (4 + 2) x 10 + (1 x 3) = 63
Although undoubtedly discovered by trial and error by the ancients, this
method is infallible for the multiplication of any two whole numbers
between 5 and 10, as the following proves by elementary (but modern)
algebra. To multiply together two numbers x and y each between 5 and 10,
close on one hand the excess (x - 5) of x over 5, and on the other the excess
(y - 5) of y over 5; the total of these two is (x - 5) + (y - 5), and 10 times
this is ((x - 5) + (y - 5)) x 10 = 10 x + 10 y - 100
7x8
TO MULTIPLY 7 BY 8:
Close (7 - 5) fingers on one hand, and
(8 - 5) on the other.
Result: 5 fingers closed in all. 3 fingers
raised on one hand and 2 on the
other.
Hence: 7x8 = 5x10 + 3x2 = 56
TO MULTIPLY 8 BY 6:
Close (8 - 5) fingers on one hand, and
(6 - 5) on the other.
Result: 4 fingers closed in all, 2 fingers
raised on one hand and 4 on the
other
Hence: 8x6 = 4x10 + 2x4 = 48
Fig. 3-35-
The number of fingers remaining extended on the first hand is 5 - (x - 5) =
10 - x , and on the other, similarly, 10 - y. The product of these two is
(10 - x) x (10 -y) = 100 - 10 x - 10 y + xy
Adding these two together, according to the method, therefore results in
(10 x + lOy - 100) + 100 - 10 x - lOy + xy = xy
namely the desired result of multiplying x by y.
6 0
He had a similar way of multiplying numbers exceeding 9. For example,
to multiply 14 by 13, he closed on one hand as many fingers as the excess of
14 over 10, namely 4, and on the other, similarly, 3, making in all 7. Then he
mentally multiplied this total (7) by 10 to get 70, adding to this the product
(4x3 = 12) to obtain 82, finally adding to this result 10 x 10 = 100 to obtain
182 which is the correct result.
By similar methods, he was able to multiply numbers between 15 and
20, between 20 and 25, and so on. It is necessary to know the squares of 10,
15, 20, 25 and so on, and their multiplication tables. The mathematical
justifications of some of these methods are as follows.
To multiply two numbers x and y between 10 and 15:
10 [(x - 10) + (y - 10)] + (x - 10) x (y - 10) + 10 2 = xy
MULTIPLYING NUMBERS BETWEEN 10
AND 15
ON THE FINGERS
(It must be known by heart that 100 is the square
of 10)
Example: 12 X 13
Close: (12 - 10) fingers on one hand, and
(12 - 10) on the other.
Result: 2 fingers closed on one hand, and 3
on the other.
Hence: 12 X 13 = 10 x (2 + 3) + (2 x 3) + 10 X 10
= 156
Fig. 3.36.
To multiply two numbers x and y between 15 and 20:
15 [(x - 15) + (y - 15)]+ (x - 15) x (y - 15) + 15 2 = xy
MULTIPLYING NUMBERS
BETWEEN 15 AND 20
ON THE FINGERS
(It must be known by heart that 225 is the
square of 15)
Example: 18 X 16
Close: (18 - 15) fingers on one hand, and
(16 - 15) on the other.
Result: 3 fingers closed on one hand,
and 1 on the other
Hence: 15 X (3 + 1) + (3 X 1) + 15 X 15
= 288
Fig. 3.37.
61
COUNTING TO THOUSANDS USING THE FINGERS
To multiply two numbers x andy between 20 and 25:
20 [(x - 20) + (y - 20)] + (x- 20) x (y- 20) + 20 2 = xy
and so on.*
It can well be imagined, therefore, how people who did not enjoy the
facility in calculation which our “Arabic" numerals allow us were none
the less able to devise, by a combination of memory and a most resourceful
ingenuity in the use of the fingers, ways of overcoming their difficulties and
obtaining the results of quite difficult calculations.
Fig. 3.38. Calculating by the fingers shown in an Egyptian funeral painting from the New
Kingdom. This is a fragment of a mural on the tomb of Prince Menna at Thebes, who lived at the
time of the 18th Dynasty, in the reign of King Thutmosis, at the end of the 15th century BCE. We
see six scribes checking while four workers measure out grain and pour bushels of corn from one
heap to another. On the right, on one of the piles of grain, the chief scribe is doing arithmetic on
his fingers and calling out the results to the three scribes on the left who are noting them down.
Later they will copy the details onto papyrus in the Pharaoh 's archives. (Theban tomb no. 69)
COUNTING TO THOUSANDS USING THE FINGERS
The method to be described is a much more developed and mathematically
more interesting procedure than the preceding one. There is evidence of
its use in China at any rate since the sixteenth century, in the arithmetical
textbook Suart fa tong zong published in 1593. E. C. Bayley (1847) attests
that it was in use in the nineteenth century, and Chinese friends of mine
from Canton and Peking have confirmed that it is still in use.
In this method, each knuckle is considered to be divided into three parts:
left knuckle, middle knuckle and right knuckle. There being three knuckles
to a finger, there is a place for each of the nine digits from 1 to 9. Those on the
little finger of the right hand correspond to the units, those on the fourth
finger to the tens, on the middle finger to the hundreds, the forefinger to
the thousands, and finally the right thumb corresponds to the tens of
thousands. Similarly on the left hand, the left thumb corresponds to the
hundreds of thousands, the forefinger to the millions, the middle finger to
the tens of millions, and so on (Fig. 3.39); finally, therefore, on the little finger
of the left hand we count by steps of thousands of millions, i.e. by billions.
Fig. 3.39. Fig 3.40.
With the right hand palm upwards (Fig. 3.40), we count on the little
finger from 1 to 3 by touching the “left knuckles” from tip to base; then
from 4 to 6 by touching the “centre knuckles” from base to tip; and finally,
from 7 to 9 by touching the “right knuckles” from tip to base. We count the
tens similarly on the fourth finger, the hundreds on the middle finger, and
so on.
In this way it is, in theory, possible to count up to 99,999 on one hand, and
up to 9,999,999,999 with both: a remarkable testimony to human ingenuity.
* The general rule being:
N ((* - AD + O' - AO) + Or - A f)(y - N)+ A' 2 = Nx + Ny- 2A/ 2 + xy - xN -yN + N 2 + A/ 2 = xy.
HOW CRO-MAGNON MAN COUNTED
62
CHAPTER 4
HOW CRO-MAGNON MAN
COUNTED
Among the oldest and most widely found methods of counting is the use of
marked bones. People must have made use of this long before they were
able to count in any abstract way.
The earliest archaeological evidence dates from the so-called Aurignacian
era (35,000-20,000 BCE), and are therefore approximately contemporary
with Cro-Magnon Man. It consists of several bones, each bearing regu-
larly spaced markings, which have been mostly found in Western Europe
(Fig. 4.1).
represent number with respect to a base. For otherwise, why would the
notches have been grouped in so regular a pattern, rather than in a simple
unbroken series?
The man who made use of this bone may have been a mighty hunter.
Each time he made a kill, perhaps he made a notch on his bone. Maybe he
had a different bone for each kind of animal: one for bears, another for
deer, another for bison, and so on, and so he could keep the tally of the
larder. But, to avoid having to re-count every single notch later, he took to
grouping them in fives, like the fingers of the hand. He would therefore
have established a true graphical representation of the first few whole
numbers, in base 5 (Fig. 4.2).
I I I I I I I I I I I I I I 1 1 I I I I
1 2 3 4 5 6 7 8 910 11 ... 15 16 ... 20
1 hand 2 hands 3 hands 4 hands
Fig. 4.2.
Fig. 4.1. Notched bones
from the Upper Palaeolithic
age.
A and C: Aurignacian.
Musec des Antiquites
nationales, St-Germain-
en-Laye. Bone C is from
Saint-Marcel (Indre, France).
B and D: Aurignacian.
From the Kulna cave ( Czech
Republic).
E: Magdalenian (19,000 -
12,000 BCE). From the
Pekarna cave (Czech
Republic). See Jelinek (1975),
pp. 435-453.
Amongst these is the radius bone of a wolf, marked with 55 notches in
two series of groups of five. This was discovered by archaeologists in 1937,
at Dolni Vestonice in Czechoslovakia, in sediments which have been dated
as approximately 30,000 years old. The purpose of these notches remains
mysterious, but this bone (whose markings are systematic, and not artisti-
cally motivated) is one of the most ancient arithmetic documents to have
come down to us. It clearly demonstrates that at that time human beings
were not only able to conceive number in the abstract sense, but also to
Also of great interest is the object shown in Fig. 4.3, a point from a
reindeer’s antler found some decades ago in deposits at Brassempouy in
the Landes, dating from the Magdalenian era. This has a longitudinal
groove which separates two series of transverse notches, each divided into
distinct groups (3 and 7 on one side, 5 and 9 on the other). The longitudinal
notch, which is much closer to the 9-5 series than to the 3-7 series, seems
to form a kind of link or vinculum (as is sometimes used in Mathematics)
joining the group of nine to the group of five.
Fig. 4.3. Notched bone from the Magdalenian era (19, 000-12, 000 BCE), found at Brassempouy
(Landes, France). Bordeaux, Museum of Aquitaine
Now what could this be for? Was it perhaps a simple tool, or a weapon,
which had been grooved to stop it slipping in the hand? Unlikely. Anyway,
what purpose would the longitudinal groove then serve? And even if this
were the case, why do we not find such markings on similar prehistoric
implements?
In fact, this object also bears witness of some activity with arithmetical
connotations. The way the numbers 3, 5, 7, and 9 are arranged, and the
frequency with which these numbers occur in many artefacts from the same
period, suggest a possible explanation.
63
Let us suppose that the longitudinal groove represents unity, and that
the transverse lines represent other odd numbers (which are all prime
except for 9 which is the square of 3).
This spike from an antler with its grooves then makes a kind of arith-
metical tool, showing a graphical representation of the first few odd
numbers arranged in such a way that some of their simpler properties are
exhibited (Fig. 4.4).
3 • *7
9-7=5-3=2
9 •' ' • 5
3 • 7
7-3 = 9- 5 = (9 + 5)-(7 + 3) = 4
9 • • 5
3 • • 7
| 3x3=9
9 • • 5
3 • • 7
i i 3 + 9 = 5 + 7 = 12
9 • • 5
Fig. 4.4. Some of the arithmetical properties of the groupings of the grooves on the bone
shown in Fig. 4.3.
As well as giving us concrete evidence for the memorisation and record-
ing of numbers, the practice of making tally marks such as described is also
a precursor of counting and book-keeping. We are therefore led to supposi-
tions such as the following.
Our distant forefathers possibly used this piece of antler for taking count
of people, things or beasts. It could perhaps have served a tool-maker to
keep account of his own tools:
HOW CRO-MAGNON MAN COUNTED
3 graters and 7 knives (in stone)
9 scrapers and 5 needles (in bone)
where the longitudinal groove linking the 5 and the 9 may, in this man’s mind,
have denoted the common material (bone) from which they were made.
Or perhaps a warrior might similarly keep count of his weapons:
3 knives and 7 daggers
9 spears with plain blade, and 5 with split blade
Or the hunter might record the numbers of different types of game
brought back for the benefit of his people:
3 bison and 7 buffalo
9 reindeer and 5 stags
We can also imagine how a herdsman could count the beasts in his
keeping, sheep and goats on the one hand, cattle on the other.
A messenger could use an antler engraved in this way to carry a
promissory note to a neighbouring tribe:
In 3 moons and 7 days we will bring
9 baskets of food and 5 fur animals
We can also imagine it being used as a receipt for goods, or a delivery
manifest, or for accounting for an exchange or distribution of goods.
Of course, these are only suppositions, since the true meaning has
eluded the scholars. And in fact the true purpose of these markings will
remain unknown for ever, because with this kind of symbolism the things
themselves to which the operations apply are represented only by their
quantity, and not by specific signs which depict the nature of the things.
Human kind was still unable to write. But, by representing as we have
described the enumeration of this or that kind of unit, the owner of the
antler, and his contemporaries, had nonetheless achieved the inventions of
written number: in truth, they wrote figures in the most primitive notation
known to history.
TAM.Y STICKS: ACCOUNTING FOR BEGINNERS
64
CHAPTER 5
TALLY STICKS
ACCOUNTING FOR BEGINNERS
Notched sticks - tally sticks - were first used at least forty thousand years
ago. They might seem to be a primitive method of accounting, but they
have certainly proved their value. The technique has remained much the
same through many centuries of evolutionary, historical, and cultural
change, right down to the present day. Although our ancestors could not
have known it, their invention of the notched stick has turned out to be
amongst the most permanent of human discoveries. Not even the wheel
is as old; for sheer longevity, only fire could possibly rival it.
Notch-marks found on numerous prehistoric cave-wall paintings alongside
outlines of animals leave no doubt about the accounting function of the
notches. In the present-day world the technique has barely changed at all.
For instance, in the very recent past, native American labourers in the
Los Angeles area used to keep a tally of hours worked by scoring a fine line
in a piece of wood for each day worked, with a deeper or thicker line to
mark each week, and a cross for each fortnight completed.
More colourful users of the device in modern times include cowboys,
who made notches in the barrels of their guns for each bison killed, and
the fearsome bounty hunters who kept a tally in the same manner for
every outlaw that they gunned down. And Calamity Jane’s father also used
the device for keeping a reckoning of the number of marriageable girls in
his town.
On the other side of the world, the technique was in daily use in the
nineteenth century, as we learn from explorer’s tales:
On the road, just before a junction with a smaller track, I came upon a
heavy gate made of bamboo and felled tree trunks, and decorated with
hexagonal designs and sheaves. Over the track itself was hung a small
plank with a set of regularly-sized notches, some large, some small, on
each side. On the right were twelve small notches, then four large ones,
then another set of twelve small ones. This meant: Twelve days march
from here, any man who crosses our boundary will be our prisoner or will
pay a ransom of four water-buffalo and twelve deals (rupees). On the left,
eight large notches, eleven middle-sized ones, and nine small ones,
meaning: There arc eight men, eleven women and nine children in our
village. [J. Harmand (1879): Laos]
In Sumatra, the Lutsu declared war by sending a piece of wood
scored with lines together with a feather, a scrap of tinder and a fish.
Translation: they will attack with as many hundreds (or thousands) of
men as there are scored lines; they will be as swift as a bird (the
feather), will lay everything waste (the tinder = fire), and will drown
their enemies (the fish). [J. G. Fevrier (1959)]
Only a few generations ago, shepherds in the Alps and in Hungary, as
well as Celtic, Tuscan, and Dalmatian herders, used to keep a tally of the
number of head in their flocks by making an equivalent number of notches
or crosses on wooden sticks or planks. Some of them, however, had a
particularly developed and subtle version of the technique as L. Gerschel
describes:
On one tally-board from the Moravian part of Walachia, dating from
1832, the shepherd used a special form of notation to separate the
milk-bearing sheep from the others, and within these, a special mark
indicated those that only gave half the normal amount. In some parts
of the Swiss Alps, shepherds used carefully crafted and decorated
wooden boards to record various kinds of information, particularly the
number of head in their flock, but they also kept separate account of
sterile animals, and distinguished between sheep and goats ....
We can suppose that shepherds of all lands cope with much the
same realities, and that only the form of the notation varies (using,
variously, knotted string or quipu [see Chapter 6 below], primitive
notched sticks, or a board which may include (in German-speaking
areas) words like Kilo (cows), Gallier (sterile animals), Geis (goats)
alongside their tallies. There is one constant: the shepherd must know
how many animals he has to care for and feed; but he also has to
know how many of them fall into the various categories - those that
give milk and those that don’t, young and old, male and female. Thus
the counts kept are not simple ones, but threefold, fourfold or more
parallel tallies made simultaneously and entered side by side on the
counting tool.
Fig. 5 . i . Swiss shepherd's tally
stick (Late eighteenth century, Saanen,
Canton of Bern). From the Museum fir
Volkerkunde, Basel ; reproduced from
Grniir (1917)
In short, shepherds such as these had devised a genuine system of
accounting.
Another recent survival of ancient methods of counting can be found in
the name that was given to one of the taxes levied on serfs and commoners
65
TAI.l.Y STICKS! ACCOUNTING LOR B K G I N N K R S
in France prior to 1789: it was called la taille, meaning “tally” or “cut”, for
the simple reason that the tax-collectors totted up what each taxpayer had
paid on a wooden tally stick.
In England, a very similar device was used to record payments of tax and
to keep account of income and expenditure. Larger and smaller notches
on wooden batons stood for one, ten, one hundred, etc., pounds sterling
(see Fig. 5.2). Even in Dickens’s day, the Treasury still clung on to this
antiquated system! And this is what the author of David Copperfield
thought of it:
Ages ago, a savage mode of keeping accounts on notched sticks was
introduced into the Court of Exchequer; the accounts were kept, much
as Robinson Crusoe kept his calendar on the desert island. In course
of considerable revolutions of time ... a multitude of accountants,
book-keepers, actuaries and mathematicians, were born and died;
and still official routine clung to these notched sticks, as if they were
pillars of the constitution, and still the Exchequer accounts continued
to be kept on certain splints of elm wood called “tallies”. Late in the
reign of George III, some restless and revolutionary spirit originated
the suggestion, whether, in a land where there were pens, ink and
paper, slates and pencils, and systems of accounts, this rigid adherence
to a barbarous usage might not border on the ridiculous? All the
red tape in the public offices turned redder at the bare mention of
this bold and original conception, and it took till 1826 to get these
sticks abolished.
[Charles Dickens (1855)]
Britain may be a conservative country, but it was no more backward than
many other European nations at that time. In the early nineteenth century,
tally sticks were in use in various roles in France, Germany, and Switzer-
land, and throughout Scandinavia. Indeed, I myself saw tally sticks in use
as credit tokens in a country bakery near Dijon in the early 1970s. This is
how it is done: two small planks of wood, called tallies, are both marked
with a notch each time the customer takes a loaf. One plank stays with
the baker, the other is taken by the customer. The number of loaves is
totted up and payment is made on a fixed date (for example, once a week).
No dispute over the amount owed is possible: both planks have the same
number of notches, in the same places. The customer could not have
removed any, and there’s an easy way to make sure the baker hasn’t added
any either, since the two planks have to match (see Fig. 5.3).
The French baker’s tally stick was described thus in 1869 by Andre
Philippe, in a novel called Michel Rondet:
The women each held out a piece of wood with file-marks on it. Each
piece of wood was different - some were just branches, others were
Fig. 5.2. English accounting tally sticks, thirteenth century. London, Society of Antiquaries
Museum
i nr-iTfn
Fig. 5.5. French country bakers' tally sticks, as used in small country towns
planed square. The baker had identical ones threaded onto a strap.
He looked out for the one with the woman’s name on it on his
strap, and the file-marks tallied exactly. The notches matched, with
Roman numerals - I, V, X - signifying the weight of the loaves that
had been supplied.
Rene Jonglet relates a very similar scene that took place in Hainault
(French-speaking Belgium) around 1900:
The baker went from door to door in his wagon, calling the housewives
out. Each would bring her “tally” - a long and narrow piece of wood,
shaped like a scissor-blade. The baker had a duplicate of it, put the two
side by side, and marked them both with a saw, once for each six-
pound loaf that was bought. It was therefore very easy to check what
was owed, since the number of notches on the baker’s and house-
wife’s tally stick was the same. The housewife couldn’t remove any
from both sticks, nor could the baker add any to both.
The tally stick therefore served not just as a curious form of bill and
receipt, but also as a wooden credit card, almost as efficient and reliable as
the plastic ones with magnetic strips that we use nowadays.
French bakers, however, did not have a monopoly on the device: the use
of twin tally sticks to keep a record of sums owing and to be settled can be
found in every period and almost everywhere in the world.
The technique was in use by the Khas Boloven in Indo-China, for
example, in the nineteenth century:
For market purchases, they used a system similar to that of country
bakers: twin planks of wood, notched together, so that both pieces
held the same record. But their version of this memory-jogger is much
more complicated than the bakers’, and it is hard to understand how
they coped with it. Everything went onto the planks - the names of the
TAI.I.Y STICKS: ACCOUNTING LOR BEGINNERS
sellers, the names of the buyer or buyers, the witnesses, the date of
delivery, the nature of the goods and the price. [J, Harmand (1880)]
As Gerschel explains, the use of the tally stick is, in the first place, to
keep track of partial and successive numbers involved in a transaction.
However, once this use is fully established, other functions can be added:
the tally stick becomes a form of memory, for it can hold a record not just
of the intermediate stages of a transaction, but also of its final result. And
it was in that new role, as the record of a completed transaction, that it
acquired an economic function, beyond the merely arithmetic function of
its first role.
The mark of ownership was the indispensable additional device that
allowed tally sticks to become economic tools. The mark symbolised the
name of its owner: it was his or her “character” and represented him or her
legally in any situation, much like a signature. Improper use of the mark
of ownership was severely punished by the law, and references to it are
found in French law as late as the seventeenth century.
The mark of ownership thus took the notched stick into a different
domain. Originally, notched sticks had only notches on them: but now they
also carried signs representing not numbers but names.
E3-EB CHH ^ fxj X~t> EEK 1\ n
Fig. 5.4. Examples of marks of ownership used over the ages. The signs were allocated to specific
members of the community and could not be exchanged or altered.
Here is how they were used amongst the Kabyles, in Algeria:
Each head of cattle slaughtered by the community is divided equally
between the members, or groups of members. To achieve this, each
member gives the chief a stick that bears a mark; the chief shuffles the
sticks and then passes them to his assistant, who puts a piece of meat
on each one. Each member then looks for his own stick and thus
obtains his share of the meat. This custom is obviously intended to
ensure a fair share for everyone. [J. G. Fevrier (1959)]
The mark of ownership probably goes back to the time before writing
was invented, and it is the obvious ancestor of what we call a signature (the
Latin verb signare actually means “to make a cross or mark”). So the mark,
the “signature” of the illiterate, can be associated with the tally stick, the
accounting device for people who cannot count.
But once you have signatures, you have contracts: which is how tally
sticks with marks of ownership came to be used to certify all sorts of
commitments and obligations. One instance is provided by the way the
Cheremiss and Chuvash tribesfolk (central Russia) recorded loans of
money in the nineteenth century. A tally stick was split in half lengthways,
6 6
each half therefore bearing the same number of notches, corresponding
to the amount of money involved. Each party to the contract took one of
the halves and inscribed his personal mark on it (see Fig. 5.4), and then
a witness made his or her mark on both halves to certify the validity and
completeness of the transaction. Each party then took and kept the half
with the other's signature or mark. Each thus retained a certified, legally
enforceable and unalterable token of the amount of capital involved (indi-
cated by the notched numbers on both tally sticks). The creditor could not
alter the sum, since the debtor had the tally stick with the creditor’s mark;
nor could the debtor deny his debt, since the creditor had the tally stick
with the debtor’s mark on it.
According to A. Conrady (1920), notched sticks similarly constituted the
original means of establishing pacts, agreements and transactions in pre-
literate China. They gave way to written formula only after the development
of Chinese writing, which itself contains a trace of the original system: the
ideogram signifying contract in Chinese is composed of two signs meaning,
respectively, “notched stick” and “knife”.
Fig. 5-5.
The Arabs (or their ancestors) probably had a similar custom, since a
similar derivation can be found in Arabic. The verb-root farada means both
“to make a notch” and “to assign one’s share (of a contract or inheritance)
to someone”.
In France, tally sticks were in regular use up to the nineteenth century as
waybills, to certify the delivery of goods to a customer. Article 1333 of the
Code Napoleon, the foundation stone of the modern French legal system,
makes explicit reference to tally sticks as the means of guaranteeing that
deliveries of goods had been made.
In many parts of Switzerland and Austria, tally sticks constituted until
recently a genuine social and legal institution. There were, first of all, the
capital tallies (not unlike the tokens used by the Chuvash), which recorded
loans made to citizens by church foundations and by local authorities.
Then there were the milk tallies. According to L. Gerschel, they worked in
the following ways:
At LUrichen, there was a single tally stick of some size on which was
inscribed the mark of ownership of each farmer delivering milk, and
opposite his mark, the quantity of milk delivered. At Tavetsch (accord-
ing to Gmiir), each farmer had his own tally stick, and marked on it the
amount of milk he owed to each person whose mark of ownership was
67
TALLY STICKS: ACCOUNTING FOR BEGINNERS
on the stick; reciprocally, what was owed to him appeared under his
mark of ownership on others' tally sticks. When the sticks were
compared, the amount outstanding could be computed.
There were also mole tallies', in some areas, the local authorities held
tallies for each citizen, marked with that citizen’s mark of ownership, and
would make a notch for each mole, or mole’s tail, surrendered. At the year's
end, the mole count was totted up and rewards paid out according to the
number caught.
Tallies were also used in the Alpine areas for recording pasture rights
(an example of such a tally, dated 1624, is said by M. Gmiir to be in the
Swiss Folklore Museum in Basel) and for water rights. It must be remem-
bered that water was scarce and precious, and that it almost always
belonged to a feudal overlord. That ownership could be rented out, sold
and bequeathed. Notched planks were used to record the sign of ownership
of the family, and to indicate how many hours (per day) of a given water
right it possessed.
4 hr 2 hr 1 hr l h hr 20 min 10 min
Finally, the Alpine areas also used Kehrtesseln or “turn tallies”, which
provided a practical way of fixing and respecting a duty roster within a
guild or corporation (night watchmen, standard-bearers, gamekeepers,
churchwardens, etc.).
In the modern world there are a few surviving uses of the notched-stick
technique. Brewers and wine-dealers still mark their barrels with Xs, which
have a numerical meaning; publicans still use chalk-marks on slate to keep
a tally of drinks yet to be settled. Air Force pilots also still keep tallies of
enemy aircraft shot down, or of bombing raids completed, by “notching”
silhouettes of aircraft or bombs on the fuselage of their aircraft.
The techniques used to keep tallies of numbers thus form a remarkably
unbroken chain over the millennia.
Fig. 5 . 6 . A water tally from Wallis (Switzerland). Basel. Museum fur Volkerkunde. See Gmur,
plate XXVI
NUMBERS ON SIRIN G S
68
CHAPTER 6
NUMBERS ON STRINGS
Although it was certainly the first physical prop to help our ancestors when
they at last learned to count, the hand could never provide more than a
fleeting image of numerical concepts. It works well enough for representing
numbers visually and immediately: but by its very nature, finger-counting
cannot serve as a recording device.
As crafts and trade developed within different communities and
cultures, and as communication between them grew, people who had not
yet imagined the tool of writing nonetheless needed to keep account of the
things that they owned and of the state of their exchanges. But how could
they retain a durable record of acts of counting, short of inventing written
numerals? There was nothing in the natural world that would do this for
them. So they had to invent something else.
In the early years of the sixteenth century, Pizarro and his Spanish
conquistadors landed on the coast of South America. There they found a
huge empire controlling a territory more than 4,000 km long, covering an
area as large as Western Europe, in what is now Bolivia, Peru, and Ecuador.
The Inca civilisation, which went back as far as the twelfth century CE, was
then at the height of its power and glory. Its prosperity and cultural sophis-
tication seemed at first sight all the more amazing for the absence amongst
these people of knowledge of the wheel, of draught animals, and even of
writing in the strict sense of the word.
However, the Incas’ success can be explained by their ingenious method
of keeping accurate records by means of a highly elaborate and fairly
complex system of knotted string. The device, called a quipu (an Inca
word meaning “knot") consisted of a main piece of
cord about two feet long onto which thinner
coloured strings were knotted in groups, these
pendant strings themselves being knotted in
various ways at regular intervals (see Fig. 6.1).
Quipus, sometimes incorrectly described as
“abacuses”, were actually recording devices that
met the various needs of the very efficient Inca
administration. They provided a means for repre-
senting liturgical, chronological, and statistical
F I g . 6 . l . A Peruvian quipu
records, and could occasionally also serve as calendars and as messages.
Some string colours had conventional meanings, including both tangible
objects and abstract notions: white, for instance, meant either “silver” or
“peace”; yellow signified “gold”; red stood for “blood” or “war”; and so on.
Quipus were used primarily for book-keeping, or, more precisely, as a
concrete enumerating tool. The string colours, the number and relative
positions of the knots, the size and the spacing of the corresponding
groups of strings all had quite precise numerical meanings (see Fig. 6.2, 6.3
and 6.4). Quipus were used to represent the results of counting (in a decimal
verbal counting system, as previously stated) all sorts of things, from mili-
tary matters to taxes, from harvest reckonings to accounts of animals slain
in the enormous annual culls that were held, from delivery notes (see
Fig. 6.5) to population censuses, and including calculations of base values
for levies and taxes for this or that administrative unit of the Inca Empire,
inventories of resources in men and equipment, financial records, etc.
1 2 3 4 5 6 7 8 9 Fig. 6.2. The first nine
Fig. 6 . 3 . The number 3, 643 as it Fig. 6 . 4 . Numerical reading of a bunch of knotted strings,
would be represented on a string in from an Inca quipu, American Museum of Natural History,
the manner of a Peruvian quipu New York, exhibit B 8713, quoted in Locke (1924): the number
658 on string E equals the sum of the numbers represented on
strings A, B, C and D.
69
NUMBERS ON STRINGS
Quipus were based on a fairly simple, strictly decimal system of positions.
Units were represented by the string being knotted a corresponding number
of times around the first fixed position-point (counting from the end or
bottom of the string), tens were represented similarly by the number of
times the string was knotted around the second position-point, the third
point served for recording hundreds, the fourth for thousands, etc. So to
“write” the number 3,643 on Inca string (as shown in Fig. 6.3), you knot
the string three times at the first point, four times at the second, six times
at the third, and three times at the fourth position-point.
Officers of the king, called quipucamayocs (“keepers of the knots”), were
appointed to each town, village and district of the Inca Empire with respon-
sibility for making and reading quipus as required, and also for supplying
the central government with whatever information it deemed important
(see Fig. 6.5). It was they who made annual inventories of the region’s
produce and censuses of population by social class, recorded the results
on string with quite surprising regularity and detail, and sent the records
to the capital.
tic. 6,s. An Inca quipucamayoc delivering his accounts to an imperial official and describing
the results of an inventory recorded on the quipu From the Peruvian Codex of the sixteenth - century
chronicler Gunman Poma de Ayala (in the Royal Library, Copenhagen), reproduced from Le
Quipucamayoc, p. 335
One of the quipucamayocs was responsible for the revenue accounts,
and kept records of the quantities of raw materials parcelled out to the
workers, of the amount and quality of the objects each made, and of
the total amount of raw materials and finished goods in the royal stores.
Another kept the register of births, marriages and deaths, of men fit for
combat, and other details of the population in the kingdom. Such
records were sent in to the capital every year where they were read by
officers learned in the art of deciphering these devices. The Inca govern-
ment thus had at its disposal a valuable mass of statistical information:
and these carefully stored collections of skeins of coloured string
constituted what might have been called the Inca National Archives.
[Adapted from W. H. Prescott (1970)]
Quipus are so simple and so valuable that they continued to be used for many
centuries in Peru, Bolivia and Ecuador. In the mid-nineteenth century, for
example, herdsmen, particularly in the Peruvian Altiplano, used quipus to
keep tallies of their flocks [M. E. de Rivero & J. D. Tschudi (1859)]. They used
bunches of white strings to record the numbers of their sheep and goats,
usually putting sheep on the first pendant string, lambs on the second, goats
on the third, kids on the fourth, ewes on the fifth, and so on; and bunches of
green string to count cattle, putting the bulls on the first pendant string,
dairy cows on the second, heifers on the third, and then calves, by
age and sex, and so on (see Fig. 6.6).
Fig. 6.6. A livestock inventory on a nineteenth-century quipu from the Peruvian Altiplano.
On bunch A (white string), small livestock: 254 sheep (string Ai), 36 lambs (A?.), 300 goats (An).
40 kids (Ar), 244 ewes (As), total = 874 sheep and goats (As). On bunch 8 (green string),
cattle: 203 bulls (Bi), 350 dairy cows (B 2 ), 235 sterile cows (Bn), total = 788 head of cattle (B4).
NUMBERS ON STRINGS
Even today native Americans in Bolivia and Peru use a very similar
device, the chimpu, a direct descendant of the quipu. A single string is used
to represent units up to 9, with each knot on it indicating one unit, as on a
quipu-, tens are figured by the corresponding number of knots tied on two
strings held together; hundreds in like manner on three strings, thousands
on four strings, and so on. On chimpus, therefore, the magnitude of a
number in powers of 10 is represented by the number of strings included
in the knot - six knots may have the value of 6, 60, 600 or 6,000 according
to whether it is tied on one, two, three, or four strings together.
These remarkable systems are not however uniquely found in Inca or
indeed South American civilisations. The use of knotted string is attested
since classical times, and in various regions of the world.
Herodotus (485-425 BCE) recounts how, in the course of one of his
expeditions, Darius, King of Persia (522-486 BCE) entrusted the rearguard
defence of a strategically vital bridge to Greek soldiers, who were his allies.
He gave them a leather strap tied into sixty knots, and ordered them to
undo one knot each day, saying:
“If I have not returned by the time all the knots are undone, take
to your boats and return to your homes!”
In Palestine, in the second century CE, Roman tax-collectors used a
“great cable”, probably made up of a collection of strings, as their register.
In addition, receipts for taxes paid took the form of a piece of string knotted
in a particular way.
Arabs also used knotted string over a long period of time not only as a
concrete counting device, but also for making contracts, for giving receipts,
and for administrative book-keeping. In Arabic, moreover, the word aqd,
meaning “knot”, also means “contract”, as well as any class of numbers
constituted by the products of the nine units to any power of ten (several
Arabic mathematicians refer to the aqd of the hundreds, the aqd of the
thousands, and so on).
70
The Chinese were also probably familiar with knotted-string numbers in
ancient times before writing was invented or widespread. The semi-
legendary Shen Nong, one of the three emperors traditionally credited with
founding Chinese civilisation, is supposed to have had a role in developing
a counting system based on knots and in propagating its use for book-
keeping and for chronicles of events. References to a system reminiscent of
Peruvian quipus can be found in the I Ching (around 500 BCE) and in the
Tao Te Ching, traditionally attributed to Lao Tse.
The practice is still extant in the Far East, notably in the Ryu-Kyu Islands.
On Okinawa, workers in some of the more mountainous areas use plaited
straw to keep a record of days worked, money owed to them, etc. At Shuri,
moneylenders keep their accounts by means of a long piece of reed or bark
to which another string is tied at the middle. Knots made in the upper half
of the main “string” signify the date of the loan, and on the lower half, the
amount. On Yaeyama, harvest tallies were kept in similar fashion; and
taxpayers received, in lieu of a written “notice to pay", a piece of string so
knotted as to indicate the amount due [J. G. Fevrier (1959)].
YEN
SEN
RIN
hundreds
tens
units
tens
units
units
3 x 100 yen
5 x 10 yen
(5 + 1) yen
(5 + 3) x 10 sen
5 sen
5 rin
Fig. 6.8. A sum of money as expressed in knotted string in the style used by workers on Okinawa
and tax-collectors on Yaeyama. The figure shows 356 yen, 85 sen and 5 rin (lyen = 100 sen, 1 sen =
10 rin). The number 5 is represented by a knot at the end of the trailing straw. See also Chapter 25,
Fig. 25.9.
The same general device can be found in the Caroline Islands, in
Hawaii, in West Africa (specifically amongst the Yebus, who live in the
hinterland of Lagos (Nigeria)), and also at the other ends of the world,
amongst native Americans such as the Yakima (eastern Washington State),
the Walapai and the Havasupai (Arizona), the Miwok and Maidu (North
& South Carolina), and of course amongst the Apache and Zuni Indians
of New Mexico.
A bizarre survival of the formerly wide role of knotted string was to
be found as late as the end of the last century amongst German flour-
millers. who kept records of their dealings with bakers by means of rope
(see Fig. 6.9 below). Similarly, knotted-string rosaries (like their beaded and
71
NUMBERS ON STRINGS
notched counterparts), for keeping count of prayers, are common to many
religions. Tibetan monks, for example, count out the one hundred and eight
unities (the number 108 is considered a sacred number) on a bunch of 108
knotted strings (or a string of 108 beads) whose colour varies with the deity
to be invoked: yellow string (or beads) for prayers to Buddha ; white string
(or white beads made from shells) for Bodhisattva-, red strings (or coral
beads) for the one who converted Tibet-, etc. A very similar practice was
current only a few decades ago amongst various Siberian tribes (Voguls,
Ostyaks, Tungus, Yakuts, etc.); and there is also a Muslim tradition, handed
down by Ibn Sa‘ad, according to which Fatima, Mohammed’s daughter,
counted out the 99 attributes of Allah and the supererogatory laudations
on a piece of knotted string, not on a bead rosary.
Fig. 6.9. German millers' counting device using knotted rope (the system in force at Baden in the
nineteenth century is illustrated)
For morning prayers ( Shahrit ) and other services in the synagogue, Jews
wear a prayer-shawl (tal it) adorned with fringes ( tsitsit ). Now, the four
corner-threads of the fringe are always tied into a quite precise number of
knots: 26 amongst “Eastern” (Sephardic) Jews, and 39 amongst “Western”
(Ashkenazi) Jews. The number 26 corresponds to the numerical value of the
Hebrew letters which make up the name of God, YHWH (see below,
Chapters 17 and 20, for more detail on letter-counting systems), and 39 is
the total of the number-values of the letters in the expression “God is One”,
YHWH EHD (see below). 39 is also the “value” of the Hebrew word
meaning “morning dew” ( tal ), and rabbis have often commented that at
prayer the religious Jew is able to hear the word of God “which falls from
his mouth as morning dew falls on the grass”.
n 1 n "
YHWH
5 + 6+5+10
= 26
Yahwe, “the Lord”
1 n k n *1 n '
YHWH EHD
4 + 8 + 1 + 5 + 6 + 5 + 10
= 39
Yahwe ehad, “the Lord is One"
•5 D
T L
30 + 9
= 39
tal, “morning dew”
Fig. 6.11.
Knotted string has thus served not only as a device for concrete
numeration, but also as a mnemotechnic tool (for recording numbers,
maintaining administrative archives, keeping count of contracts, calendars,
etc.). Although knotted string does not constitute a form of writing in the
strict sense, it has performed all of writing’s main functions - to preserve
the past and to ensure the survival of contracts between members of the
same society. Numbers on strings can therefore be considered for our
purposes as a special form of written numbers.
Fig. 6.10. The bands and fringe of prayer-shawl
NUMBER, VALUE AND MONEY
72
CHAPTER 7
NUMBER, VALUE AND MONEY
At a time when people lived in small groups, and could find what they
needed in the nature around them, there would have been little need for
different communities to communicate with each other. However, once
some sort of culture developed, and people began to craft objects of use or
desire, then, because the raw resources of nature are unequally distributed,
trade and exchange became necessary.
The earliest form of commercial exchange was barter, in which people
exchange one sort of foodstuff or goods directly for another, without
making use of anything resembling our modern notion of “money”. On
occasion, if the two parties to the exchange were not on friendly terms,
these exchanges took the form of silent barter. One side would go to an
agreed place, and leave there the goods on offer. Next day, in their place or
beside them, would be found the goods offered in exchange by the other
side. Take it or leave it: if the exchange was considered acceptable, the
goods offered in exchange would be taken away and the deal was done.
However, if the offer was not acceptable then the first side would go away,
and come back next day hoping to find a better offer. This could go on for
several days, or even end without a settlement.
Among the Aranda of Australia, the Vedda of Ceylon, Bushmen and
Pygmies of Africa, the Botocoudos in Brazil, in Siberia, in Polynesia - such
transactions have been observed. But with growth in communication, and
the increasing importance of trade, barter became increasingly inconve-
nient, depending as it does on the whims of individuals or on interminable
negotiations.
The need grew, therefore, for a stable system of equivalences of value. This
would be defined (much as numbers are expressed in terms of a base) in
terms of certain fixed units or standards of exchange. With such a system it
is not only possible to evaluate the transactions of trade and commerce, but
also to settle social matters - such as “bride price” or “blood money” - so
that, for instance, a woman would be worth so many of a certain good as a
bride, the reparation for a robbery so many. In pre-Hellenic Greece, the
earliest unit of exchange that we find is the ox. According to Homer’s
Iliad (XXIII, 705, 749-751; VI, 236; eighth century BCE), a “woman good
for a thousand tasks” was worth four oxen, the bronze armour of Glaucos
was worth nine, and that of Diomedes (in gold) was worth 100. And, in
decreasing order of value, are given: a chased silver cup, an ox, and half
a golden talent. The Latin word pecunia (money), from which we get “pecu-
niary”, comes from pecus, meaning “cattle”; and the related word peculium
means “personal property”, from which we also get "peculiar". In fact the
strict sense of pecunia is “stock of cattle”. The English word fee has come to us
partly from Old English feoh meaning both “cattle” and “property” which
itself is believed to be derived via a Germanic root from pecus (compare
modern German Vich, “livestock”), and partly from Anglo-French fee which
is probably also of similar Germanic derivation. Like the Sanskrit rupa
(whence “rupee”), these words remind us of a time when property, recom-
pense, offerings, and ritual sacrifices were evaluated in heads of cattle. In
some parts of East Africa, the dowry of a bride is counted in cattle. The
Latin capita (“head”) has given us “capital”. In Hebrew, keseph means both
“sheep” and “money”; and the root-word made of the letters GML stands
for both “camel” and “wages”.
In ancient times, however, barter was a far from simple affair. It was
surrounded by complicated formalities, which were probably associated
with mysticism and magical practices, as is confirmed by ethnological study
of contemporary “primitive” societies and by archaeological findings. We
may imagine, therefore, that in pastoral societies the concept of the “ox
standard” grew out of the "ox for the sacrifice” which itself depended on the
intrinsic value attributed to the animal.
L. Hambis (1963-64), describing certain parts of Siberia, says “Buying
and selling was still done by barter, using animal pelts as a sort of monetary
unit; this system was employed by the Russian government until 1917 as a
means of levying taxes on the people of these parts.” In the Pacific islands,
on the other hand, goods were valued not in terms of livestock but in terms
of pearl or sea-shell necklaces. The Iroquois, Algonquin, and other north-
east American Indians used strings of shells called wampum. Until recently,
the Dogon of Mali used cowrie shells. One Ogotemmeli, interviewed by
M. Griaule (1966), says “a chicken is worth three times eighty cowries, a
goat or a sheep three times eight hundred, a donkey forty times eight
hundred, a horse eighty times eight hundred, an ox one hundred and
twenty times eight hundred.” “But” continues Griaule, “in earlier times the
unit of exchange was not the cowrie. At first, people bartered strips of cloth
for animals or goods. The cloth was their money. The unit was the palm’
of a strip of cloth twice eighty threads wide. So a sheep was worth eight
cubits of three ‘palms’ . . . Subsequently, values were laid down in terms of
cowries by Nommo the Seventh, Master of the Word."
With some differences of detail, practices were similar in pre-Columbian
Central America. The Maya used also cotton, cocoa, bitumen, jade, pots,
pearls, stones, jewels, and gold. For the Aztecs, according to J. Soustelle,
73
NUMBER, VALUE AND MONEY
Fig. 7.1. Tunic worn in the nineteenth century by members of the Tyal tribe in Formosa. More
than 2,500 precious stones are attached in bands, at the edges and on either side of the centre line.
Such tunics were used as “money " in buying livestock and in the trade in young women. The bands
of precious stones could be detached separately to serve as pocket money for everyday purchases.
New York, Chase Manhattan Bank Museum of Money
“certain foodstuffs, goods or objects were employed as standards of value
and as tokens of exchange: the quachtli (a piece of cloth) and ‘the load’ (20
quachtlis)-, the cocoa bean used as ‘small change’ and the xiquipilli (a bag of
8,000 beans); little T-shaped axes of copper; feather quills filled with gold.”
The same kind of economy was practised in China prior to the adoption of
money in the modern sense. In the beginning, foodstuffs and goods were
exchanged, their value being expressed in terms of certain raw materials, or
certain necessities of life, which were adopted as standards. These might
include the teeth and horns of animals, tortoise shells, sea-shells, hides, or
fur pelts. Later, weapons and utensils were adopted as tokens of value:
knives, shovels, etc. These would at first have been made of stone, but later,
from the Shang Dynasty, of bronze (sixteenth to eleventh century BCE).
However, regular use of such kinds of items was cumbersome and not
always easy. As a result, metal played an increasingly important role, in the
form of blocks or ingots, or fashioned into tools, ornaments or weapons,
until finally metal tokens were adopted as money in preference to other
forms, for the purposes of buying and selling. The value of a merchandise
was measured in terms of weight, with reference to a standard weight of
one metal or another.
Fig. 7.2. Bronze “knife” from the Zhou possibly buy a fowl; five or six would be the
period, used as a unit of barter in China; price of a slave. New York, Chase Manhattan
approximately 1000 BCE. Beijing Museum Bank Museum of Money
Thus it was that “When Abraham purchased the Makpelah Cave, he
weighed out four hundred silver shekels for Ephron the Hittite.”* Later on
Saul, seeking his father’s she-asses, sought the help of a seer for which he
gave one quarter of a shekel of silver (I Samuel IX, 8). Similarly, the fines
laid down in the Code of the Alliance were stipulated in shekels of silver, as
also was the poll tax (Exodus XXX, 12-15) [A. Negev (1970)].
In the Egypt of the Pharaohs likewise, foodstuffs and goods were often
valued, and paid for, with metal (copper, bronze, sometimes gold or silver)
measured out in nuggets or in flakes, or given in the form of bars or rings
which were measured by weight. The principal standard of weight was the
deben, equivalent to 91 grams of our measure. For certain purchases, value
was determined in certain fractions of the deben. For example, in the Old
Kingdom (2780-2280 BCE) the shat, one twelfth of a deben, was used (equi-
valent, therefore, to 7.6 grams). In the New Kingdom (1552-1070 BCE)
the shat gave way to the qat, one tenth of the deben or 9.1 grams.
In a contract from the Old Kingdom we can see how value was expressed
in terms of the shat. According to this, the rent of a servant was to be paid
as follows, the values being in shats of bronze:
8 bags of grain
value
5
shats
6 goats
value
3
shats
silver
value
5
shats
Total
value
13
shats
* The Old Testament shekel is equivalent to 11.4 grams of our measure.
NUMBER, VALUE AND MONEY
74
As another example, the following account from the New Kingdom
shows debens of copper being used as a standard of value.
Sold to Hay by Nebsman the Brigadier:
1 ox, worth 120 debens of copper
Received in exchange:
2 pots of fat, value 60 debens
5 loin-cloths in fine cloth, worth 25 debens
1 vestment of southern flax, worth 20 debens
1 hide, worth 15 debens
In this example we can see how goods could be used in payment as well as
metal tokens in the marketplace of ancient times. That ox, for instance, cost
120 debens of copper, but not one piece of real metal had changed hands:
60 of the debens owing had been settled by handing over 2 pots of fat, 25
more with 5 loin-cloths, and so on.
Although goods had been exchanged for goods, therefore, this was not
a straightforward barter. It in fact reflected a real monetary system.
Thenceforth, by virtue of the metal standard, goods were no longer
bartered at the whim of the dealers or according to arbitrary established
practice, but in terms of their “market price”.
There is a letter dating from around 1800 BCE which gives a vivid
illustration of these matters. It comes from the Royal Archives of the town
of Mari, and was sent by Iskhi-Addu, King of Qatna, to Isme-Dagan, King
of Ekallatim. Iskhi-Addu roundly reproaches his “brother” for sending a
meagre “sum” in pewter, in payment for two horses worth several times
that amount.
Thus [speaks] Iskhi-Addu thy brother:
This should not have to be said! But speak I must, to console my
heart. . . . Thou hast asked of me the two horses that thou didst desire,
and I did have them sent to thee. And see! how thou hast sent to me
merely twenty rods of pewter! Didst thou not gain thy whole desire
from me without demur? And yet thou dare’st send me so little
pewter! . . . Know thou that here in Qatna, these horses are worth six
hundred shekels of silver. And see, how thou hast sent me but twenty
rods of pewter! What will they say of this, when they hear of it?
An understandable indignation, since a shekel of silver was worth three
or four rods of pewter at the time.
It should not be thought, though, that “money”, in the modern sense
of the word, was used in payment in those times. It was not a “coinage”
in the sense of pieces of metal, die-cast in a mint which is the prerogative
of the State, and guaranteed in weight and value. The idea of a coinage
sound in weight and alloy did not come about until the first millennium
BCE, most probably with the Lydians. Until that time, only a kind of
“base-weight” played a role in transactions and in legal deeds, acting as
a unit of value in terms of which the prices of individual items of merchan-
dise, or individual deeds, could be expressed. On this basis, this or that
metal was first counted out in ingots, rings, or other objects, and then its
weight, in units of the “base-weight”, was determined, and in this way could
be used as “salary”, “fine”, or “exchange”.
Let us go back a few thousand years and, in the description of Maspero,
observe a market from Egypt of the Pharaohs.
Early in the morning endless streams of peasants come in from the
surrounding country, and set up their stalls in the spots reserved for
them as long as anyone can remember. Sheep, geese, goats and wide-
horned oxen are gathered in the centre to await buyers. Market
gardeners, fishermen, fowlers and gazelle hunters, potters and crafts-
men squat at the roadside and beside the houses, their goods heaped
in wicker baskets or on low tables, fruits and vegetables, fresh-baked
bread and cakes, meats raw or variously prepared, cloths, perfumes,
jewels, the necessities and the frivolities of life, all set out before the
curious eyes of their customers. Low and middle class alike can provide
for themselves at lower cost than in the regular shops, and take advan-
tage of it according to their means.
The buyers have brought with them various products of their own
labours, new tools, shoes, mats, pots of lotion, flasks of drink, strings
of cowrie shells or little boxes of copper or silver or even golden rings
each weighing one deben* which they will offer to exchange for the
things they need.
For purchase of a large beast, or of objects of great value, loud, bitter
and protracted arguments take place. Not only the price, but in what
species the price shall be paid, must be settled, so they draw up lists
whereon beds, rods, honey, oil, pick-axes or items of clothing may
make up the value of a bull or a she-ass.
Fig. 7.4. Brass ingot formerly used as
monetary standard in the black slave market
of the West African coast. New York, Chase
Manhattan Bank Museum of Money
* Maspero uses tabnou , here replaced by the the more precise term deben.
75
NUMBER, VALUE AND MONEY
The retail trading does not involve so much complicated reckoning.
Two townsmen have stopped at the same moment in front of a fellah
with onions and corn displayed in his basket.* The first's liquid assets
are two necklaces of glass pearls or coloured enamelled clay; the
second one has a round fan with a wooden handle, and also one of
those triangular fans which cooks use to boost the fire.
Fig. 7 . 5 . Market scenes in an Egyptian funeral painting of the Old Kingdom, Fifih or Sixth
Dynasty (around 2500 BCE). The painting adorns the tomb of Feteka at the northern end of the
necropolis of Saqqara (between Abusir and Saqqara). See Lepsius (1854-59), vol. II, page 96
(Tomb no. 1), and Porter & Moss (1927-51), vol. 3 part 1, page 351.
“This necklace would really suit you,” calls the first, “it's just your
style!”
“Here is a fan for your lady and a fan for your fire,” says the
other.
Still, the fellah calmly and methodically takes one of the necklaces
to examine it:
“Let’s have a look, I’ll tell you what it’s worth.”
With one side offering too little, and the other asking for too much,
they proceed by giving here and taking there, and finally agree on the
number of onions or the amount of grain which will just match
the value of the necklace or the fan.
Further along, a shopper wants some perfume in exchange for a pair
of sandals and cries his wares heartily:
“Look, fine solid shoes for your feet!”
* Some of the scenes described can be seen on an Egyptian funeral painting from the Old Kingdom,
reproduced here in Fig. 7.5.
But the merchant is not short of footwear just now, so he asks for
a string of cowries for his little jars:
“See how sweet it smells when you put a few drops around!” he says
winningly.
A woman passes two earthen pots, probably of ointment she has
made, beneath the nose of a squatting man.
“This lovely scent will catch your fancy!”
Behind this group, two men argue the relative worth of a bracelet
and a packet of fish-hooks; and a women with a small box in her hand
is negotiating with a man selling necklaces; another woman is trying to
get a lower price on a fish which the seller is trimming for her.
Barter against metal requires two or three more stages than simple
barter. The rings or the folded sheets which represent debens do not
always have the standard content of gold or silver, and may be of short
weight. So they must be weighed for each transaction to establish
their real value, which offers the perfect opportunity for those con-
cerned to enter into heated dispute. After they have passed a quarter
of an hour yelling that the scales do not work, that the weighing has
been messed up, that they have to start all over again, they finally
weary of the struggle and come to a settlement which roughly satisfies
both sides.
However, sometimes someone cunning or unscrupulous will
adulterate the rings by mixing their precious metal with as much
false metal as possible short of making their trickery apparent. An
honest trader who is under the impression that he received a payment
of eight gold debens, who was in fact paid in metal which was one
third silver, has unwittingly lost almost one third of his part. Fear of
being cheated in this way held back the common use of debens for a
long time, and caused the use of produce and artisanal objects in
barter to be maintained.
At the end of the day, the use of money (in the modern sense of the term)
became established once the metal was cast into small blocks or coins,
which could be easily handled, had constant weight, and were marked with
the official stamp of a public authority who had the sole right to certify
good weight and sound metal.
This ideal system of exchange in commercial transactions was invented
in Greece and Anatolia during the seventh century BCE. (In China the
earliest similar usage occurred also at about the same time, apparently,
around 600-700 BCE, during the Chow Dynasty.) Who might have first
thought of it? Some consider that Pheidon, king of Argos in the
Peloponnese, introduced the system in his own city and in /Egina, around
650 BCE. However, the majority of scholars agree that the honour of the
NUMBER, VALUE AND MONEY
76
invention should go to Asia Minor under the Greeks, most probably
to Lydia.
Be that as it may, the many advantages of the use of coins led to its rapid
adoption in Greece and Rome, and amongst many other peoples. The rest
is another story.
Fig. 7 .6. Greek coins.
Left: silver tetradrachma
from Agrigento. around
415 BCE.
Right: tetradrachma from
Syracuse, around 310 BCE.
Agrigento Museum
By learning how to count in the abstract, grouping every kind of thing
according to the principle of numerical base, people also learned how to
estimate, evaluate and measure all sorts of magnitudes - weights, lengths,
areas, volumes, capacities and so on. They likewise managed to conceive
ever larger numbers, though they could not yet attain the concept of
infinity. They worked out many technical procedures (mental, material and
later written), and laid the early foundations of arithmetic which, at first,
was purely practical and only later became abstract and led on to algebra.
The way also opened up for the devising of a calendar, for a systemisa-
tion of astronomy, and for the development of a geometry which was at
first based straightforwardly on measurement of length, area and volume,
before becoming theoretical and axiomatic. In short, the grasp of these
fundamental data allowed the human race to attempt the measurement
of its world, little by little to understand it better and better, to press
into humanity’s service some of their world’s innumerable secrets, and to
organise and to develop their economy.
77
writing: the INVENTION OF SUMER
CHAPTER 8
NUMBERS OF SUMER
writing: the invention OF SUMER
Writing, as a system enabling articulated speech to be recorded, is beyond
all doubt among the most potent intellectual tools of modern man. Writing
perfectly meets the need (which every person in any advanced social group
feels) for visual representation and the preservation of thought (which of its
nature would otherwise evanesce). It also offers a remarkable method of
expression and of preservation of communication, so that anyone can keep
a permanent record of words long since spoken and flown. However, it is
much more than a mere instrument.
By recording speech in silent form, writing does not merely conserve it,
but also stimulates thought such as, otherwise, would have remained
latent. The simplest of marks made on stone or paper are not just a
tool: they entomb old thoughts, but also bring them back to life. As
well as fixing language, writing is also a new language, silent perhaps,
which lays a discipline on thought and, in transcribing it, organises
it. . . . Writing is not only a means of durable expression: it also gives
direct access to the world of ideas. It faithfully represents the spoken
word, but it also facilitates the understanding of thought and gives
thought the means to traverse both space and time. [C. Higounet
(1969)]
Writing, therefore, in revolutionising human life, is one of the greatest of
all inventions. The earliest known writing appeared around 3000 BCE, not
far from the Persian Gulf, in the land of Sumer, which lay in Lower
Mesopotamia between the Tigris and Euphrates rivers. Here also were
developed the earliest agriculture, the earliest technology, the first towns
and cities, by the Sumerians, a non-Semitic people of still obscure origins.
As evidence of this we have numerous documents known as “tablets”
which were used as a kind of “paper” by the inhabitants of this region. The
oldest of these (which also carry the most archaic form of the writing) were
discovered at the site of Uruk,* more precisely at the archaeological level
designated as Uruk IVal
These tablets are, in fact, small plaques of dry clay, roughly rectangular
* The royal city of Uruk was situated south of Lower Mesopotamia on the Iraqi site of Warka (now about
twenty km north of the Euphrates). It has given its name to the epoch in which, it is presumed, the Sumerian
people first appeared in the region and in which writing was invented in Mesopotamia.
ATU 111
(E)
ATU 111 ATU 264
Fig. 8 . 1 . Archaic Sumerian tablets, discovered at Uruk (level IVa). They are among the earliest
known instances of Sumerian writing. Several of these tablets are divided by horizontal and vertical
lines into panels which contain numbers and signs representing writing (which already seem to
follow a standard pattern). These indicate a degree of precise analytical thought, composed of
separate elements brought together, as in articulate speech. The Iraqi Museum, Baghdad
in outline and convex on their two faces (see Fig. 8.1). On one side, some-
times on both, they bear hollowed-out markings of various shapes and
sizes. These marks were made on the clay while still soft by the pressure of
a particular tool. As well as these hollow markings we may also find outline
drawings made with a pointed tool, representing all kinds of things or
t The best known of the Sumerian archaeological sites, and the first to be excavated, Uruk has served to
establish a “time scale” for this civilisation. In certain sectors deep excavation has revealed a series of strata
to which archaeologists refer to determine approximate dates for their finds: the ordering of the different
layers, from top to bottom, corresponds to the different stages in the history of the civilisation.
NUMBERS OF SUMER
78
beings. The hollow markings correspond to the different units in the
Sumerian sequence of enumeration (in the archaic graphology); they are,
therefore, the most ancient “figures” known in history (see Fig. 8.2). The
drawings are simply the characters in the archaic writing system of Sumeria
(Fig. 8.3).
Some of these tablets also have symbolic motifs in relief, made by rolling
cylindrical seals over the surface of the tablet, from one end to the other.
ft
€
w
w
C
0
Narrow
notch
Small
circular
indentation
Thick notch
Thick notch with
small circular
indentation
Large circular
indentation
Large circular
indentation with
small circular
indentation
Fig. 8.2. The shapes of archaic Sumerian numbers
These tablets seem to have served as records of various quantities
associated with different kinds of goods - invoices, as it were, for supplies,
deliveries, inventories, or exchanges. Let us have a closer look at the draw-
ings on these tablets, and try to discern the principal character of this
writing system. Some of these drawings are very realistic and show the
essential outlines of material objects, which may be quite complex (Fig. 8.3).
On occasion, the drawings are much simplified, but still strongly evoke
their subject. For example, the heads of the ox, the ass, the pig, and the dog
are drawn in a concrete though very stylised way, and the drawing of the
animal’s head stands for the animal itself.
More often, however, the original object is no longer directly recognis-
able; the part stands for the whole, and effect represents cause, in a stylised
and condensed symbolism. A woman, for instance, is represented by a
schematic drawing of the triangle of pubic hair (Fig. 8.3 F), and the verb to
impregnate by a drawing of a penis (Fig. 8.3 E).
Generally speaking, as a result of these abbreviations and the subtly
simplified relation between representation and object represented, the
latter mostly eludes us. The symbols are simple geometric drawings, and
the represented objects (where we can determine what they are, by seman-
tic or palaeographic means) have little apparently in common with their
representations. Consider the sign for a sheep, for example (Fig. 8.3 U):
what might this drawing possibly represent, a circle surrounding a cross?
A sheep-pen? A brand? We have no idea.
What is striking about these drawings is their constant and definite
character* in which each particular symbol exhibits little variation of form.
* This means that the design has been finalised once and for all, so that "writing” implies choosing and
setting up a repertoire of generally accepted and recognised symbols.
A
B
c
D
f
I?P
f
bird
reed
head, chief;
summit, thigh
haunch
E
F
G
H
0
V
1
mountain,
penis, fertilise
pubis, woman
palm tree, date
foreign land
1
j
K
L
9
®
fountain,
water or
eye, to look
well, water-butt
stream, wave
fish
M
N
p
Q
= 3 %.
hand, fist
plough
pig, boar
P‘g
R
s
T
U
A
y
©
ass, horse
ox
dog
sheep
V
w
X
Y
9
I
goat
stock-pound
man
fire, fire, light
Fig. 8.3. Pictograms from archaic Sumerian writing
Comparing this with the number of variations which will emerge in
subsequent periods, we are obliged to see in this constancy and regularity
the mark of true writing - in the sense of a fully worked-out system which
everyone has adopted - and therefore to consider that we are seeing the
very origin of writing or, at any rate, its earliest stages, based no doubt on
earlier usages but bearing this essential new feature of being a generally
accepted uniform practice.
We find ourselves contemplating, therefore, a system of graphical
79
writing: the invention OF SUMER
Fig . 8.4. Some examples of the evocative “logical aggregates" used in archaic Sumerian writing
symbols intended to express the precise thoughts which occur in speech.
However, it is still not writing in the strict and full sense:* we are still in the
“prehistory”, or rather the “protohistory”, of the development of writing
(that is, in the pictographic stage).
All of these symbols, whether we know what they mean or not, are
graphical representations of material objects.
But we should still not conclude that they can only represent material
objects. Each object can be used to symbolise not only the activities or
actions directly implied by the object, but also related concepts. The
leg, for example, can also represent “walking”, “going” or “standing up”; the
hand can stand for “taking”, “giving”, “receiving” (Fig. 8.3 M); the rising
sun for “day”, “light” or “brightness”; the plough for “ploughing”, “sowing”,
“digging” (Fig. 8.3 N) and, by extension, “ploughman”, “farmer”, and so on.
The scope of each ideogram can also be extended by a device which had
already, at this time, been long applied to symbolism. Two parallel lines can
represent the idea of “friend” and “friendship”; two lines crossing each
other, the idea of “enemy” or “hostility”. The Sumerians gave great place
to this idea of enlarging the possible range of meanings of their drawings
by combining two or more together to represent new ideas, or aspects of
* In the strict sense of the word, a mere visual representation of thought by means of symbols of material
objects cannot be considered true writing, since it is more closely related to spoken words than to thought
itself. For it to be considered true writing, it would in addition need to be a systematic representation of
spoken language, since writing, like language, is a system and not a random sequence of items. Fevrier (1959)
says: "Writing is a system for human communication using well-defined conventional symbols for the repre-
sentation of language, which can be transmitted and received, which can be equally well understood by both
parties, and which are associated with the words of the spoken language.”
reality otherwise hard to express. The combination mouth + bread thereby
expresses “eat”, “devour”; mouth + water expresses “drink”; mouth + hand
expresses “prayer” (in accordance with Sumerian ritual); and eye + water
denotes “tears”, “weeping”.
In the same way, an egg beside fowl suggests the act of “giving birth”,
strokes underneath a semi-circle suggest darkness falling from the heavenly
vault, “night”, “the dark”. In that flat, lowland country, where “mountain”
was synonymous with foreign lands and enemy country (Fig. 8.3 H), the
juxtaposition woman + mountain meant “foreign woman” (literally “woman
from the mountains”) and therefore, by extension, “female slave” or “maid-
servant” (since women were brought to Sumer, bought or captured, to serve
as slaves). The same association of ideas gave rise to the combination
man + mountain to denote a male slave (Fig. 8.4 F).
Human thought could therefore be better expressed by this system of
pictograms and ideograms than by a purely representational visual art.
This system was a systematic attempt to express the whole of thought in
the same way as it was represented and dissected in spoken language. But
it was still far from perfect, being a long way yet from being able to denote
with precision, and without ambiguity, everything that could be expressed
in spoken language. Because it depended excessively on the material world
of objects which could be drawn as pictures, it required a very large number
of different symbols. In fact, the total number of symbols used in this
first age of writing in Mesopotamia has been estimated to be about two
thousand.
Furthermore, not only was this writing system difficult to manipulate,
it was also seriously ambiguous. If, for example, plough can also mean
“ploughman”, how are we supposed to know which one is meant? Even for
one and the same word, how can its various nuances be distinguished -
nuances which language can meticulously encapsulate and which are essen-
tial to complete understanding of the thought (including such qualities as
gender, singular and plural, quality, and the countless relationships
between things in time and in space)? How can one distinguish the many
ways in which actions vary with time?
This writing was certainly a step towards representing spoken language,
but it was limited to what could be expressed in images, that is to say to the
immediately representable aspects of objects and actions, or to their imme-
diately cognate extensions. For such reasons the original Sumerian writing
remains, and will no doubt always remain, undecipherable. Consider
the bull’s head in Fig. 8.1 D. Is it really “the head of a bull”? Or is it - more
plausibly - “a bull”, a unit of livestock (“one ox”), one of the many products
one can obtain from cattle (leather, milk, horn, meat)? Or does it represent
some person who may have had a name on the lines of “Mr Bull” (thus
NUMBERS OE SUMER
80
being the equivalent of a signature)? Only the few people immediately
implicated would be in a position to know what exactly was intended by the
bull’s head on this particular tablet.
In these circumstances, Sumerian writing at this stage of development is
better thought of as an aide-memoire than as a written record in the proper
sense of the term: something which served to help people recall what they
already exactly knew (possibly missing out some essential detail), rather
than something which could exactly express this to someone who had never
known it directly.
Such a scheme answered the purposes of the time well enough. Apart
from a few “lists of symbols”, all of the known archaic Sumerian tablets
carry summaries of administrative actions or of exchanges, as we can see
from the totalled numbers which can be found at the end of the document
(or on its other side). All of these tablets are, therefore, accounts (in the
financial sense of the term). Pure economic necessity therefore played,
beyond doubt, a leading role in the story:* the emergence of this writing
system was undoubtedly inspired by the necessities of accounting and
stocktaking, which caused the Sumerians to become aware of the fact that
the old order, which was still based on a purely oral tradition, was running
out of steam and that a completely new approach to the organisation of
work was called for.
As P. Amiet explains, “Writing was invented by accountants faced with
the task of noting economic transactions which, in the rapidly developing
Sumerian society, had become too numerous and too complex to be merely
entrusted to memory. Writing bears witness to a radical transformation of
the traditional way of life, in a novel social and political environment
already heralded by the great constructions of the preceding era.” At that
time the temples were solely responsible for the economy of all Sumer,
where continual over-production required a very centralised system of
redistribution which became increasingly complicated, a situation which
undoubtedly gave rise to the invention of writing. But accounting is simply
the recording, by rote or by writing, of operations which have already taken
place, and which concern solely the displacements of objects and of people.
According to J. Bottero, archaic Sumerian writing is perfectly adapted to
this function, which is the reason why its earliest form - which had a
profound effect on later developments - was such as to serve above all as an
aide-memoire.
In order, however, to become completely intelligible, and above all
in order to attain the status of “writing” in the true sense of the word
* Does the development of this writing have solely an economic explanation? Did not different needs (reli-
gious, divinatory, even literary) also play a part? Did people not communicate with each other at a distance
in writing, for instance? There are those who think so; but so far no archaeological find has lent support to
such possibilities.
(i.e. capable of recording unambiguously whatever could be expressed in
language), this archaic picto-ideography was therefore obliged to make
great advances not only in clarity and precision, but also in universality of
reference.
This transition began to occur around 2800-2700 BCE, at which time
Sumerian writing became allied to spoken language (which is the most
developed way of analysing and communicating reality).
The idea at the root of this development was to use the picture-signs, no
longer merely pictorially or ideographically, but phonetically, by relating
them to spoken Sumerian, somewhat as in our picture-puzzles, where a
phrase is punningly represented by objects whose names form parts of the
sounds in the spoken phrase. For example, a picture showing a needle and
thread being used to sew a bunch of thyme, a goalkeeper blocking a goal-
kick, and the digit “9”, could (in English) represent the saying “A stitch in
time saves nine”.
Thus a picture of an oven is at this time (2800-2700 BCE) no longer used
to represent the object, but rather to represent the sound ne, which is the
Sumerian word for “oven”. Likewise, a picture of an arrow (in Sumerian ti
stands for the sound ti; and since the word for “life” in Sumerian is also
pronounced ti the arrow picture also stands for this word. As Bottero
explains: “Using the pictogram of the arrow ( ti ) to denote something quite
different which is also pronounced ti (‘life’) completely breaks the primary
relation of the image to the object (arrow) and transfers it to the phoneme
(ti); to something, that is, which is not situated in the material world but is
inherent solely in spoken language, and has a more universal nature. For
while the arrow, purely as a pictogram, can only refer to the object ‘arrow’
and possibly to a limited group of related things (weapon, shooting,
hunting, etc.), the sound ti denotes precisely the phoneme, no matter where
it may be encountered in speech and without reference to any material
object whatever, and corresponds solely to this word, or to this part of
a word (as in ti-bi-ra, ‘blacksmith’). The sign of the arrow is therefore
no longer a pictogram (it depicts nothing) but a phonogram (evoking a
phoneme). The graphical system no longer serves to write things, but to
write words, and it no longer communicates one single idea, but the whole
of speech and language.”
This represents an enormous advance, because such a system is now
capable of representing the various grammatical parts of speech: pronouns,
articles, prefixes, suffixes, nouns, verbs, and phrases, together with all the
nuances and qualifications which can hardly, if at all, be represented in any
other way. “As such,” adds Bottero, “even if this now means that the reader
must know the language of the writer in order to understand, the system
can record whatever the spoken language expresses, exactly as it is
81
THE SUMERIANS
expressed: the system no longer serves merely as a record to assist memory
and recall, but can also inform and instruct.”
It is not our business to go into the specific details of the language for
which the Sumerians developed their graphical system, once they had
reached the phonetic stage in the above way. But we may echo Bottero in
saying that Sumerian writing (enormous advance though it was), because
it was born of a pictography designed to aid and extend the memory,
remained fundamentally a way of writing words: an aide-memoire devel-
oped into a system, enhanced by the extension into phonetics, but not
essentially transformed by it. (After the entry of the phonetic aspect, the
Sumerians in fact kept many of their archaic ideograms of which each one
continued to denote a word designating a specific entity or object, or even
several words connected by more or less subtle relations of meaning,
causality or symbolism.)
THE SUMERIANS
(Adapted from G. Rachet’s Dictionnaire de I’archeologie)
The geographical origins of the Sumerians remain a topic of controversy.
Though some would have them originate from Asia Minor, it seems rather
that they arrived in Lower Mesopotamia from Iran, having come from
central Asia.
Their language, which remains imperfectly known, was agglutinative,
like the early Asiatic (pre-Semitic and pre-Indo-European) languages, and
the Caucasian and Turco-Mongolian languages of today. In any case, wher-
ever they came from was mountainous, as is shown by two things which
they brought with them to south Mesopotamia: the ziggurat, a relic of
ancient mountain religions, and stone-carving; whereas the Mesopotamian
region is bare of stone.
Their most likely date of arrival in Mesopotamia can be placed in the
so-called Uruk period, during the second half of the fourth millennium BCE,
either during the Uruk IV period, or that of Uruk V. Quite possibly they
arrived gradually, in minor waves, thereby leaving no archaeological traces
for the whole of the Uruk period. It certainly seems that this city, home of
the epic hero Gilgamesh, had been the primordial centre of the culture they
bore. And it is certain the so-called Jemdet Nasr period began under their
initiative, at the end of the fourth millennium BCE, to be followed by the
pre-Sargonic era or Ancient Dynasty which saw the first culmination of
Sumerian civilisation.
These periods were marked by three cultural manifestations: the devel-
opment of glyptics (where cylinders engraved with parades of animals, and
various scenes of a religious nature are dominant among the tablets);
the development of sculpture with relief on stone vases, animals and
personages in the round, themes treated with great mastery and with a
force which did not exclude elegance, the masterpiece of this period being
the mask, known as the Lady of Warka, imbued with a delicate realism;
finally, the emergence of writing which, if it has not given us annals, allows
us to identify the gods to whom the temples were dedicated and to learn the
names of certain personages, in particular those which have been found in
the royal tombs of Ur.
The towns of the land of Sumer: Ur, Uruk, Lagas, Umna, Adab, Mari, Kis,
Awan, Aksak, were constituted as city-states or, as Falkenstein has said, city-
temples, which fought incessantly to exert a hegemony which they
exercised more or less by turns. Up to the Archaic Dynasty II, we nowhere
find a palace, since the king was in reality a priest, vicar of the god, who
lived in the precincts of the temple, the Gir-Par, of which it seems we have
an example in the edifice of Nippur.
The priest-king bore the title of EN, “Lord”; it is only during the Archaic
Dynasty II that the title of king, Lugal, emerges, and at the same time
the palace, witness to the separation of State and priesthood, and the
emergence of a military monarchy. The earliest known palace is that of
Tell A at Kis, and the first personage who bore the title of Lugal was in fact
a king of Kis, Mebaragesi (around 2700 BCE). The furnishings of the
tombs of Ur, which date from subsequent centuries, reveal the high
level of material civilisation which the Sumerians had attained. The
metallurgists had acquired a great mastery of their art and the sculptors
had produced fine in-the-round works. We see a parallel development of
urbanisation and of monumental building: the oval temple of Khafaje, the
square temple of Tell Asmar, the temple of Ishtar at Mari, the temple of
Inanna at Nippur. The expansion of the Sumerian cities was brusquely
arrested in the twenty-fourth century BCE by the formation of the
Semitic empire of Akkad. But the Akkadians assimilated the Sumerian
culture and spread it beyond the land of Sumer. Savage tribes from the
neighbouring mountains, Lullubi and Guti, put an end to the Akkadian
Empire and ravaged the countryside until the king of Uruk, Utu-Hegal,
overthrew the power of the Guti and captured their king, Tiriqan. Now an
age of Sumerian renaissance began, with the hegemony of Lagas and above
all of Ur.
At the beginning of the second millennium BCE, the Sumerians were
once again dominant with the dynasties of Isin and of Larsa, but after the
triumph of Babylon, under Hammurabi, Sumer disappeared politically; but
nevertheless the Sumerian language remained a language of priests, and
many features of their civilisation, assimilated by the Babylonian Semites,
were to survive across the Mesopotamian culture of Babylon.
NUMBERS OF SUMER
82
THE SEXAGESIMAL SYSTEM
Let us now pass to the numbers themselves. The Sumerians did not count
in tens, hundreds and thousands, but adopted instead the numerical base
60, grouping things by sixties and by powers of 60.
We ourselves have vestiges of this base, visible in the ways we express
time in hours, minutes and seconds, and circular measure in degrees,
minutes and seconds. For instance, if we have to set a digital timepiece to
9; 08; 43
then we know that this corresponds to 9 hours, 8 minutes and 43 seconds,
being time elapsed since midnight; and this can be expressed in seconds as
follows;
9 x 60 2 + 8 x 60 + 43 = 32,923 seconds.
Likewise, when a ship’s officer determines the latitude of a position he will
express it as, for instance: 25°; 36'; 07", and everyone then knows that the
position is
25 x 60 2 + 36 x 60 + 7 = 92,167"
north of the Equator.
With the Greeks, and later the Arabs, this was used as a scientific
number-system, adopted by astronomers. Since the Greeks, however, with
few and belated exceptions, this system has been used solely to express
fractions (e.g. minutes and seconds as subdivisions of an hour). But in more
distant times, as excavations in Mesopotamia have revealed, it gave rise to
two quite separate number-systems which were used for whole numbers as
well as fractions. One was the system used solely for scientific purposes by
the Babylonian mathematicians and astronomers, later inherited by the
Greeks who in turn passed it down to us by way of the Arabs. The other,
more ancient yet and which we are about to discuss, was the number-
system in common use amongst the Sumerians, predecessors of the
Babylonians, and exclusively amongst them.
THE SUMERIAN ORAL COUNTING METHOD
60 is certainly a large number to use as base for a number-system, placing
considerable demands on the memory since - in principle at least - it
requires knowledge of sixty different signs or words to stand for the
numbers from 1 to 60. But the Sumerians overcame this difficulty by using
10 as an intermediary to lighten the burden on the memory, as a kind
of stepping-stone between the different sexagesimal orders of magnitude
(1, 60, 60 2 , 60 3 , etc.).
Ignoring sundry variants, the Sumerian names for the first ten numbers,
according to Deimel, Falkenstein and Powell, were
1 ges (or as or die)
6 as
2 min
7 imin
3 eS
8 ussu
4 limmu
9 Him mu
5 id
10 u
They also gave a name to each multiple of 10 below 60 (so, up this point,
it was a decimal system):
Fig. 8.5B.
10
u
20
nis
30
usu
40
nismin for nimin or nin)
50
ninnu
60
ges for gesta)
Apart from the case of 20 ( nis seems to be independent of min = 2 and
of u = 10), these names are in fact compound words. The word for 30, there-
fore, is formed by combining the word for 3 with the word for 10:
30 = usu < *es.u = 3 x 10
(where the asterisk indicates that an intermediate word has been restored).
In the same way, the word for 40 is derived by combining the word for
20 with the word for 2:
40 = nismin = nis. min = 20 x 2 .
The variants of this are simply contractions of nismin:
40 = nin < ni.(-m).in = ni.(-s).min < nismin.
The word for 50 comes from the following combination:
50 = ninnu < *nimnu = niminu = nimin.u = 40 + 10.
In the words of F. Thureau-Dangin, the Sumerian names for the numbers
20, 40 and 50 seem like a sort of “vigesimal enclave” in this system. Note,
by the way, that the word for 60 (ges) is the same as the word for unity. No
doubt this was because the Sumerians thought of 60 as a large unity.
Nevertheless, to avoid ambiguity, it was sometimes called gesta.
The number 60 represents a certain level, above which, in this oral
numeration system, multiples of 60 up to 600 were expressed by using 60
as a new unit:
83
60
ges
360
ges- as
(= 60 x 6)
120
ges- min
(= 60 x 2)
420
ges-imin
(= 60 x 7)
180
ges-es
(= 60 x 3)
480
ges-ussu
(= 60 x 8)
240
ges-limmu
(= 60 x 4)
540
ges-ilimmu
(= 60 x 9)
300
ges-ia
(= 60 x 5)
600
ges-u
(= 60 x 10)
Fig. 8.5c.
The next level is reached at 600, which is now treated as another new
unit whose multiples were used up to 3,000:
600
ges-u
2,400
ges-u-limmu
(= 600 x 4)
1,200
ges-u-min
(= 600 x 2) 3,000
ges-u- id
(= 600 x 5)
1,800
ges-u-es
(= 600 x 3) 3,600
sar
(= 60 2 )
Fig. 8.5D.
The number 3,600 (sixty sixties) is the next level, and it is given a new
name ( sar ) and in turn becomes yet another new unit:
sar
3,600
(=
60 2 )
sar-as
21,600
(= 3,600 x 6)
sar- min
7,200
(=
3,600 x 2)
sar-imin
25,200
(=3,600x7)
sar-es
10,800
(=
3,600 x 3)
sar-ussu
28,800
(= 3,600 x 8)
sar-limmu
14,400
(=
3,600 x 4)
sar-ilimu
32,400
(= 3,600 x 9)
sar-ia
18,000
(=
3,600 x 5)
sdr-u
36,000
(= 3,600 x 10)
Fig. 8.5E.
The following levels correspond to the numbers 36,000, 216,000,
12,960,000, and so on, proceeding in the same sort of way as above:
36,000
sdr-u
(= 60 2 x 10)
144,000
sdr-u-limmu
(= 36,000x4)
72,000
sdr-u- min
(= 36,000x2)
180,000
sdr-u- id
(= 36,000 x 5)
108,000
sar-u-es
(= 36,000 x 3)
216,000
sargal
(= 60 3 )
(literally: “big 3,600”)
Fig. 8.5F.
216,000
432,000
sargal
sargal-min
(= 60 3 )
(= 216,000 x 2)
1.296.000
1.512.000
sargal-as
sargal-imin
(= 216,000 x 6)
(= 216,000 x 7)
1,080,000
sdrgal-id
(= 216,000 x 5)
2,160,000
sargal -u
(= 216,000 x 10)
Fig. 8.5G.
THE SUMERIAN ORAL COUNTING METHOD
2.160.000 sargal- u (= 60 3 x 10) 8,640,000 sargal-u-limmu (=2,160,000x4)
4.320.000 sargal-u-min (=2,160,000x2) 10,800,000 sargal-u-id (=2,160,000x5)
6.480.000 sargal-u-es (=2,160,000x3)
12,960,000 sargal-su-nu-tag (= 60 4 )
(“Unit greater than big sar")
Fig. 8.5H.
FROM THE ORAL TO THE WRITTEN
NUMBER-SYSTEM
When, around 3200 BCE, the Sumerians devised a numerical notation,
they gave a special graphical symbol to each of the units 1; 10; 60; 600
(= 60 x 10); 3,600 (= 60 2 ); 36,000 (= 60 2 x 10), that is to say to each term
in the sequence generated by the following schema:
1
10
10x6
(10 x 6) x 10
(10 x 6 x 10) x 6
(10 x 6 x 10 x 6) x 10
They therefore mimicked the names of the different units in their oral
system which, as we have seen, used base 60 and proceeded by a system of
levels constructed alternately on auxiliary bases of 6 and of 10 (Fig. 8.6).
Fig . 8.6. The structure of the Sumerian number-system, which was a sexagesimal system
constructed upon a base of 10 alternating with a base of 6 (thus activating in turn two divisors of
the base 60: 10 x 6 = 60)
NUMBERS OF SUMF. R
84
THE VARIOUS FORMS OF SUMERIAN NUMBERS
In the archaic epochs, unity was represented by a small notch (sometimes
elongated), 10 by a circular indentation of small diameter, 600 (= 60 x 10)
by a combination of these two, 3,600 (= 60 2 ) by a circular indentation
of large diameter, and 36,000 (= 3,600 x 10) by the smaller circular
indentation within the larger circular indentation (Fig. 8.2 and 8.6).
To start with, these symbols were impressed on the tablets in the
following orientation:
° ® 'O O ®
1 10 60 600 3,600 36,000
Fig. 8.7.
However, starting in the twenty-seventh century BCE, these became
rotated anticlockwise through 90°. Thus the non-circular symbols thence-
forth no longer pointed from top to bottom but from left to right:
0 0 D IS> O @
1 10 60 600 3,600 36,000
Fig. 8.8.
After the development of the cuneiform script, these number-symbols
took on a completely new form, angular and with much sharper outlines.
• The number 1 was thereafter represented by a small vertical wedge
(instead of a small cylindrical notch);
• the number 10 was represented by a chevron (instead of the small
circular impression);
• the number 60 was represented by a larger vertical wedge (instead
of a wide notch);
• the number 600 by this larger vertical wedge combined with the
chevron of the number 10;
• the number 3,600 by a polygon formed by joining up four wedges
(instead of the larger circle);
• the number 36,000 by the polygon for 3,600, with the wedge for 10
in its centre;
and, finally, the number 216,000 (the cube of 60, for which a special symbol
was introduced into cuneiform script) was represented by combining the
polygon for 3,600 with the wedge for 60 (see Fig. 8.9).
1
10
60
600
3,600
36.000
216,000
ARCHAIC
NUMBERS
(known from around
3200-3100 BCE)
VERTICAL ARRANGEMENT
0
0
0
O
HORIZONTAL ARRANGEMENT
(probably from the third millennium BCE)
D
0
D
w
&
E>
O
@
CUNEIFORM
NUMBERS
(known since at least
the 27th century BCE
r
r
T
<
<
<
r
r
T
*
(?
*>
?>
*>
$
4 >
&
*$>
&
Fig, 8.9. The development of the shapes of numbers originating in Sumeria. The change from
the archaic to the cuneiform shapes resulted from the replacement of the "old stylus", which was
cylindrical at one end, and pointed at the other, by the “flat " stylus shaped something like a modern
ruler. This new writing instrument conduced its users to break the curves into a series of wedges or
chevrons. See Deimel (1924, 1947) and Labat (1976 and in FPP).
Clay as Mesopotamian “paper” and how to write on it
In Mesopotamia, stone is rare; wood, leather and parchment are difficult to
preserve, and the soil consists of alluvial deposits. The inhabitants of this
region therefore took what came to hand for the purpose of expressing their
thoughts or for recording the spoken word, and what they had to hand was
clay. They had used this raw material since very early times for modelling
figurines, for sculpture, and for glyptics*, and later most ingeniously put it
* Close examination of archaeological finds leads one to believe that the usages of clay held no secrets for
the Mesopotamians, four thousand years BCE. This is an important consideration for the history of writing
in this region, since it effectively implies that they were fully aware of the possibilities of the medium. The
85
THE VARIOUS FORMS OF SUMERIAN NUMBERS
to diverse uses, especially for the purpose of writing, for more than three
thousand years, in more than a dozen languages^. To borrow a phrase from
J. Nougayrol (1945), you might say that these people created “civilisations
of clay”.
The originality of Mesopotamian graphics directly reflects the nature of
this material and the techniques available to work with it, and we have an
interest in devoting some attention to this; what follows will allow us to
better trace the evolution of the forms of figures and written characters
which originated in Sumer.
We have seen how Sumerian figures were hollow marks of different
shapes and sizes (Fig. 8.2), while the written characters were real drawings
representing beings and objects of every kind (Fig. 8.3). Originally, there-
fore, there were fundamental differences of technique between the
production of the one and the production of the other. The number-signs,
like the motifs created using cylindrical or stamp-like seals, were produced
by impression ; the written characters on the other hand were traced.
For these purposes the Sumerians used a reed stem (or possibly a rod
made of bone or ivory), which at one end was shaped into a cylindrical
stylus, while the other end was sharpened to a point somewhat like a
modern pen (Fig. 8.10).
The pictograms were made by pressing the pointed end quite deeply into
the clay, still fresh, of the tablets (Fig. 8.11). To draw a line, the same
NARROW REED STYLUS WIDE REED STYLUS
Pointed end,
used for drawing lines
Cylindrical stylus,
used to make the impressions
for the numbers
4 mm 1 cm
Fig . 8.io. Reconstruction of the writing instruments of the Sumerian scribes (archaic era)
character, possibly religious but certainly symbolic, of the motifs appearing on these vases and jugs, their
repeated occurrence and their systematic stylisation, must not only have accustomed their creators to
express a number of thoughts and ideas in this way but also to subsume these into ever simpler and more
concise designs.
t At the dawn of the second millennium BCE, at the time when writing emerged in Sumer, the use of clay
for tablets’ intended to bear conventional signs was already widespread in the region. This point also
is very important, for it clarifies one reason why Sumerian vvriting moved on to a systematic phase: consid-
ering the difficulty of sculpture, carving, and painting, and the fact that they demand time for their
execution, the universal adoption of clay throughout Mesopotamia is readily explained by the ease
with which it can be worked (compared with wood, bone, or stone, whether for engraving, embossing,
impression, moulding, or cutting).
pointed end was pressed in as before, and then drawn parallel to the surface
through the required distance. Of
course, this would often result in
a wavy line, and could give rise to F ig. s.n. How the
spillover on either side, because of archaic Sumerian
the softness of the material. pictograms were drawn
on soft clay tablets
For the numbers, on the other
hand, the Sumerians made these by making an imprint on the soft damp
clay with the other end of the instrument, the end shaped into a circular
stylus. This was done with the stylus held at a certain angle to the surface of
the clay. They had two styluses of different diameters: one about 4 mm, the
other about 1 cm (Fig. 8.10). According to the angle at which they held the
stylus, either a circular imprint, or a notch, would be obtained, and its size
would depend on the diameter of the stylus (Fig. 8.12):
• a circular imprint of smaller or larger diameter if the stylus was held
perpendicular to the surface of the clay;
• a notch, narrow or wide, if the stylus was held at an angle of 30° -
45° to the surface; the imprint would be more elongated if the angle
was small.
ACTION RESULT
I
bl
]'
Narrow stylus applied at an
angle of 45°
O
Small notch
I
P
«
ii
90"
< o
Narrow stylus applied
perpendicularly
O
Small circular indentation
1
fj
j’
Wide stylus applied at an
angle of 45*
O
Wide notch
1
“1
-
K
J 90°
(I) : Wide stylus applied at
45°
(II) : Narrow stylus applied
perpendicularly
Wide notch with
small indentation
I
n
fl
d
j-
Wide stylus applied
perpendicularly
O
barge circular indentation
I
3
d
90- L
Id"
J 90
s*
(I) : Wide stylus applied
perpendicularly
(II) : Narrow calamus applied
perpendicularly
©
Large circular
indentation with small
circular indentation
Fig . 8.12. How the archaic numbers were impressed on soft clay tablets
Ml M II K R S OK SUMEH
86
Why Sumerian writing changed direction
In the very earliest times, the signs used in Sumerian writing were drawn
on the clay tablets in the natural orientation of whatever they were supposed
to represent: vases stood upright, plants grew upwards, living things were
vertical, etc. Similarly, the non-circular figures for numbers were also
vertical (the stylus being held sloping towards the bottom of the tablet).
These signs and figures were generally arranged on the tablets in
horizontal rows which, in turn, were subdivided into several compartments
or boxes (Fig. 8.1, tablet E). Within each box, the figures were generally at
the top, starting from the right, while the drawings used for writing were
at the very bottom, like this:
oo
0®
©OO
111
000
000
©oooo
© © O
m
X
00
00
<$'
V
Fig. 8.13.
Now, if we examine the arrangement of figures and drawings on one of
the tablets of the so-called Uruk period (around 3100 BCE), we find that
where one of the boxes is not completely full the empty space is always on
the left of the box (see the second box from the right in the top row of the
tablet in Fig. 8.14).
This proves that the scribes of the earliest times wrote from right to left
and from the top to the bottom. The non-circular figures were vertical, and
Fig . 8.14. Sumerian tablet from Uruk, from around 3100 BCE. Iraqi Museum, Baghdad
the drawings had their natural orientations. In short, in the beginning
Sumerian writing was read from right to left and from top to bottom.
This arrangement long persisted on Mesopotamian stone inscriptions.
It can be seen especially on the Stele of the Vultures (where the text is
arranged in horizontal bands, and the boxes succeed each other from right
to left and from top to bottom), in the celebrated Code of Hammurabi
(whose inscription, which is read from right to left and from left to right,
is arranged in vertical columns), and in several legends later than the
seventeenth century BCE.
It went quite differently in the case of clay tablets, however: that is, in the
case of everyday writings. Starting around the twenty-seventh century BCE,
the signs used for writing, and the figures used for numbers, underwent
a rotation through 90° anticlockwise.
Fig. 8 . 15 . Sumerian tablet (Telia, about 3500 BCE). Bibliotheque nation ale, Paris, Cabinet des
Medailles (CMH 870 F). See de Gcnouillac (1909), plate IX
To verily this, consider the tablet in Fig. 8.15, and look at in the direction
I — » II indicated by the long arrow in the Fig., after turning it 90° clockwise
so that I — > II is from right to left and at the top. Then we can see that if
a compartment is not full up, the empty space is at the bottom, and not at
the left. Likewise, in the original position of the tablet, the empty space is
at the right.
According to C. Higounet (1969), this would be due to a change in the
orientation with which the tablets were held.
With the small tablets of the earliest times, holding the tablet obliquely
in the hand made it easier to trace drawings in columns from top to
bottom. But, when the tablets became larger, the scribes had to place them
upright in front of them, and the signs became horizontal, and the writing
went in lines from left to right.
Be that as it may, thenceforth the drawings and the non-circular figures
had an orientation 90° anticlockwise from their original one (Fig. 8.16);
87
THE VARIOUS FORMS OF SUMERIAN NUMBERS
"turned sideways, they became less pictorial, and therefore more liable to
undergo a certain systematisation.” [R. Labat]
FISH r
This did not occur all at once, however. It is not seen at all around 2850
BCE. It begins to appear in the archaic tablets of Ur (2700-2600 BCE), and
in those of Fara (Suruppak), where the majority of the signs are made up of
impressed lines, while many other tablets of the same period continue to
show the curved lines traced by the older method.
HEAD
OX
Fig. 8 . i 6 . Anticlockwise rotation, through a quarter turn, of the Sumerian signs and numbers
The emergence of the cuneiform signs
The radical transformation which the Sumerian characters underwent
after the Pre-Sargonic era (2700-2600 BCE) is due simply to a change of
implement.
While the drawings used in writing had originally been traced out with
the pointed end of the stylus, this changed when they had the idea of using
instead, for this, the method which had always been used for the figures
denoting numbers, namely impressing the marks on the clay. Instead of
using a pointed stylus for tracing lines, they preferred to use a reed stem (or
a rod of bone or ivory) whose end was trimmed in such a way that its tip
formed a straight edge, and no longer a circle or a point. This edge was then
pressed into the clay, to achieve cleanly, at one stroke, a line segment of
a certain length; this clearly was much more rapid than drawing it with a
pointed tool.
Of course this new type of stylus made characters of quite a different
shape, with sharper lines and an angular appearance; these signs are called
cuneiform (from the Latin cuneus, “a wedge”) (Fig. 8.17).
The angular shapes of the imprints made by such a stylus on the clay
naturally led to greater stylisation of the shapes of the various signs. Curves
were broken up, and where necessary were replaced by a series of line
segments, so that a picture was reduced to a collection of broken lines. In
this new form of Sumerian writing, a circle, for example, became a polygon,
and curves were replaced by polygonal lines (Fig. 8.18).
CHEVRON
Fig. 8.17. Impressing cuneiform
signs on soft clay. The vertical wedge
was made by pressing lightly on the
clay with one of the corners of the
" beak ” of the stylus (the heavier
the pressure, the larger the wedge).
ARCHAIC SIGNS
CUNEIFORM SIGNS
Uruk period
(about 3100 BCE)
Jemdet-Nasr era
(about 2850 BCE)
Pre-Sargonic Era
(about 2600 BCE)
Third Ur Dynasty
(about 2000 BCE)
STAR
DIVINITY
*
*
*
*
EYE
0 - 4 -
<-f A-
HAND
$
& M
M I
M
BARLEY
¥ ¥
WWWW
jUffrrrr
Wm
LEG
K
pn . r
FIRE
TORCH
ft
BIRD
IF
V
HEAD
SUMMIT
CHIEF
& p
P
<tw
Fig. 8.18.
NUMBERS OF SUMER
At the beginning of this change in form, the signs nevertheless remained
very complex, since people wished to preserve as much as possible of
the detail of the original drawings, and because in the majority of cases the
objective was still to achieve the outline of a concrete object. But, after a
long period of adaptation, from the end of the third millennium BCE the
scribes only kept what was essential and therefore made their marks much
more rapidly than before.
And this is how the signs in Sumerian writing finally lost all resemblance
to the real objects which they were meant to represent in the first place.
THE SUMERIAN WRITTEN COUNTING METHOD
Starting with these basic symbols, the first nine whole numbers were
represented by repeating the sign for unity as often as required; the
numbers 20, 30, 40, and 50 by repeating the sign for 10 as often as required,
the numbers 120, 180, 240, etc. by repeating the symbol for 60, and so on.
Generally, since the system was based on the additive principle, a number
was represented by repeating, at the level of each order of magnitude, the
requisite symbol as often as required.
For example, a tablet dating from the fourth millennium BCE (Fig. 8.1,
tablet C) carries the representation of the number 691 in the following
form:
Fig. 8.19.
3
I? ©SI
600
60
1 10 10 10
Likewise, on a tablet from Suruppak, from around 2650 BCE, the
number 164,571 is represented as follows (Fig. 8.20 and 12.1):
® © © ©
36,000 drawn 4 times over = 36,000 x 4 = 144,000
o o o o o
3,000 drawn 5 times over = 3,600 x 5 = 18,000
^ ^ ^
600 drawn 4 times over = 600 x 4 = 2,400
O Ed
60 drawn 2 times over = 60 x 2 = 120
»oe° oP
10 drawn 5 times over = 10 x 5 = 50
1 drawn 1 time = =1
164,571
Fig. 8.20.
88
w
nm
nw
IKfflr
30 8
60 50 7
180 40 1
240 40 1
120 10 9
4
38
117
221
281
139
Fig. 8.21A.
TRANSLATION
Fig. 8 . 2 1 b . Sumerian tablet from about 2000 BCE, giving a tally of livestock by means of
cuneiform signs and numbers. Translation by Dominique Charpin. See de Genouillac (1911),
plate V, no. 4691 F
Similarly, for the cuneiform representation, on a tablet dating from the
second dynasty of Ur (about 2000 BCE), found in a warehouse at Drehem
(Asnunak Patesi), various numbers are represented as shown on Figs. 8.21
A and B.
Finally, and in the same way, on a tablet contemporary with this last one,
but from a clandestine excavation at Tello, we find the numbers 54,492 and
199,539 also expressed in cuneiform symbols:
36,000 drawn 1 time = 36,000
3,600 drawn 5 times over = 18,000
60 drawn 8 times over = 480
10 drawn 1 time = 10
1 drawn 2 times over = 2
$»><><><> OFF
OO OO
36,000 drawn 5 times over = 180,000
3,600 drawn 5 times over = 18,000
600 drawn 2 times over = 1,200
60 drawn 5 times over = 300
10 drawn 3 times over = 30
1 drawn 9 times over = 9
Fig. 8 . 22 . Barton (1918), Table Hlb 24, no. 16
89
THE SUMERIAN WRITTEN COUNTING METHOD
We may observe in passing that the Sumerians grouped the identical
repeated symbols in such a way as to facilitate the grasp, in one glance, of the
values of the assemblages within each order of magnitude. Considering just
the representations of the first nine numbers, these groupings were initially
made according to a dyadic or binary principle (Fig. 8.23) and later according
to a ternary principle in which the number 3 played a special role (Fig. 8.24).
ARCHAIC NUMBERS
1
2
3
4
5
6
7
8
9
T
rr
ITT
▼
ft
TT
fflr
lyr
ftf
m
w
tCQT
fflf
wr
vvvyy
w
Fig . 8.23. The dyadic (binary) principle of representing the nine units
CUNEIFORM NUMBERS
T
TT
TIT
Y
W
1
¥
m
w
w
Fig. 8 . 24 . The ternary principle of representing the nine units
Thus the Sumerian numbering system sometimes required inordinate
repetitions of identical marks, since it placed symbols side by side to repre-
sent addition of their values. For example, the number 3,599 required a
total of twenty-six symbols!
For this reason, the Sumerian scribes would seek simplification by often
using a subtractive convention, writing numbers such as 9, 18, 38, 57, 2,360
and 3,110 in the form:
20 - 2
18
O
10 - 1
9
00 r
00
40-2
38
60-3
57
f©o~
f§) ED |oo
2.400 - 40
2,360
(cf. Fig. 8.26)
3,120 - 10
3,110
(cf. Fig. 8.26)
r~ r-
The sign: | or | , which represented the sound LA, was the precise
equivalent of our “minus”.
Fig. 8 . 25 .
Fig. 8 . 26 . Sumerian tablet from Suruppak (Fara), 2650 BCE. Istanbul Museum. See Jestin
(1937), plate LXXX1V, 242 F
From the pre-Sargonic era (about 2500 BCE), certain irregularities start
to appear in the cuneiform representation of numbers. As well as the
subtractive convention just described, the multiples of 36,000 can be found
represented as shown in Fig. 8.27, instead of simply repeating the symbol
for 36,000 once, twice, or three, four, or five times.
72,000 108,000 144,000 180,000
Fig. 8 . 27 . See Deimel
NUMBERS OF SUMER
90
These forms evidently correspond to the arithmetical formulae
72,000 = 3,600 x 20 (instead of 36,000 + 36,000)
108.000 = 3,600 x 30 (instead of 36,000 + 36,000 + 36,000)
144.000 = 3,600 x 40 (instead of 36,000 + 36,000 + 36,000 + 36,000)
180.000 = 3,600 x 50
(instead of 36,000 + 36,000 + 36,000 + 36,000 + 36,000).
In this, the Sumerians were doing nothing other than what we would
today refer to as “expressing in terms of a common factor”. Observing that
the symbol for 3,600 is itself made up of the symbol for 360 with the
symbol for 10, they also, after their fashion, made the number 3,600 a
common factor so that, for instance, instead of representing 144,000 in
the form
(3,600 x 10) + (3,600 x 10) + (3,600 x 10) + (3,600 x 10)
they used instead the simpler form
3,600 x (10 + 10 + 10 + 10).
Another special point arising in the cuneiform notation concerned the
two numbers 70 (= 60 + 10) and 600 (= 60 x 10), since both involved
juxtaposing the symbol for 60 and the symbol for 10. This can clearly lead
to ambiguity, since for 70 they are combined additively, and for 600 multi-
plicatively. This ambiguity was not present, however, in the archaic
notation:
Fig. 8.28a. Fig. 8.28b. 600
They were, however, able to eliminate any possible confusion. In the case
of 70, they placed a dear separation between the wedge (for 10) and the
chevron (for 60) so as to indicate addition (Fig. 8.29 A), while for 600 they
put them in contact so as to form an indivisible group, to represent multi-
plication (Fig. 8.29 B).
Y< Y<
60 + 10 60 x 10
70 Fig. 8.29A. 600 Fig. 8.29B.
A different problem arose with the representation of the numbers 61, 62,
63, etc. In the beginning, the number 1 was represented by a small wedge,
and the number 60 by a larger wedge, and so there was no ambiguity:
Tt
Tt7
fm
Tw
Yw
Tut
Twf
Yffff
TffiF
60 1
60 2
60 3
60 4
60 5
60 6
60 7
60 8
60 9
61
62
63
64
65
66
67
68
69
Fig. 8.30.
Later, however, 1 and 60 came to be represented by the same size of
vertical wedge, and it was very difficult to distinguish between 2 and 61,
or between 3 and 62, for example:
TT TT
YYY ITT
1.1 60.1
2 61
1.1.1 60.1.1
3 62
Fig. 8.31.
Therefore they had the idea of distinctly separating the unit symbols for
the sixties from those for the units.
T T
T TT
T TfT
T W
T W
Tf
T W
Tf
t m
60 1
61
60 2
62
60 3
63
60 4
64
60 5
65
60 6
66
60 7
67
60 8
68
60 9
69
Fig. 8.32.
This particular problem with the cuneiform sexagesimal notation was
the root of a most interesting simplification to which we shall return in
Chapter 13.
For a long time, the cuneiform characters (known since at least twenty-
seven centuries BCE) coexisted with the archaic numeral signs (Fig. 8.9). On
certain tablets contemporary with the kings of the Akkad Dynasty (second
half of the third millennium BCE), we see the cuneiform numbers side by
side with their archaic counterparts. The intention, it seems, was to mark a
distinction of rank between the people being enumerated: the cuneiform
figures were for people of higher social standing, and the others for slaves
or common people [M. Lambert, personal communication]. The cuneiform
number-symbols did not definitively supplant the archaic ones until the
third dynasty of Ur (2100-2000 BCE).
91
neugebauer’s hypothesis
CHAPTER 9
THE ENIGMA OF THE
SEXAGESIMAL BASE
In all of human history the Sumerians alone invented and made use of a
sexagesimal system - that is to say, a system of numbers using 60 as a base.
This invention is without doubt one of the great triumphs of Sumerian civil-
isation from a technical point of view, but it is nonetheless one of the greatest
unresolved enigmas in the history of arithmetic. Although there have been
many attempts to make sense of it since the time of the Greeks, we do not
know the reasons which led the Sumerians to choose such a high base. Let us
begin with a review of the explanations that have been put forward so far.
theon of Alexandria’s hypothesis
Theon of Alexandria, a Greek editor of Ptolemaic texts, suggested in the
fourth century CE that the Sumerians chose base 60 because it was
the “easiest to use” as well as the lowest of “all the numbers that had the
greatest number of divisors”. The same argument also cropped up 1,300
years later in Opera mathematica, by John Wallis (1616-1703), and again, in
a slightly different form, in 1910, when Lofler argued that the system arose
“in priestly schools where it was realised that 60 has the property of having
all of the first six integers as factors”.
FORMALEONl’s AND CANTOR’S HYPOTHESES
In 1789 a different approach was suggested by the Venetian scholar
Formaleoni, and then repeated in 1880 by Moritz Cantor. They held that
the Sumerian system derived from exclusively “natural” considerations: on
this view, the number of days in a year, rounded down to 360, was the
reason for the circle being divided into 360 degrees, and the fact that
the chord of a sextant (one sixth of a circle) is equal to the radius gave rise
to the division of the circle into six equal parts. This would have made 60
a natural unit of counting.
lehmann-haupt’s hypothesis
In 1889, Lehmann-Haupt believed he had identified the origin of base 60
in the relationship between the Sumerian “hour” ( danna ), equivalent to
two of our current hours, and the visible diameter of the sun expressed in
units of time equivalent to two minutes by current reckoning.
neugebauer’s HYPOTHESIS
In 1927 O. Neugebauer proposed a new solution which located the source
of base 60 in terms of systems of weights and measures. This is how the
proposal was explained by O. Becker and J. E. Hoffmann (1951):
It arose from the combination of originally quite separate measure-
ment units using base 10 and having (as in spoken language, and like
the Egyptian systems) different symbols for 1, 10, and 100 as well as
for the “natural fractions", 1/2, 1/3, and 2/3. The need to combine the
systems arose particularly for measures of weight corresponding to
measures of price or value. The systems were too disparate to be
harmonised by simple equivalence tables, and so they were combined
to give a continuous series such that the elements in the set of higher
values (B) became whole multiples of elements in the set of lower
values (A). Since both sets of values had the structure 1/1, 1/2, 2/3, 1,
2, 3 ... 10, the relationship between the two sets A and B had to allow
for division by 2 and by 3, which introduced factor 6. So from the
decimal structure of the original number-system, the Sumerians ended
up with 60 as the base element of the new (combined) system.
On the other hand, F. Thureau-Dangin (1929) took the view that this
entirely theoretical explanation cannot be a correct account of the origin of
Sumerian numbering, because it is “undoubtedly the case that base 60 only
occurs in Sumerian weights and measures because it was already available
in the number-system”.
OTHER SPECULATIONS
The Mesopotamians, according to D. J. Boorstin (1986), got to 60 by
multiplying the number of planets (Mercury, Venus, Mars, Jupiter, and
Saturn) by the number of months in the year: 5 x 12 is also a multiple of 6.
In 1910, E. Hoppe tried to refute, then to adapt Neugebauer’s hypothesis:
in this view, the Sumerians would have seen that base 30 provided for most
of their needs, but chose the higher base of 60 because it was also divisible
by 4. He subsequently proposed another explanation, based on geometry:
the sexagesimal system, he argued, must have been in some relationship
to the division of the circle into six equal parts instead of into four right
angles, which made the equilateral triangle, instead of the square, the
fundamental figure of Sumerian geometry. If the angle of an equilateral
triangle is divided into 10 “degrees”, in a decimal numbering system, then
the circle would have 60 degrees, thus giving the origin of base 60 for the
developed numbering system.
However, as was pointed out by the Assyriologist G. Kewitsch (1904),
neither astronomy nor geometry can actually explain the origin of a
THE ENIGMA OF THE S F. X A G E S I M A L BASF.
92
number-system. Hoppe's and Neugebauer’s speculations are far too theor-
etical, presupposing as they do that abstract considerations preceded
concrete applications. They require us to believe that geometry and astron-
omy existed as fully-developed sciences before any of their practical
applications. The historical record tells a very different story!
I once knew a professor of mathematics who likewise tried to persuade his
students that abstract geometry was historically prior to its practical applica-
tions, and that the pyramids and buildings of ancient Egypt “proved” that
their architects were highly sophisticated mathematicians. But the first
gardener in history to lay out a perfect ellipse with three stakes and a length
of string certainly held no degree in the theory of cones! Nor did Egyptian
architects have anything more than simple devices - “tricks”, “knacks” and
methods of an entirely empirical kind, no doubt discovered by trial and error
- for laying out their ground plans. They knew, for example, that if you took
three pieces of string measuring respectively three, four, and five units in
length, tied them together, and drove stakes into the ground at the knotted
points, you got a perfect right angle. This “trick” demonstrates Pythagoras’s
theorem (that in a right-angled triangle the square on the hypotenuse equals
the sum of the squares on the other two sides) with a particular instance in
whole numbers ((3 x 3) + (4 x 4) = 5 x 5), but it does not presuppose knowledge
of the abstract formulation, which the Egyptians most certainly did not have.
All the same, the Sumerians’ mysterious base 60 has survived to the
present day in measurements of time, arcs, and angles. Whatever its origins,
its survival may well be due to the specific arithmetical, geometrical and
astronomical properties of the number.
kewitsch’s hypothesis
Kewitsch speculated in 1904 that the sexagesimal system of the Sumerians
resulted from the fusion of two civilisations, one of which used a decimal
number-system, and the other base 6, deriving from a special form of
finger-counting. This is not easily acceptable as an explanation, since there
is no historical record of a base 6 numbering system anywhere in the world
[F. Thureau-Dangin (1929)].
BASE 12
On the other hand, duodecimal systems (counting to base 12) are widely
attested, not least in Western Europe. We still use it for counting eggs and
oysters, we have the words dozen and gross (= 12 x 12), and measurements
of length and weight based on 12 were current in France prior the
Revolution of 1789, in Britain until only a few years ago, and still are in
the United States.
The Romans had a unit of weight, money, and arithmetic called the as,
divided into 12 ounces. Similarly, one of the monetary units of pre-
Revolutionary France was the sol, divided into 12 deniers. In the so-called
Imperial system of weights and measures, in use in continental Europe
prior to the introduction of the metric system (see above, pp. 42-3), length
is measured in feet divided into 12 inches (and each inch into 12 lines and
each line into 12 points, in the obsolete French version).
The Sumerians, Assyrians, and Babylonians used base 12 and its
multiples and divisors very widely indeed in their measurements, as the
following table shows:
LENGTH
1 ninda
1 ninni
1 su
“perch”
12 cubits
10 x 12 ells
2/12ths of a cubit
WEIGHT
lgin
“shekel”
3 x 12 su
(8.416 grams)
AREA
1 bur
1 sar
150 x 12 sar
12 x 12 square cubits
(35.29 centiares)
VOLUME
lgur
1 pi
1 banes
1 ban
1 sila
25 x 12 sila
3 x 12 sila
3x6 sila
6 sila
(842 ml)
The Mesopotamian day was also divided into twelve equal parts (called
danna), and they divided the circle, the ecliptic, and the zodiac into twelve
equal sectors of 30 degrees.
Moreover, there is clear evidence on tablets from the ancient city of Uruk
[see Green & Nissen (1985); Damerov & Englund (1985)] of several different
Sumerian numerical notations, which must have been used concurrently
with the classical system (see Fig. 8.9, recapitulated in Fig. 9.1 below),
amongst which there are the measures of length shown in Fig. 9.2.
1 10 60 120 1,200 7,200
(= 12 x 5) (= 12 x 10) (= 12 x 10 x 10) (= 12 x 10 x 10 x 6)
Fig. 9.1.
93
AN ATTRACTIVE HYPOTHESIS
Fig . 9 . 2 . Archaic Sumerian tablets from Uruk, showing a numerical notation that is different
from the standard one. (Numerous tablets of this kind prove that the Sumerians had several parallel
systems), Date: c. 3000 BCE. Baghdad, Iraqi Museum. Source: Damerov & Englund (1985)
To sum up, base 12 could well have played a major role in shaping the
Sumerian number-system.
AN ATTRACTIVE HYPOTHESIS
The major role given to base 10 in Sumerian arithmetic is similarly well-
attested: as we saw in Chapter 8, it was used as an auxiliary unit to
circumvent the main difficulty of the sexagesimal system, which in theory
requires sixty different number-names or signs to be memorised. This is all
the more interesting because the Sumerian word for “ten”, pronounced u,
means “fingers”, strongly suggesting that we have a trace of an earlier
finger-counting system of numerals.
This makes it possible to go back to Kewitsch’s hypothesis and to give it
a different cast: to suppose that the choice of base 60 was a learned solution
to the union between two peoples, one of which possessed a decimal system
and the other a system using base 12. For 60 is the lowest common multiple
of 10 and 12, as well as being the lowest number of which all the first six
integers are divisors.
Our hypothesis is therefore this: that Sumerian society had to begin with
both decimal and duodecimal number-systems; and that its mathemati-
cians, who reached a fairly advanced degree of sophistication (as we can see
from the record of their achievements), subsequently devised a learned
system that combined the two bases according to the principle of the LCM
(lowest common multiple), producing a sexagesimal base, which had the
added advantage of convenience for numerous types of calculation.
This is a very attractive and quite plausible hypothesis: but it fails as a
historical explanation of origins on the obvious grounds that it presupposes
too much intellectual sophistication. For we must not forget that most
historically and ethnographically attested base numbers arose for reasons
quite independent of arithmetical convenience, and that they were chosen
very often without reference to a structure or even to the concepts of
abstract numbers.
ARE THERE MYSTICAL REASONS FOR BASE 60?
Sacred numbers played a major role in Mesopotamian civilisations;
Sumerian mathematics developed in the context of number-mysticism; and
so it is tempting to see some kind of religious or mystical basis for the sexa-
gesimal system.
Sumerian mathematics, like astrology, cannot be disentangled from
numerology, with which it has reciprocal relations. From the dawn of
the third millennium BCE, the number 50 was attributed to the temple
of Lagas, son of the earth-god, and this shows that from the earliest
times numbers had “speculative” meanings. The Akkadians brought
number-symbolism into Babylonian thought, making it an essential
element of the Name, the Individual and the Work. Alongside their
scientific or intellectual functions, numbers became part of the way the
Mesopotamians conceived the structure of the world. For example,
the numeral sar or saros (= 3,600) is written in cuneiform as a sign
which is clearly a deformation of the circle [see Fig. 8.9], and it also
means “everything”, “totality”, “cosmos”. In Sumerian cosmogony, two
primordial entities, the “Upper Totality” or An-Sar and the “Lower
Totality” or Ki-Sar came together to give birth to the first gods.
Moreover, the full circle of 360° is divided into degrees, whose basic
unit of 1/360 is called Ges - and the symbol for Ges is precisely what is
used to signify “man” and thus for elaborating the names of masculine
properties. The higher base unit or sosse (= 60) is also pronounced Ges
[see Fig. 8.5], and its sign (with an added asterisk or star) is the figure
of the “Upper God”, or heaven, whose name is pronounced An(u), by
virtue of the ideogram that defines it as a divinity and as heaven. So the
celestial god, 60, is the father of the earth-god, 50; the god of the Abyss
is 40, two thirds of 60. The moon-god is 30 (it has been suggested,
without any evidence, that the moon-god has this number in virtue
of the number of days in the lunar cycle); and the sun-god has the
number 20, which is also the determining number of “king” . . .
[Adapted from M. Rutten (1970)]
It seems plausible, in this context, to think that base 60 commended
itself to the mystic minds of Sumerians because of their cult of the “Upper
God” Anu, whose number it was.
There are many attested examples, in Australia, Africa, the Americas,
THE. ENIGMA OF THE SEXAGESIMAL BASF
94
and Asia, of number-systems with a base (most often, base 4) that has
mystical ramifications. However, the Sumerian system is much more devel-
oped than any of these, and presupposes complete familiarity with abstract
number-concepts. For this reason it does not seems right to consider
Sumerian mysticism as the origin of the Sumerian base 60. Things should
rather be looked on the other way round: it is far more probable that 60 was
the “number” of the Upper God Anu precisely because it was already the
larger of the units of Sumerian arithmetic.
THE PROBABLE ORIGIN OF THE
SEXAGESIMAL SYSTEM
So where does base 60 come from? Here is what I believe to be the solution
to this enigma.
It is necessary to suppose (without a great deal of material evidence) that
the Mesopotamian basin had one or more indigenous populations prior to
Sumerian domination. A second essential premise (but one that is not at all
controversial) is that the Sumerians were immigrants, that they came from
somewhere else, more than probably in the fourth millennium BCE. Though
we know very little about the indigenous population, and almost nothing
about the prior cultural connections of the Sumerians, who seem to have
broken all ties with their previous environment, we can speculate with a fair
degree of confidence that these two cultures possessed different counting
systems, one of which was duodecimal, and the other quinary.
Let us look again at Sumerian number-names.
123456789 10
ges min es limmu id as imin ussu ilimmu u
Ges (1) is a word that also means “man”, “male” and “erect phallus”; min
(2) also means “woman”; and es (3) is also the plural suffix in Sumerian
(rather like -s in English). The symbolism of these number-names is both
apparent and very ancient indeed, taking us back to “primitive” perceptions
of man as vertical (in distinction to all other animals) and alone, of woman
as the “complement” of a pair (man and woman, or woman and child), and
of “the many” beginning at three. (In Pharaonic Egypt as in the Hittite
Empire, plurals were indicated by writing the same hieroglyph three times
over, or by adding three vertical bars after the sign; in classical Chinese,
the ideogram for “forest” consists of three ideograms for “tree", whereas the
concept “crowd” was represented by a triple repetition of the ideogram for
“man”.) So the semantic meanings of the names of the first three numbers
of Sumerian is a trace of those lost ages when people had only the most
rudimentary concepts of number, counting only “one, two, and many”.
More importantly, however, the names of the numbers in spoken
Sumerian also carry unmistakable traces of a quinary system. As, six, looks
like an elision of id and ges, “five (and) one”; imin, seven, is more certainly
a contraction of id and min, “five and two”; ilimmu is clearly related to id
and limmu, “five and four”. In other words, Sumerian number-names derive
from a vanished system using base 5. We speculate therefore that one of
the two populations involved had a quinary counting system, and that in
contact with a civilisation using base 12, the sexagesimal system was
invented or chosen, since 5 x 12 = 60.
As we have already seen, the quinary base is anthropomorphic and
derives from learning to count on the fingers of one hand and using the
other hand as a “marker” when counting beyond 5. However, the origin of
base 12 is far less obvious. My own view is that it was probably also based
on the human hand.
Each finger has three articulations (or phalanxes): and if you leave out
the thumb (as you have to, since you use it to check off the phalanxes
counted), you can get to 12 using only the fingers of one hand, as in
Fig. 9.3 below:
Repeating the device once over, you get from 13 to 24, then from 25 to
36, and so on. In other words, with a finger-counting device of this kind,
base 12 seems the most natural for a numbering system.
This hypothesis is difficult to prove, but phalanx-counting of this type
does exist and is in use today in Egypt. Syria, Iraq, Iran, Afghanistan,
Pakistan, and some parts of India. Sumerians could therefore easily have
used it at the dawn of their civilisation.
How then can we explain the fact that u, the Sumerian word for “10”,
95
THE PROBABLE ORIGIN
means “fingers”, and that there is no trace of a duodecimal system in
spoken numbers, and no special word for the dozen (“12” is u-min,
meaning “ten-two”)?
My view is that spoken Sumerian numbers carry no trace of either base
12 or base 10: in other words, the name of the number 10 is not evidence of
a lost decimal number-system, but merely the metaphoric expression of a
universal human perception of human anatomy, the fact that there are
ten fingers in all on the two hands.
At all events, my hypothesis has the advantage over all other
speculations of giving a concrete explanation for the mysterious origin of
base 60. As we saw in Chapter 3, basic finger-counting techniques, supple-
mented by mental effort (which quickly becomes quite “natural” once the
principle of the base has been grasped), has often opened the way to arith-
metical elaborations far superior to the original rudimentary system
involved. From this, we can assert that the origin of base 60 could well have
been connected to the finger-counting scheme shown in Fig. 9.4, currently
in use across a broad band stretching from the Middle East to Indo-China.
This particular device makes 60 the principal base, with 5 and 12 serving
as auxiliaries. This is how it is done:
Using your right hand, you count from 1 through 12 by pressing the tip
of your thumb onto each of the three phalanxes (articulations) of the four
opposing fingers. When you reach a dozen on the right hand, you check it
offby folding the little finger of your left hand. You return to the right hand
and count from 13 through 24 in similar fashion, then fold down the ring
finger of your left hand, then count from 25 through 36 again on the right
hand. The middle finger of the left hand is folded down to mark off 36, and
you proceed to count from 37 to 48 on the right, then folding down the left
index finger. Repeating the operation once more, you get to 60, and fold
down the last remaining finger of the left hand (the thumb). As you can’t
count any higher numbers with this system, 60 is the obvious base.
LEFT HAND RIGHT HAND
36
My hypothesis can therefore be told as a story. As a result of the
symbiosis of two different cultures, one of which used a quinary finger-
counting method, and the other a duodecimal base deriving from a system
of counting the phalanxes with the opposing thumb, 60 was chosen as the
new higher unit of counting as it represented the combination of the two
prior bases.
Since 60 was a pretty large number to use as a base, arithmeticians
looked for an intermediate number to use, so as to mitigate the difficulties
that arise from people’s limited capacity to memorise number-names. Base
5 was too small compared to 60 - it would have required very long number-
strings to express intermediate numbers; so 10 was chosen, a number
provided by nature, so to speak, and of an ideal magnitude for the task in
hand. Why not base 12? It has many advantages over 10, but it would prob-
ably have disoriented those accustomed to the quinary base, for whom 10,
being twice the number of fingers on one hand, must have seemed more
natural.
Since 6 is the coefficient required to turn 10 into 60, the Sumerian
system, by its own dynamic, or rather, because of the inherent properties of
the numbers involved, became a kind of compromise between 6 and 10,
which served as the alternate and auxiliary bases of the sexagesimal system.
Only subsequently could it have been observed that the resulting base had
very valuable arithmetical properties as well as advantages for astronomy
and geometry, which could only have been discovered as mastery of the
counting tool and of the applied sciences progressed. Those properties
and advantages came to seem so considerable and numerous that the
Sumerians gave the main units the names of their own gods.
That is, in my view, the most plausible explanation of base 60. All the
same, it should only be taken as a story - a story for which no archaeologi-
cal proof or even evidence exists, as far as I know. However, if it were the
true story of the origin of the sexagesimal system, then it would give
added support to the anthropomorphic origin of the other common and
historically-attested bases (5, 10 and 20), and thus underline the huge
importance of human fingers in the history of numbers and counting.
Fig. 9.4-
WRITTEN NUMERALS IN ELAM AND MESOPOTAMIA
96
CHAPTER lO
THE DEVELOPMENT OF
WRITTEN NUMERALS IN ELAM
AND MESOPOTAMIA
As we have seen, by the fourth millennium BCE clay was already a
traditional material, not only for building work, but also and above all as
the basic medium for the expression of human thought. In this period, the
Mesopotamian peoples were entirely at ease with clay in a wide range of
applications, and they used it to throw earthenware and ceramic vessels
and figurines, to mix mortar, to mould bricks, and to shape seals, beads,
jewels, and so on. It is therefore not unreasonable to suppose that the
inhabitants of Sumer, long before they devised their written numerals and
writing system, made diverse kinds of clay or earthenware objects or tokens
with conventional values in order to symbolise and to manipulate numbers.
FROM PEBBLES TO ARITHMETIC
Concrete arithmetic (which as we shall see most certainly existed in the
region of Mesopotamia) necessarily derived from the archaic “heap of
pebbles” counting method, put to numerical use. The “pebbles" method is
attested in every corner of the globe and clearly played a major role in the
history of arithmetic - for pebbles first allowed people to learn how to
perform arithmetical operations.
The English word calculus, like the French calcul (which has a more general
meaning of “arithmetic”, “counting operation”, or “calculation”) comes
directly from the Latin calculus, which means ... a pebble, and, by exten-
sion, a ball, token, or counter. The Latin word is related to calx, calcis,
which, like the similar-sounding Greek word khaliks, means “rock” or “lime-
stone”, and which has numerous etymological derivations in modern
European languages, from German Kalkstein, “limestone”, to English calcium
and French calcul, in the sense of “kidney stones”.
Because the Greeks and Romans taught their children to count and to
perform arithmetical operations with the help of pebbles, balls, tokens, and
counters made of stone (probably limestone, which is lighter and easier
to fashion than marble or granite) their word for “doing pebbles” ( calcula-
tion ) has come to refer to all and any of the elementary arithmetical
operations - addition, subtraction, multiplication, and division.
Greek and Arabic both have their own independent etymological proofs
of the origins of arithmetic in the manipulation of stones. Greek psephos
means both “stone” and “number”; Arabic haswa, meaning “pebble”, has
the same root as ihsa, which means “a count (of things)” and “statistics”.
At its simplest, the pebble method is extremely primitive: even more
than the basic forms of notched-stick counting described in Chapter 1, it
represents the “absolute zero” of number-techniques. It can only supply
cardinal numbers, requires no memorisation and no abstraction, and uses
exclusively the principle of one-for-one correspondence.
However, once abstract counting has been mastered, the pebble method
is sufficiently adaptable to allow great strides to be made. In some African
villages, accounts of marriageable girls (and of boys of military age) were
kept until quite recently by this method. On reaching puberty each village
girl gave a metal bangle to the local matchmaker, who threaded it onto a
strap alongside other similar bangles; when her marriage was imminent,
the girl would take back her bangle, leaving the matchmaker with an
accurate and immediate account of the number of “matches” left to make.
It is a most convenient way of performing a subtraction in the absence
of any knowledge of arithmetic as such.
In Abyssinia (now Ethiopia) tribal warriors had a similar device. On
leaving for a foray each warrior placed a pebble on a heap, and on return
to the village, each survivor removed his pebble. The number of unclaimed
pebbles provided the precise total of losses in the skirmish. Exactly the
same device is portrayed in the opening sequence of Eisenstein’s film, Ivan
The Terrible (part I): we see each soldier in the army of Ivan IV Vassilievich,
Tsar of all the Russias, placing a metal token on a tray before setting off for
the siege of Kazan.
In the course of time it became apparent that the device could not be
taken very far, and did not satisfy many perfectly ordinary requirements.
For instance, you need to collect a thousand pebbles just to count up to
1,000! But once the principle of a base in a numbering system had arisen,
pebbles could be used more imaginatively.
In some cultures, the idea arose of replacing natural pebbles with pieces
of stone of various sizes and attributing a conventional value of a different
order of magnitude to each size. So, in a decimal numbering system, a unit
of the first order might be represented by a small pebble, a unit of the
second order (tens) by a slightly larger pebble, a unit of the third order
(hundreds) by a larger stone, a unit of the fourth order (thousands) by an
even bigger one, and so on. To represent the other numbers in the series by
this method, you just needed an appropriate number of pebbles of the
appropriate size.
It was a practical device, but not yet quite serviceable, because it was
hard to find a sufficiency of pebbles or stones of identical sizes and shapes.
97
WHAT DO THE COUNTING TOKENS MEAN?
For some societies there was an additional and quite crucial obstacle if they
inhabited regions where stone was uncommon.
The pebble method was therefore perfected by recourse to malleable
earth, a material far better suited to making regular counting tokens. That
is what happened in Elam and Mesopotamia in prehistoric times, prior to
the invention of writing and of written numerals.
MESOLITHIC AND NEOLITHIC TOKENS
IN THE MIDDLE EAST
In several archaeological sites in the Middle East, in places as far from each
other as Anatolia, the Indus Valley, the shores of the Caspian Sea, and the
Sudan (Fig. 10.2), researchers have unearthed thousands upon thousands
of small objects in a wide variety of sizes and regular geometrical shapes,
such as cones, discs, spheres, pellets, sticks, tetrahedrons, cylinders, and so
on (Fig. 10.1). These are the objects to which we shall apply the generic
term of calculi.
Some of these calculi are inscribed with parallel lines, crosses, and other
similar patterns (Fig. 10.1 B, C, D, E, M, O). Others are decorated with carved
or moulded figurines that are visible representations of different kinds of
beings or things (jars, cattle, dogs, etc.). Finally, there are some that have
neither pattern nor figurine (Fig. 10.1 A, F, G, H, I, J, L, N, P, Q, R, S, T, U).
The oldest calculi found so far, dating from the ninth to the seventh
millennium BCE, come from Beldibi (Anatolia), Tepe Asiab (Mesopotamia),
Ganj Dareh Tepe (Iran), Khartoum (Sudan), Jericho (West Bank)) and Abu
Hureyra (Syria). The most recent, dating from the second millennium BCE,
were found at Tepe Hissar (Iran), Megiddo (Israel), and Nuzi (Mesopotamia).
Most of the calculi were found scattered around at ground level. Others,
however, were found inside or next to egg-shaped or spherical hollow clay
balls or bullae. However, although bullae are not found prior to the fourth
millennium BCE, hundreds of them have been unearthed at Tepe Yahya
(Iran), Habuba Kabira (Syria), Uruk (Mesopotamia), Susa (Iran), Chogha
Mis (Iran), Nineveh (Mesopotamia), Tall-i-Malyan (Iran) and Nuzi
(Mesopotamia).
WHAT DO THE COUNTING TOKENS MEAN?
Denise Schmandt-Besserat has put together all that is known about these
tokens and argues that for the Middle Eastern civilisations of the ninth to
the second millennium BCE they constituted three-dimensional pictograms
of the specific goods or produce which they served to account for in
commercial exchanges.
A Tepe Yahya
Q
B Jarmo
%
C Tepe Hissar
m
D Ganj Dareh Tepe
©
E Susa
F Susa
£
G Susa
Q
H Caydnii Tepesi
©
I Susa
4
J Tepe Guran
4
K Tepe Gawra
I
L Khartoum
Q
M Ur
N Ganj Dareh Tepe
m
O Susa
P Susa
©
Q. Uruk
a
R Beldibi
S Uruk
T Susa
&
u
Ganj Dareh Tepe
Fig. i o . i . Selection of tokens found at various sites
In other words, she believes that the tokens were shaped so as to
represent or symbolise the very things that they “counted", using actual or
schematic images of, for instance, pots, heads of cattle, and so on, and in
some cases were marked with dots or lines to indicate their places in a
numbering series (one rectangular plaque has 2x5 dots on it, for example,
and one cow’s head figurine has 2x3 dots marked on it).
WRITTEN NUMERALS IN ELAM AND MESOPOTAMIA
98
TYPE OF TOKEN
s
D
z.
z
cylinders
discs
spheres and
pellets
cones (various
sizes)
sticks
hollow clay balls
or bullae
M I L
BCE
SITE
REGION
9th
Beldibi
Anatolia
★
★
★
★
★
Tepe Asiab
Mesopotamia
★
★
★
★
★
9th— 8th
Ganj Dareh Tepe
Iran
★
★
★
★
★
8th
Khartoum
Sudan
★
★
★
8th— 7th
Cayonii Tepesi
Anatolia
★
★
★
★
★
7th— 6th
Jericho
West Bank
★
★
Tell Ramad
Syria
★
★
★
★
Ghoraife
Syria
★
★
★
Suberde
Anatolia
★
★
★
★
Jarmo
Mesopotamia
★
★
★
★
★
Tepe Guran
Iran
★
★
★
Anau
Iran
★
★
★
6th
Tell As Sawwan
Mesopotamia
★
★
Can Hasan
Anatolia
★
★
★
Tell Arpichiya
Mesopotamia
★
★
6th— 5tli
Chaga Sefid
Iran
★
★
★
★
★
Tal-i-Iblis
Iran
★
★
★
4th
Tepe Yahya
Iran
★
★
★
★
Habuba Kabira
Iran
★
★
★
★
Warka (Uruk)
Mesopotamia
★
★
★
★
★
Susa
Iran
★
★
★
★
★
★
Chogha Mis
Iran
★
★
★
★
★
Nineveh
Mesopotamia
★
4th-3rd
Tall-i-Malyan
Iran
★
★
★
★
Tepe Gawra
Iran
★
3rd
/emdet Nasr
Mesopotamia
★
★
Kis
Mesopotamia
★
★
★
Tello
Mesopotamia
★
★
★
★
Fara
Mesopotamia
★
★
★
3rd-2nd
Tepe Hissar
Iran
★
★
★
2nd
Megiddo
Israel
★
Nuzi
Mesopotamia
★
Fig. io. 2 . Middle Eastern archaeological sites with finds of clay objects of various shapes and
sizes, some of which are known to have been used for arithmetical operations and for accounts
It is an attractive idea, and if it could be proved it would show that there
was a very sophisticated accounting system in the Middle East in the
earliest periods of the prehistoric record in that area.
However, it is only a hypothesis, and there is no solid evidence to
support it. It presupposes the existence of a sufficiently complex market
economy to have created the need felt for such an elaborate counting
system. Schmandt-Besserat nonetheless takes her argument even further,
and claims that this “three-dimensional symbolic system” is the origin of
Sumerian pictograms and ideograms, that is to say is the source of the
earliest writing system in the world.
Her conclusions derive from the discovery of a very large number of
objects of various shapes (discs, spheres, cones, cylinders, and triangles)
inscribed with exactly the same motifs - parallel lines, concentric circles,
crossed lines - as are found on Sumerian tablets of the Uruk period,
where a cross inside a circle stands for “sheep”, three parallel lines inside
a circle stands for “clothes”, and so on (Fig. 10.3). The signs of Sumerian
writing, she says, are simply two-dimensional reproductions of the three-
dimensional tokens.
This important claim is nonetheless somewhat specious because it
presupposes that there was a completely common and standardised set
of traditions and conventions over a huge geographical area throughout
a period of several thousand years - and what we know of the area and
period suggests on the contrary that its cultures were very diverse. It is
quite wrong to "explain” Sumerian pictograms by the shape of tokens
found as far afield as Beldibi, Jericho, Khartoum, or Tepe Asiab, dating
from eras as varied as the fourth, sixth, and ninth millennia BCE, since
the cultures of these places in those periods probably had nothing whatso-
ever to do with developments in Sumer itself. (S. J. Liebermann gives a
full critique.)
However, Schmandt-Besserat’s general idea is not unacceptable,
provided it is handled more methodically, by studying, not the entire collec-
tion of tokens in existence, but each subset of them in the context of its
particular culture, in its specific location, at a particular period.
There must be major reservations about the overall conclusion concern-
ing the origins of Sumerian writing. If it is ever demonstrated satisfactorily
that there was a proper “system” of three-dimensional representation in
these early Middle Eastern cultures, we will certainly find not one, but
many different “systems” in the area. If we do establish a derivation from
one such system to Sumerian writing, we are unlikely to establish it for
more than a small number of individual signs.
All the same, these three-dimensional tokens must have meant something
for their inventors and users, even if they did not form part of a system
99
FROM TOKENS TO CALCULI
Token shape
Sumerian sign
Known meaning of the sign
a
7
jar, pot, vase
oil, grease, fat
•
e
sheep
4
7
bread, food
m
©
leather
&
As
clothing
Fig. 10 . 3 . Comparison of tokens and their allegedly corresponding pictograms in early Sumerian
writing (from D. Schmandt-Besserat, 1977)
in the proper sense. They are obviously connected to ancient practices of
symbolisation, which we can see in use on painted ceramics and in glyptics.
One conclusion that might well come out of these speculations if sufficient
evidence is found is that these tokens represent perhaps the final interme-
diate stage in the evolution of purely symbolic expressions of thought into
formal notations of articulated language.
MULTIFUNCTION OBJECTS
The variety of the tokens is so great, their geographical locations so diverse,
and their chronological origins are so widely separated that they could not
possibly have belonged to a single system.
Even within a single period and place, they did not all serve the same
purposes.
We can all the same make a few plausible guesses about their meaning if
we bear in mind the specific nature of the cultures to which they belong.
Some of the tokens, for example, that have holes in the middle and were
found threaded on string, were probably objects of personal decoration.
Such “necklaces” may also have served as counting beads, much like
rosaries, allowing priests to count out gods or prayers. Other tokens decor-
ated with the heads of animals may have been amulets, invoking the spirits
of the animals represented, in terms of superstitions about the protective
values of the different species (warding off the evil eye, illnesses, accidents,
etc.). And since clay was plentiful and easy to shape, we can suppose that
a fair number of these tokens were playing pieces, for ancient games like
fives, draughts, chess, and so on.
FROM TOKENS TO CALCULI
However, the most interesting tokens from our point of view, and whose
function is not in any doubt, are those small clay objects of varying shapes
and sizes found inside the hollow clay balls called bullae. They were in
use in Sumer and Elam (a region contiguous to Mesopotamia, covering
the western part of the Iranian plateau and the plain to the east of
Mesopotamia proper) from the second half of the fourth millennium BCE,
and they served both as concrete accounting tools and also, as we shall see,
as calculi which permitted the performance of the various arithmetical oper-
ations of addition, subtraction, multiplication, and even division. The
Assyrians and the Babylonians called these counting tokens abnu (plural:
abnati), a word used to mean: 1. stone, 2. stone object, 3. stone (of a fruit),
4. hailstone, 5. coin [from R. Labat (1976) item 229]. Long before that, the
Sumerians had called them imna, meaning “clay stone” (S. J. Liebermann
in AJA). We will call them calculi, remembering that by this term we refer
exclusively to the tokens found inside or close to hollow clay balls, the bullae.
THE FORMAL ORIGINS OF SUMERIAN NUMERALS
Archaic Sumerian numerals suggest very strongly at first glance that they
derive from a pre-existing concrete number- and counting system, but
they also seem to have obviously formal origins. The various symbols used
(Fig. 8.2, repeated in Fig. 10.4 below) look very much like some of the
calculi "copied” onto clay tablets once writing had been invented: specifi-
cally, the little cone, the pellet, the large cone, the perforated large cone,
the sphere, and the perforated sphere. To put things the other way round
(Fig. 10.4):
• the fine line representing the unit in archaic Sumerian numerals
looks a two-dimensional representation of the small cone token;
• the small circular imprint representing the tens looks like a pellet-
shaped token;
• the thick indentation for 60 looks like a large cone;
• the thick dotted indentation for 600 looks like a large perforated
cone;
» the large circular imprint (3,600) looks like a sphere;
• the large dotted circular imprint (36,000) looks like a perforated
sphere.
These resemblances are so obvious that the relationship would have
to be accepted even if there were no other proof. But as we shall see, the
archaeological record contains more than adequate confirmation of these
identifications.
WRITTEN NUMFRAI.S IN ELAM AND MESOPOTAMIA
100
SPOKEN
NUMERALS
CALCULI
WRITTEN NUMERALS
Number-
names
_
Archaic
Cuneiform
Mathematical
structure
1
ges
a
small
cone
0
T
1
10
u
«
pellet
•
10
60
ges
a
large
cone
T
10.6
(= 60)
600
gesu
a
perforated
large cone
3
K
10 . 6.10
(= 60.10)
3,600
sar
Q
sphere
O
10 . 6 . 10.6
(= 60 2 )
36,000
sar-u
©
perforated
sphere
©
10 . 6 . 10 . 6.10
(= 60 2 .10)
216,000
sargal
?
?
$
10 . 6 . 10 . 6 . 10.6
(= 60 3 )
Archaeological
From mid-
From
From
date (BCE)
4th millennium
c. 3200
c. 2650
Fig. 10.4. Number-names, numerals and calculi of Sumerian civilisation. The calculi come from
several Mesopotamian sites (Uruk, Nineveh, Jemdet Nasr, Kis, Ur, Tello, Surrupak, etc.
THE HOLLOW CLAY BALLS FROM
THE PALACE OF NUZI
It was in 1928-29 that Mesopotamian calculi were first properly identified,
when the American archaeologists from the Oriental Research Institute in
Baghdad excavating the Palace of Nuzi (a second-millennium BCE site near
Kirkuk, in Iraq) came across a hollow clay ball clearly containing “some-
thing else”, inscribed with cuneiform writing in Akkadian (Fig. 10.5) which
in translation reads as follows:
Abnati (“things”) about sheep and goats:
21 ewes which have lambed 6 she-goats that
6 female lambs have had kids
8 adult rams 1 he-goat
4 male lambs [2] kids
The sum of the count is 48 animals. When the clay ball was opened, it was
found to contain precisely 48 small, pellet-shaped, unbaked clay objects
(which were subsequently mislaid). It seemed logical to assume that these
tokens had previously been used to count out the livestock, despite the
difficulty of distinguishing between the different categories by this system
of reckoning.
Fig. 10 . 5 . Hollow clay ball
or bulla found at the Palace of
Nuzi, 48mm x 62mm x 5 Omm.
Fifteenth century BCE. From
the Harvard Semitic Museum.
Cambridge, MA (inventory no.
SMN1854)
The archaeologists might have thought nothing of their discovery
without a chance occurrence that suddenly explained the original purpose
of the find. One of the expedition porters had been sent to market to buy
chickens, and by mistake he let them loose in the yard before they had been
counted. Since he was uneducated and did not know how to count, the
porter could not say how many chickens he had bought, and it would have
been impossible to know how much to pay him for his purchases had he not
come up with a bunch of pebbles, which he had set aside, he said, “one for
each chicken”. So an uneducated local hand had, without knowing it,
repeated the very same procedures that herdsman had used at the same site
over 3,500 years before.
Thirty years later, A. L. Oppenheim at the University of Chicago carried
out a detailed study of all the archaeological finds at Nuzi, and discovered
that the Palace kept a double system of accounting. The cuneiform tablets
of the Palace revealed the existence of various objects called abnu (“stones”)
that were used to make calculations and to keep a record of the results. The
texts written on the tablets make clear reference to the "deposit” of abnu, to
“transfers” of the same, and to “withdrawals”. The meticulous cuneiform
accounts made by the Palace scribes were “doubled”, as Schmandt-Besserat
explains, by a tangible or concrete system. One set of calculi may for instance
have represented the palace livestock. In spring, the season of lambing, the
appropriate number of new calculi would have been added: calculi repre-
senting dead animals would have been withdrawn; perhaps calculi were even
moved from one shelf to another when animals were moved between
flocks, or when flocks moved to new pasture, or when they were shorn.
101
THF. HOLLOW CLAY BALLS FROM NUZ
The hollow clay ball was therefore probably made by a Palace accountant
for recording how many head of livestock had been taken to pasture by
local shepherds. The shepherds were illiterate, to be sure, but the accoun-
tant must have known how to count, read, and write: he was probably a
priest, as he possessed the great privilege of Knowledge, and must have
been one of the managers of the Nuzi Palace’s goods and chattels. The proof
of this lies in the Akkadian word sangu, which means both “priest" and
“manager of the Temple’s wealth”; it is written in cuneiform in exactly the
same way as the verb manu, which means “to count”.
When shepherds left for pasture, the functionary would make as many
unbaked clay pellets as there were sheep, and then put them inside the clay
“purse”. Then he would seal the purse and mark on it, in cuneiform, an
account of the size of the flock, which he then signed with his mark. When
the shepherd came back the purse could be broken open and the flock
checked off against the pellets inside. There could be no disputing the
numbers, since the signed account on the outside certified the size of
the flock as far as the masters of the Palace were concerned, and the calculi
provided the shepherd with his own kind of certified account.
The later discovery of an oblong accounting tablet shaped like the base of
the hollow clay ball in the ruins of the same palace, but from a higher (and
therefore more recent) stratum, gave further support to Oppenheim’s views.
The story now moves to Paris, where, at the Musee du Louvre, there are
about sixty of these hollow clay balls brought back c. 1880 by the French
Archaeological Mission to Iran, which had been excavating the city of
Susa (about 300 km east of Sumer, in present-day southwestern Iran, Susa
was the capital of Elam and then of the Persian Empire under Darius). Up
until recently the only interest that had been shown in them concerned
the imprints of cylinder-seals with which most of them are decorated
(Fig. 10.10). Several of the bullae had been broken during shipment to
Paris, other had been found broken. All the same, some of them were
intact, and sounded like rattles when shaken. X-ray photography showed
that they contained calculi - but not all of the same uniform type. When
some of them were very carefully opened, they were found to contain clay
discs, cones, pellets, and sticks (Fig. 10.6)
As P. Amiet then argued, these “documents”, since they came from a site
dated about 3300 BCE, proved that Elam had an accounting system far
more elaborate than that of Nuzi with its plain “unit counters”, and had
it 2,000 years earlier. In other words, this counting system had survived
for two millennia, but had regressed over that period, losing the use of
a base, and retreating to a rudimentary and purely cardinal method.
It was therefore correctly assumed that the counting system of Susa
consisted of giving tangible form to numbers by the means of various calculi
which symbolised numerical values both by their own number and by their
respective shapes and sizes, which corresponded to some order of magni-
tude within a given number-system (for example, a stick was a unit of the
first order of magnitude, a pellet for a unit of the second order, a disc for
a unit of the third order, and so on).
More recent finds in Iran (Tepe Yahya, Chogha Mis, Tall-i-Malyan,
Sahdad, etc.), in Iraq (Uruk, Nineveh, Jemdet Nasr, Kis, Tello, Fara, etc.),
and in Syria (Habuba Kabira) have proved Oppenheim and Amiet to be
correct. What they have also shown is that the system was not restricted
to Elam, but that similar accounting methods were used throughout
the neighbouring region, including Mesopotamia. These methods are
thus even more ancient than the one used for the accounting tablets of the
Uruk period.
FROM CLAY BALLS TO ACCOUNTING TABLETS
It then seemed very likely that the archaic accounting tablets of Sumer were
directly descended from the clay calculi-md-bulla accounting system. The
archaic Sumerian figures obviously were related to the calculi', and, unlike
the later, perfectly rectangular tablets that were made to a standard pattern,
the archaic counting tablets are just crude oblong or roughly oval slabs
(Fig. 8.1 C above). So there really had been a point in time when the stones
were supplanted by their own images in two-dimensional form, and the
hollow clay balls replaced by these flat clay slabs. But this remained only
a conjecture in the absence of all the archaeological evidence needed to
reconstitute the intermediate stages of the supposed development and of
evidence to allow firm datings.
In the 1970s, the French Archaeological Delegation to Iran (DAFI),
under the direction of Alain Le Brun, excavated the Acropolis of Susa,
and established a far more accurate
and substantiated stratigraphy of
Elamite civilisation than had previ-
ously been possible, and, in 1977-
78, important finds were made
which make the transition compre-
hensible in archaeological terms.
A word of warning, however: the
development we describe below is
attested only at Susa. Nonetheless
there are good reasons for believ-
ing that much the same thing
happened at Sumer.
Fig. io.6. Sketch of the contents of an
unbroken bulla, as revealed by X-rays
WRITTEN NUMERALS IN ELAM AND MESOPOTAMIA
The first reason is that Elamite civilisation is pretty much contemporary
with Sumer, and flourished in very similar fashion in precisely similar
circumstances in the second half of the fourth millennium BCE. For that
reason various aspects of Elamite civilisation are used as reference points
(or as potentially applicable models) for the civilisation of Uruk. All the
same the Elamites retained many features that are distinct from those of
their Mesopotamian neighbours.
Side 1
Fig. 10 . 7 . Proto-Elamite tablet (Susa,
level unknown), c. 3000 BCE.J. Schell (1905)
identified this as an inventory of stallions
(erect manes), mares (flat manes), and colls
(no manes), with the numbers of each indi-
cated by various indentation-marks. Side 2
bears the imprint of a cylinder-seal repres-
enting standing and resting goats. Paris,
Musee du Louvre, Sb 6310
The second reason is that the Elamites, like the Sumerians, were fully
conversant with the use of clay for expressing human thought visually and
symbolically, and later on in using it to represent articulated language.
For we know that the Elamites acquired a writing system around 3000
BCE, the earliest traces of which are the clay “tablets” (Fig. 10.7) found
at several Iranian sites, mainly at Susa, at archaeological level XVI. Like
archaic Sumerian tablets, they bear on one side (sometimes both sides)
a number of numerical signs alongside more or less schematic drawings,
and occasionally the imprint of a cylinder-seal.
And finally, as we have seen, the system of calculi and bullae was used in
Elam as well as Sumer since at least 3500-3300 BCE.
Such manifest analogies between the two civilisations lead us to hope
that new archaeological finds at Sumerian sites will one day establish once
and for all the relationship between Sumer and Elam.
WHO WERE THE ELAMITES?
The oldest Iranian civilisation arose in the area now called Khuzestan.
Its people called themselves Haltami, which the Bible transforms into Elam.
The origins of Elam are as ill understood as its language, despite the efforts
of many linguists to decipher it. We know only that the name of Elam
means "land of God”. Elamite appears to be an agglutinative language, like
Sumerian and other Asianic languages; some linguists think it belongs to
the Dravidian group (southern India) and is related to Brahaoui, which is
currently spoken in Baluchistan. It should be noted that from the beginning
of the third millennium BCE there appear to have been close relations
between Elam and Tepe Yahya (Kirman), which is located on a possible
migration route from India. The Elamite tablets found there have been
dated as late fourth millennium BCE.
It seems most likely that the Elamites arrived and settled in the area that
was to bear their name in the fifth millennium BCE, joining a farming culture
of which the earliest traces date from the eighth millennium BCE. The
earliest pieces of Susan art are decorated ceramics, showing archers and
beasts of prey (Tepe Djowzi), and horned snakes (Tepe Bouhallan), and Susa,
which became a full-blown city in the fourth millennium, seems to have been
the most important Elamite town. Painted ceramics were abandoned during
what Amiet calls the earlier period of “proto-urban” Elamite civilisation.
Throughout its history, Mesopotamia had relations with Elam, from
which it imported wood, copper, lead, silver, tin, building stone, and rare
stones such as alabaster, diorite, and obsidian, but from the start of the
third millennium BCE relations were intense. The periods are divided as
follows: from 3000 to 2800 BCE, the palaeo-Elamite period; from 2800 to
2500 BCE, the Sumero-Elamite period (subdivided into early and late,
during which Sumerian influence is very noticeable); from 2500 to 1850
BCE, the Awan Dynasty, interrupted by an Akkadian conquest, was
replaced by the dynasty of Shimash.
Susa became the central city in the second millennium BCE, and Elamite
civilisation reaches its apogee in the middle of the thirteenth century BCE
under the reign of Untash Gal who built Tchoga-Zanbil. During the first
millennium BCE, Elam is closely connected to the Kingdom of Anshan
which, from the sixth century BCE, became one of the key points in the
Achaemenian Persian Empire.
THE STAGES OF ELAMITE ACCOUNTING
With the help of the latest discoveries made by DAFI, we can now
reconstruct the stages in the development of accounting systems in Elam.
We begin in the second half of the fourth millennium BCE, in an advanced
urban society where trading is increasing every day. And with an active
economy, there is a pressing need to keep durable records of sales and
purchases, stock lists and tallies, income and expenditure . . .
103
First stage: 3500 - 3300 BCE
Levels: Susa XVIII; Uruk IVb. For sources, see Fig. 10.4, 10.8, and 10.10
Susan officials have an accounting system through which they can
represent any given number (for example, a price or a cost) by a given
number of unbaked clay calculi each of which is associated with an order
of magnitude according to the following conventions:
1
Q
©
Stick
Pellet
Disc
1
10
100
Scale in cm
0 12 3
& ^
Small cone Large perforated
cone
300 3,000
Fig. io.8. The only calculi found in or very near hollow clay balls at the Acropolis of Susa.
The values shown derive from the decipherment explained in Chapter 11 below. From DAFI 8,
plate 1 (Susa, level XVI 1 1)
Intermediate numbers are represented by using as many of each type of
calculus as required. For example, the number 297 calls for 2 discs, 9 pellets,
and 7 sticks:
Fig. 10.9-
Fig. io.io. Exterior of a bulla marked with a cylinder-seal.
Susa, c. 3300 BCE. From the Musee du I.ouvre, item Sb 1943
You then place these objects with conventional values (whose use is not
entirely dissimilar to our current use of coins or standard weights) into a
hollow ball, spherical or ovoid in shape (Fig. 10.10), the outside of which is
then marked by a cylinder-seal, so as to authenticate its origin and to guar-
antee its accuracy. For in Elam, as in Sumer, men of substance each had
their own individual seal - a kind of tube of more or less precious stone on
which a reversed symbolic image was carved. The cylinder-seal, invented
around 3500 BCE, was its owner’s representative mark. The owner used it
to mark any clay object as his own, or to confer his authority on it, by
rolling the cylinder on its axis over the still-soft surface (Fig. 10.11).
THF. STAGES OF ELAMITE ACCOUNTING
Let us imagine we are at the Elamite capital of Susa. A shepherd is about
to set off for a few months to a distant pasture to graze a flock of 297 sheep
that a wealthy local owner has entrusted to him. The shepherd and the
owner call on one of the city’s counting men to record the size of the flock.
After checking the actual number of sheep, the counting master makes a
hollow clay ball with his hands, about 7 cm in diameter, that is to say hardly
bigger than a tennis ball. Then, through the thumb-hole left in the ball, he
puts inside it 2 clay discs each standing for 100 sheep, 9 pellets that each
stand for 10 sheep, and 7 little sticks, each one representing a single animal.
Total contents: 297 heads (Fig. 10.9).
When that is done, the official closes up the thumb-hole, and, to certify
the authenticity of the item he has just made up, rolls the owner’s cylinder-
seal over the outside of the ball, making it into the Elamite equivalent of
a signed document. Then to guarantee the whole thing he rolls his own
cylinder-seal over the ball. This makes it unique and entirely distinct from
all other similar-looking objects.
Fig. lo.ii. Cylinder-seal imprints from accounting documents found at Susa
The counting master then lets the bulla dry and stores it with other
documents of the same kind. With its tokens or calculi inside it, the bulla
is now the official certification of the count of sheep that has taken place,
and serves as a record for both the shepherd and the owner. On the
shepherd’s return from the pastures, they will both be able to check
whether or not the right number of sheep have come back - all they need
to do is break open the ball, and check off the returning sheep against the
tokens that it contains.
At about the same period, the Sumerians used a very similar system:
hollow clay balls have been found at Warka at the level of Uruk IVb, at
Nineveh and Habuba Kabira (Fig. 10.4). The Sumerians, however, were
accustomed to counting to base 60, using tens only as a supplemen-
tary system to reduce the need for memorisation (Fig. 8.5, 8.6 above), and
the tokens that they used were also shaped rather differently. At Sumer,
WRITTEN NUMERALS IN ELAM AND MESOPOTAMIA
the small cone stood for 1, the pellet for 10, the large cone for 60, the perfo-
rated large cone stood for 600, the sphere represented 3,600, and the
perforated sphere meant 36,000 (Fig. 10.4).
It was a sophisticated system for the period, since values were regularly
multiplied by 10 by means of the perforation. By pushing a small circular
stylus through the cone signifying 60, or through the sphere signifying
3,600, the values of 600 (60 x 10) and 36,000 (3,600 x 10) were obtained.
The hole or circle was thus already a virtual graphic sign for the pellet, with
a value of 10.
Let us now imagine ourselves in the market of the royal city of Uruk,
capital of Sumer. A cattle farmer and an arable farmer have made a deal to
exchange 15 head of cattle against 795 bags of wheat. However, the live-
stock dealer has only got 8 head of cattle at the market, and the grain seller
has only 500 bags of wheat immediately available. The deal is done
nonetheless, but to keep things above board there has to be a contract. The
cattle man agrees to deliver a further 7 cattle by the end of the month, and
the arable farmer promises to supply 295 bags of grain after that year’s
harvest. To make a firm record of the agreement, the cattle man makes
a clay ball and puts in 7 small cones, one for each beast due, then closes
the ball and marks its surface with his own cylinder-seal, as a signature. The
arable farmer, for his part, makes another clay ball and puts in it 4 large
cones, each one standing for 60 bags of wheat, 5 pellets each standing for
10 bags, and 5 small cones for the 5 remaining bags due, then seals and
signs the clay ball in like manner. Then a witness puts his own “signature"
on the two documents, to guarantee the completeness and accuracy of
the transaction. Finally, the two traders exchange their bullae and go their
separate ways.
So although this remains an illiterate society, it possesses a means of
recording transactions that has exactly the same force and value as written
contracts do for us today.
At a time when cities were still relatively small, and where trade was still
relatively simple, business relations were conducted by people who knew
each other, and whose cylinder-seals were unambiguously identifiable. For
that reason, the nature of a transaction recorded in a bulla is implicit in the
identity of the seal(s) upon it: the symbolic shapes on the outside of the clay
ball tell you whether you are dealing with this farmer or that miller, with
a particular craftsman or a specific potter. As for the numbers involved in
the transaction, they are unambiguously recorded by the nature and
number of the calculi inside.
Cheating is therefore ruled out. Each party to the deal possesses the
record of what his partner owes him, a record certified by his business
partner’s own identity, in the form of his seal.
104
Second stage: 3300 BCE
Level: Susa XVIII. For sources see Fig. 10.13
The great defect of the system in place was that the hollow clay balls had to
be broken in order to verify that settlements conformed to the contracts.
To overcome this, the idea arose of making various imprints on the outer
surface of the bullae (alongside the imprints of the necessary cylinder-seals)
to symbolise the various tokens or calculi that are inside them. Technically,
the device harks back to the more ancient practice of notching, but it is
quite altered in its significance by the new context.
The corresponding marks are: a long, narrow notch, made by a stylus
with its point held sideways on to the surface, to represent the stick; a small
circular imprint, made by the same stylus pressed in vertically, to represent
the pellet; a large circular imprint, made by a larger stylus or just by press-
ing in a finger-tip, to represent a disc; a thick notch, made by a large stylus
held obliquely, to represent a cone; and a thick notch with a circular
imprint to represent a perforated cone.
!•' 8 °C f]CJ f
long narrow small circular large circular thick notch perforated
notch imprint imprint thick notch
Fig. io.12. Numerical markings on bulla e found at Susa
This constitutes a kind of resume of the contract, or rather a graphic
symbolisation of the contents of each accounting “document”.
Henceforth, an Elamite bulla containing (let us say) 3 discs and 4 sticks
(making a total of 3 x 100 + 4 = 304 units) carries on its outer face,
alongside the cylinder-seal imprints, 3 large circular indentations and 4
narrow lines. No longer is it necessary to break open the clay balls simply
to check a sum or to make an inventory - because the information can now
be “read” on the outside of the bullae.
The cylinder-seal imprint or imprints show the bulla’s origin and
guarantee it as a genuine document, and the indentations specify the
quantities of beings or things involved in the accounting operation.
Third stage: c. 3250 BCE
Level: Susa XVIII. See Fig. 10.15 below
These indentations thus constitute real numerical symbols, since each of
them is a graphic sign representing a number. Together they make up a
genuine numbering system (Fig. 10.14). So why carry on using calculi and
putting them in bullae, when it’s much simpler to represent the corre-
sponding values by making indentations on slabs of clay? Mesopotamian
105
and Elamite accountants very quickly realised that of the two available
systems, one was redundant, and the calculi were rapidly abandoned. The
spherical or ovoid bullae came to be replaced by crudely rounded or oblong
clay slabs, on which the same information as was formerly put on the casing
of the bullae was recorded, but on one side only.
The cylinder-seal imprint remained the mark of authenticity on these
new types of accounting records, whose shape, at the start, roughly imitates
that of a bulla. The sums involved in the transaction are represented on the
soft clay by graphic images of the calculi that would previously have been
Bulla A corresponding Calculi Cylinder-seal
DAFI 8, bulla 13
Fig. 3.2 DAFI 8, plate I DAFI 8, Fig. 6.13
Bulla B Calculi Cylinder-seal
DAFI 8, bulla 4,
Fig. 3.1 and plate III DAFI 8, plate I DAFI 8, Fig. 7.8
Bulla C Calculi Seal
DAFI 8, bulla 2, DAFI 8, plate I DAFI 8, Fig. 3.3
Fig. 3.3 and plate 1.3
Scale in cm — i
0 12 3
Fig. 10.13. Bullae containing the same number of calculi as are symbolised on the outer surface
by indentations next to the cylinder-seal imprints. Susa, level XVIII (approx. BCE 3300), excavated
by DAFI in 1977-1978. Similar bullae have been found at Tepe Yahya and Habuba Kabira, but
none so far at Uruk.
THE STAGES OF ELAMITE ACCOUNTING
enclosed in a bulla. This stage therefore marks the appearance of the first
“accounting tablets” in Elam.
It should be noted that the three stages laid out above occurred in a rela-
tively short period of time, since all the evidence for them is attested at the
same archaeological level (Susa XVIII), in the same room, and on the same
floor level. The imprint of the same cylinder-seal on one bulla and two
tablets (see bulla C in Fig. 10.13 and tablet B in Fig. 10.15 below, for
example) seems to confirm that both systems existed side by side at least
for a time.
found inside bullae and
on the ground at the
Acropolis of Susa
(see also Fig. 10.6, 10.8,
and 10.10)
found on the outer side
of bullae of the second
kind and on the number-
tablets excavated at Susa
(see Fig. 10.13, 10.15,
and 10.16)
I B I
found on the so-called
proto-Elamite tablets
(Fig. 10.7 and 10.17)
Narrow and
long notches
Small circular
imprints
Discs
rge circular
prints
Thick notches
Perforated
Thick notches
with a small
circular imprint
“Winged” circular
imprint
SUSA XVIII
SUSA XVIII and XVII
SUSA XVI, XV and XIV, etc.
Fig . io.)4. The indentations made on the outer side of the bullae imitate the shape of the calculi
that are enclosed. Moreover, these marks resemble not only the number-tablets found at Susa but
also the figures on the proto-Elamite tablets of later periods.
WRITTEN NUMERALS IN E I. A M AND MESOPOTAMIA
106
Fourth stage: 3200-3000 BCE
Levels: Susa XVII; Uruk IVa. See Fig. 8.1 above and 10.16 below
This stage sees only a slow refinement of the system in place already: exactly
the same types of information are included on the accounting tablets of the
fourth period as on those of the third. However, the tablets themselves
become less crudely shaped, the numbers are less deeply indented in the
clay, and their shapes become more regular. In addition, the cylinder-seals
are now imprinted on both sides of the tablet, and not just on the “top”.
However, like the earlier bullae and crude tablets, this stage of develop-
ment is still not “writing” in the proper sense. The notation records only
numerical and symbolic information, and the things involved are described
only in terms of their quantity, not by signs specifying their nature. Nor is
the nature of the operation indicated by any of these documents: we have
Fig. 10 . 15 . Roughly circular or oblong tablets containing indented numerical marks (similar to
those found on bullae,) alongside one or two cylinder-seal imprints. Items dated c. 3250 BCE, from
Susa level XVIII, excavated dy DA FI in 1977-1978.
no idea if they are records of a sale, a purchase, or an allocation, nor can we
know the names, the numbers, the functions, or the locations of any of the
parties to the transaction. We have already made the assumption that
the cylinder-seals, since they indicate the identities of the contracting
parties, would also have indicated the type of transaction in a society where
Side 1 Side 2
F i g . 10 . 16 . Numerical tablets from Susa level XVII , c. 3200- 3000 BCE, excavated by DA FI
in 1972
107
THF. STAGES OF ELAMITE ACCOUNTING
people were known to one another. This makes very clear just how concise,
but also how imprecise are the purely symbolic visual notations of these
documents, which constitute the trace of the very last stage in the prehis-
tory of writing. Cylinder-seal imprints do in fact disappear from the tablets
as soon as pictograms and ideograms make their appearance.
Fig. 10.17. The first proto-Elamite tablets. They are less crude, rectangular tablets giving written
name-signs alongside the corresponding numbers. From Susa, c. 3000-2800 BCE; excavated by
DAFI in 1969-1971 (Cf. A. Le Brun)
At Sumer, writing emerged at the same time as these regular tablets from
Elam. The first Uruk tablets date from 3200-3100 BCE (Fig. 8.1 above) and,
although they remain exclusively economic documents, they use a notation
(archaic Sumerian numerals) which is founded not on making a “picture” of
a vague idea, but on something much more precise, analytical and articu-
lated. In tablet E of Fig. 8.1, for example, you can see how the document is
divided into horizontal and vertical lines, marking out squares in which
pictograms are placed beside groups of numbers. Sumerian tablets are thus
ahead of the Susan ones of the same period: Sumer has something like
writing, and Susa has only symbols.
Fifth stage: 3200-2900 BCE
Level: Susa XVI. See also Fig. 10.17, tablets A, B, C
The tablets from this period are thinner and more regularly rectangular
(standardised), but most significantly they carry the first signs of “proto-
Elamite” script alongside numerical indentations. The purpose of the signs
is to specify the nature of the objects involved in the transaction associ-
ated with the tablet. On several tablets found at Susa XVI, there are no
cylinder-seal imprints.
Sixth stage: 2900-2800 BCE
Level: Susa XV and XIV. See Fig. 10.17, D, E and F
In this period, the proto-Elamite script on the tablets grows to cover more
of the surface than the number-signs. Could this mean that the script might
hold the key to the grammar of the language? Is proto-Elamite the earliest
alphabetic script? We do not know, as it remains to be deciphered.
THE PROBLEMS OF SO-CALLED
PROTO-ELAMITE SCRIPT
This script appeared at the dawn of the third millennium BCE and spread
from the area around Susa to the centre of the Iranian plateau. It remained
in use in Elam until around 2500 BCE, when it was supplanted by
cuneiform writing systems from Mesopotamia, whence derived Elamite
script proper, whose final form was neo-Elamite.
How did proto-Elamite arise? Some scholars believe the Elamites
invented it, independently of the Sumerians. This presupposes that it
resulted from a similar set of steps, starting from identical circumstances,
and following the same generic idea based on earlier rudimentary trials in
the area. That is not implausible, especially in the light of the developments
we have just charted.
WRITTEN NUMERALS IN ELAM AND MESOPOTAMIA
108
Other scholars take the opposite view, namely that proto-Elamite script
was inspired by Sumerian. This is also quite plausible, even if the nature of
the “inspiration" must have been quite a distant one. Some of the proto-
Elamite signs look as if they might be related to specific Sumerian
pictograms and ideograms, but most of the signs are too different to allow
any systematic comparison of the two scripts. On the other hand, it may
well be that the Sumerian invention of writing inspired their neighbours
the Elamites (Uruk and Susa are less than two hundred miles apart) to
invent a writing of their own. Sumerian accounting tablets are one or two
centuries older than their Elamite equivalents, and there is no doubt in
which direction the invention flowed.
It seems probable that writing would have been invented in Susa even
without the example or inspiration of Sumer, since all the social and
economic dynamics that led to the invention of writing elsewhere were
present amongst the Elamites. For as the history of numbers shows, people
in similar circumstances and faced with similar needs often do make very
similar inventions, even when separated by centuries and continents.
Be that as it may, proto-Elamite script remains a mystery. The signs
almost certainly represented beings and things of various kinds, but the
forms used are simplified and conventionalised to a point where guessing
their meaning is impossible. We also know next to nothing about the
language which this script represents.
0
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2
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i
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90
9
£
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Fig. 10.18. The signs of proto-Elamite script. References: Mecquenen; Scheil; Meriggi
109
THE INVENTION OF THE BALANCE SHEET
CHAPTER 11
THE DECIPHERMENT OF A
FIVE-THOUSAND-YEAR-OLD
SYSTEM
In 1981, when I published the first edition of The Universal History of
Numbers, the number-signs in the proto-Elamite script (Fig. 11.1) still
presented major problems.
A table drawn up by W. C. Brice (1962), and later also referred to by
A. Le Brun and F. Vallat (1978), clearly shows how these number-symbols
received very varied, indeed contradictory, interpretations over the years on
the part of the majority of epigraphists and specialists in these questions.
Despite the great difficulties, I decided to apply myself to the task. In
1979 I began my researches which, one year later, culminated in the
complete decipherment of these number-signs, after close examination
of a large number of invoice tablets which had been discovered by the
French Archaeological Mission to Iran at the end of the last century. These
documents may be found in the collections of the Louvre and the Museum
of Teheran.
We shall come shortly to the method which I followed. But, in order to
appreciate it, we must first make yet another visit to the land of Sumer . . .
THE INVENTION OF THE BALANCE SHEET
IN SUMERIA
The period from 3200 to 3100 BCE saw, as we have observed, the
beginnings of written business accounts.
At first, however, the system was primitive. The documents held only
one kind of numerical record at a time: one tablet for 691 jugs, for example
(Fig. 8.1 C above), another tablet for 120 cattle (Fig. 8.1 D), another for 567
sacks of corn, another for 23 chickens, yet another for 89 female slaves
imported from abroad, and so on.
But from around 3100 BCE as business transactions and distributions of
goods became increasingly numerous and varied, the inventories and the
accounts for each transaction also grew more complex and voluminous,
and the accountants found they had to cut down on the cost of clay. From
this time on the pictures and the numbers took up increasing amounts
of space on the tablets. Onto a single rectangular sheet of clay, divided
into boxes by horizontal and vertical lines, were recorded inventories of
livestock in all their different kinds (sheep, fat sheep, lambs, lambkins,
ewes, goats, kids male and female or half-grown, etc.) in all necessary detail.
A single tablet, too, was used to summarise an agricultural audit in which
all the different kinds of species were distinguished.
Fig. li.i. Theproto-
Elamite number-signs
A
>
B
#
C
•
•©•
0«
D
03
E
)
F
0
G
©
H
o
I
3
J
3
K
L
12=3
M
3
N
0
0
P3B
F
G
H
M
N
I
©
o
a
3
System proposed by Scheil
See MDPvi (1905)
1
10
100
1,000
10,000
System proposed by Scheil
See MDP xvn (1923)
1
10
100
60
600
System proposed by Scheil
See MDP xvu (1923)
1
10
100
600
6,000
System proposed by Langdon
SeeJRAS (1925)
1
?
100
1,000
10,000
System proposed by Scheil
See MDP xxvi (1935)
1
10
100
1.000
10,000
System proposed by de Mecquenem
See MDP xxxi (1949)
1
10
100
300
1,000
Fig. u. 2 . Various contradictory conclusions drawn over the years concerning the values of the
proto-Elamite number-signs
THE DECIPHERMENT OF A F I V E -T H O U S A N D - Y E A R - O I. D SYSTEM
But then: the balance sheet was invented. Now people wrote on both sides
of the tablet: the “recto” side bore the details of a transaction, the “verso”
the totals under the various headings.
The idea took hold, and with refinement proved to be of the greatest
usefulness. At Uruk, in 2850 BCE, a proposal of marriage has been made.
The girl’s father and the father of the future spouse have just agreed on the
“bride price”. When the ceremony has taken place, the bride’s father will
receive from the other 15 sacks of barley, 30 sacks of corn, 60 sacks of
beans, 40 sacks of lentils, and 15 hens. But, in view of the frailties of human
memory and in order to avoid any quarrels later, the two men betake them-
selves to one of the religious leaders of the town in order to draw up the
contract in due form and give the force of law to the engagement.
Having taken note of all the elements of the marriage contract, the
notary then fashions a roughly rectangular tablet of clay, and takes up his
“tracing tools”.
For writing, he uses two ivory sticks of different cross-section, pointed at
one end and, at the other, fashioned into a kind of cylindrical stylus
(Fig. 8.10 above). The pointed ends are used to draw lines or to trace
pictograms on the soft clay (Fig. 8.11 above), and the cylindrical styluses are
used to mark numbers by pressing at a given angle on the surface of the
tablet. According to the angle between stylus and the tablet, the impression
made on the soft clay will be either a notch or a circular imprint, whose size
will depend on the diameter of the stylus which is used. As in Fig. 8.12
above, this will be a narrow or a wide notch, according as the wide or
narrow stylus is used, if the angle is 30-45°; or it will be a circular
imprint of small or large diameter, according to the stylus, if it is applied
perpendicular to the surface of the tablet.
Then, holding the tablet with its long side horizontal, the scribe draws four
vertical lines, thereby dividing it into five sections, one for each item in the
contract. At the bottom of the rightmost division he draws a “sack of barley”,
in the next a “sack of corn”, then a “sack of beans”, then a “sack of lentils”, then
finally in the leftmost division he draws a “hen”. Then he places the corre-
sponding numerical quantities: in the first division, a small circular imprint
for the number 10, and 5 small notches each worth 1, thus making up the
total of 15 sacks of barley; in the second, three imprints of 10 for the number
30; in the third, he marks the number 60 with a large imprint, and so on.
On the back of the tablet, he makes the summary, that is, the totals of
the inventory according to the numbers on the front, namely “145 sacks
(various)” and “15 hens”.
This done, the two men append their signatures to the bottom of the
tablet, but not as used to be done by rolling a cylinder-seal over it. Instead,
they use the pointed end of the stylus to trace conventional signs which
no
represent them. Then, having given the document into the safekeeping of
the notary, they part.
HOW THE SUMERIAN NUMBERS
WERE DECIPHERED
The story reconstituted in the preceding section was not imaginary: it was
achieved on the basis of the document shown in Fig. 11.3, which provides
detailed evidence of how the Sumerian scribes used to note on one side
of the tablets the details of the accounting, and on the other side a kind of
summary of the transaction in the form of totals under different headings.
Translation
Side 1 Side 2 Side 1 Side 2
Fig . 11.3. Sumerian " invoice " discovered at Uruk, said to be from the Jemdet Nasr era
(c. 2850 BCE). Iraqi Museum, Baghdad. ATU 637
But it is precisely this feature which has enabled the experts to decipher
various ancient number-systems such as Sumerian, hieroglyphic or linear
Cretan, and so on. The values of the numbers could therefore be deter-
mined with certainty by virtue of applying a large number of checks and
verifications to these totals.
Observing, for example, that on the front of some tablet there were ten
narrow notches here and there, while on the back there was a single small
circular imprint, and then finding this correspondence confirmed in a
sufficient number of similar cases, they can conclude that the narrow notch
denotes unity and the small circular imprint denotes 10.
« 2 > =1 • =10
Now suppose that we are trying to discover the unknown value, which
we shall denote by x, of the wide notch:
-xl
Ill
SIMILAR PRACTICE O E THE ELAMITE SCRIBES
Of course, lacking any other indication, and in the absence of a bilingual
“parallel text” (linguistic or mathematical), the value of this number would
have long remained a mystery. But a happy chance has placed into our
hands the tablet shown in Fig. 11.3, which bears the three numbers
described above of which two have already been deciphered, which will
indeed be our “Rosetta Stone”.
We begin, of course, by ignoring the count of the 15 hens (one small
circular imprint and 5 narrow notches, together with the pictogram of the
bird), since this is reproduced exactly on the reverse of the document. So we
shall only bother with the details of the inventory of sacks (goods denoted
by the same writing sign throughout). Adding up the numbers on side 1,
we therefore obtain
DDD •• • •
• DP • P ti
10 + 5 + 30 + x + 40 = x + 85
while on side 2 we find
• ODD
• DD
2x + 20 + 5 = 2x + 25
On equating these two results, we obtain the equation
x + 85 = 2x + 25
which, on reduction, finally gives the result we are seeking, namely
= x = 60
the tablet in Fig. 11.4 A which refers to a similar accounting operation.
The goods in question are represented by writing signs (whose meaning,
in many cases, still eludes us). But the numbers associated with the
various goods are clearly indicated by groups of number-signs. The subse-
quent diagram (Fig. 11.4 B) shows what we shall from now on call the
“rationalised transcription” of the original tablet.
Fig. 11.4 a. Accounting tablet from Susa. Louvre. See MDP, VI , diagram 358
NUMBERS
WRITING
IJSSSSOOl
«s »))»
m
A
SIDE I
*=»))!! *8800(1(7
SIDE 2
Fig. n. 4B.
SIDE 1
SIDE 2
A Egft.
However, we are only entitled to draw this conclusion as to the value of
the sign in question if the value so determined gives consistent results for
several other tablets of similar kind. And this turns out to be the case.
SIMILAR PRACTICE OF THE ELAMITE SCRIBES
It was precisely by observing similar practice on the part of the Elamite
scribes, and carrying out systematic verifications of the same kind on a
multitude of proto-Elamite tablets (some of the most important of which
will be shown below) that I was able, myself, to arrive at the solution of this
thorny problem.
Some of these tablets can lead us to it, even though the values of the
proto-Elamite numbers may remain unknown. Consider for example
Now we see, on the front of the tablet:
• the wide notch twice;
• the large circular impression twice;
• the small circular impression 9 times;
• the narrow, lengthened notch once;
• a circular arc twice;
• and a peculiar number (Fig. 11.1 D) once only.
This, moreover, is exactly what we also find on the reverse of the tablet.
The number which is shown on side B therefore corresponds to the grand
total of the inventory on the front.
In the same way, on the tablet shown in Fig. 11.5, the front and the
reverse both show six narrow notches.
THE DECIPHERMENT OF A F I V E -T H O U S A N D - Y E A R- O L D SYSTEM
SIDE 1 SIDE 2
Fig. 11 . 5 . Tablets from Susa. Teheran Museum. See MDP, XXVI, diagram 437
DETERMINING THE VALUES OF THE
PROTO-ELAMITE NUMBERS
Now consider the tablet shown in Fig. 11.6. In the present state of the
tablet, on the front side the narrow notch occurs only 18 times, and
the smaller circular impression occurs 3 times, while on the reverse the
narrow notch occurs 9 times and the circular impression 4 times.
If we proceed by analogy with the Sumerian numbers of similar form,
attributing value 1 to the narrow notch and value 10 to the circular imprint,
then the total from the front of the tablet (18 + 3 x 10 = 48) and the total
from the reverse (9 + 4 x 10 = 49) differ by 1. We may conjecture that this
difference is the result of a missing piece broken off from its left-hand side,
which would have damaged the numerical representation in the last line of
the top face.
Since, moreover, there are similar tablets* on which we find exactly
equal totals on the two sides, we may conclude that this explanation for the
discrepancy is in fact correct.
Therefore we may definitively fix the value of the narrow notch as 1, and
the value of the small circular impression as 10.
Fig. u. 6 a.
Accounting tablet
from Susa. Teheran
Museum. See MDP,
XXVI, diagram 297
Fig. ii.6b.
SIDE 1
* See, for example, tablet 353 of MDP, VI (Louvre: Sb 3046).
112
SIDE 1 SIDE 2
Fig. 11 . 7 . Accounting tablet from Susa. Louvre. See MDP, XVII, diagram 3
Now we must take account of the fact that the Elamites set their numbers
down from right to left (in the same direction as their writing), starting
with the highest-order units and proceeding left towards the lower-
order units. Furthermore, close examination of the tablets shows that the
Elamite scribes used two different systems for writing numbers, both of
which were based on the notion of juxtaposition to represent addition.
These two systems made use, in general, of different symbols (Fig. 11.10
and 11.11).
For the first of these two proto-Elamite systems, it is pretty clear that the
number-signs were always written in the following order, from right to left
and from highest value to lowest value (Fig. 11.8).
I !?', «*» ) J • o 1 d $
ABC DEFGHMNP
Fig. ii. 8 .
The number-signs of the second system always occur as follows, again
from right to left and in decreasing order (Fig. 11.9).
J 0 Z I
F G I J K L O
_ Variants Variants
Fig. n. 9-
The above shows, therefore, that
• on the one hand, the numbers labelled A, B, C, D, and E (which
always occur to the left of the narrow notch which represents 1)
correspond to orders of magnitude below 1, that is to say to fractions;
• on the other hand, H, M, N, and P, and also I (or J), K (or L) and
O correspond to orders of magnitude above 10 (since they always
occur to the right of the small circular impression representing 10)
(Fig. 11.10 and 11.11).
In the end, therefore, by working out the totals on many other tablets,
1 was able to obtain the following results which, as we shall see below, can
be confirmed in other ways.
113
DETERMINING THE VALUES OF THE PROTO-ELAMITE NUMBERS
| p; CO ))) (JUU ••• O 133 <$>(*>
<
MDP
XXVI
diagram 362
► « ) J « o jjffij
< -TaJ
j MDP
' XXVI
diagram 362
H))55 ««
<
MDP
XXVI
diagram 259
IP® OUR*
<
MDP
XXVI
diagram 5
MDP
XXVI
diagram 20
3? So
<
MDP
XXVI
diagram 150
$ # :k » ) 9
<
MDP
XXVI
diagram 362
Fig. 11 . 10 . Instances of number taken from accounting tablets, which show how the earliest proto -
Elamite number-system worked
"85 B
<
MDP
XVII
105
naoooo c
J oodo k
<
500 MDP
aS xvu
45
999 °eoe 00030 MDP
0000 XXVI
< .... 156
NTBEIBBB mdp
8080 t;
<
5JPSS00 ^ 41 MDP
00 XXVI
< ■* “ 156
10 E
<
MDP
XVII
275
trurratm MDP
D23 nama xxvi
< 27
ooEBtni MDP III
oo B3JES1 xxv ' 1)
< - - 27 <
53 EE3 MDP
XXVI
27
Fig . 11 . 11 . Instances from accounting tablets which illustrate the second proto-Elamite number-
system
A =
1
120
j b
'?(
B = —
60
)
E
1
5
For the number E (the circular arc), for example, I considered the
tablet shown in Fig. 11.12 which, as can be seen from its rationalised
transcription, bears two kinds of inventory:
UPPER SIDE
LOWER SIDE
TTTHPj
Fig. ii. 12 a. Accounting tablet
from Susa. Louvre. Ref MDP,
XVII, diagram 17
) t ) 1
v > 1
) I ) S>
v > K
) 1
v ) im
) i M
> lit
) x > V
UPPER SIDE
3 X 33
LOWER SIDE
Fig. ii.12b.
• one, associated with the script character J , which has 10 circular
arcs on the top face and 2 narrow notches on the reverse;
• the other, associated with the ideogram J , which has 5 circular arcs
on the top face and 1 narrow notch on the reverse.
Therefore, denoting by x the unknown value of the number (E) in
question, these two inventories give, according to the totals of the two
sides, the two equations
x+x+x+x+x+x+x+x+x+x=2
x+x+x+x+x=l
namely
lOx = 2
5x = 1
which is precisely how it was possible to determine the value 1 for the
circular arc.
Now let us try to evaluate the large circular imprint and the wide notch
(H and M in Fig. 11.1). Because they look just like the Sumerian signs
THE DECIPHERMENT OF A FIVt-THOUSAND-YEAR-OLll SYSTEM
associated with 60 and 3,600 respectively (Fig. 8.7 and 9.15 above), we are
at first tempted to conclude that the same values should be attributed to
them in the present case. But when we examine the proto-Elamite tablets
we find that this cannot be true. As we have seen, the Elamites set their
numbers down from right to left, in decreasing order of magnitude and
always commencing with the highest. Therefore, if these signs had the
Sumerian values, the large circular impression should come before the wide
notch in writing numbers. But this is not the case, as can be seen from
Fig. 11.10 for example.
The document shown in Fig. 11.13 leads without difficulty to the
ascertainment of the value of the proto-Elamite large circular impression.
UPPER SIDE
Fig. ii. 13 a. Accounting tablet
from Susa. Teheran Museum. Ref.
MDP, XXXI, diagram 3 FiG.11.13B.
Ignoring the two circular arcs and the doubled round imprint which are
on both sides of the tablet, we find
• 9 small circular impressions and 12 narrow notches on the upper face;
• 1 large circular impression and 2 narrow notches on the lower.
Therefore, if we now evaluate these numerical elements on the two faces
of the tablet, bearing in mind what we have already found out, we obtain
the following:
Upper 9 x 10 + 12 = 102
Lower lxx + 2= x + 2
Since these must be equal, we find the equation x + 2 = 102, whose
solution is that x = 100.
Now consider the tablet shown in Fig. 11.14, on which we find
• 20 small circular impressions, and 2 large ones, on the upper face;
• 1 wide notch and one large circular impression, on the lower.
Let us now give the value 100 to the large circular impression, as we have
114
just determined, and denote by y the value of the wide notch. We then
obtain the following totals:
Upper 20 x 10 + 2 x 100 = 400
Lower 1 x y + 100 -y + 100
Since these also, as before, must be equal, we obtain the equation
y + 100 = 400, whose solution is thaty = 300.
From the preceding arguments, therefore, we attribute the value 100 to
the large circular impression, and the value 300 to the wide notch.
UPPER SIDE LOWER SIDE
Fig. 11.14. Tablet from Susa. Teheran Museum. Ref MDP, XXVI, digram 118
Of course, this would not allow us to conclude that these values
correspond to a general reality unless we also find at least one other tablet
which gives completely concordant results. This is, however, precisely the
case for the tablets shown in Fig. 11.15 and 11.16.
UPPER SIDE LOWER SIDE
Fig. m.isa. Tablet from Susa. Louvre, Ref. MDP, VI, diagram 220
UPPER SIDE
°oooo^Jf 300 + 9 x 10 390
300 + 100 400
2x300 + 3x10 + 3 633
1,423
LOWER SIDE
4 x 300 + 2 x 100 + 2 x 10 + 3 . . . 1,423
Fig. 11.15B.
115
UPPER SIDE LOWER SIDE
Fig. ii.i6a. Tablet from Susa. Teheran Museum, Ref MDP, XXVI, diagram 439
00
2 x 100
.. 200
oo qr m
300 + 2 x 100
.. 500
1
88
300
. . 300
JJS80O
H8
2 x 100 + 4 x 10 + 4
. . 244
ooo^J
300 + 3x 10
... 330
0006 O
Is
100 + 9x10
... 190
1,764
LOWER SIDE
5 x 300 + 2 x 100 + 6 x 10 + 4 .
. 1,764
Fig. 11.16B.
In conclusion, the results established so far (which from now on will be
considered definitive) are the following:
1'f
'?<
0
*0°
0 O
03 j)
0
o
(5
1
1
1 1
1 10 100
300
60
30
10 5
Fig. 11.17.
Therefore, of the eleven number-signs of the proto-Elamite system, nine
have been deciphered.
Now let us consider the delicate problem of the following two number-
signs:
n $
N P
Fig. 11.18.
As we have already shown in Fig. 11.2, these two numbers have been
interpreted in the most diverse ways since the beginning of this century
(the number labelled N, for example, has been assigned to 600, to 6,000,
DETERMINING THE VALUES OF THE P ROTO- EL AM I TE NUMBERS
to 10,000, or even to 1,000). To try to have a better understanding of
the situation, we shall consider the tablet shown in Fig. 11.19 A. According
to V. Scheil, this is “an important example of an exercise in agricultural
accounting”. As far as I know, this is the only preserved intact proto-
Elamite document which contains both the entire set of number-signs of
the first system and also a grand summary total.
On this tablet, we find:
• on the top face, a series of twenty numerical entries (corresponding
to an inventory of twenty lots of the same kind denoted, it would seem,
by the script character at the right of the top line);
• on the reverse, the corresponding grand total (itself preceded by the
same written character).
UPPER SIDE LOWER SIDE
Fig. 11. 19 a. Accounting tablet from Susa. Ref MDP, XXVI , diagram 362
Fig. 11.19B.
THE DECIPHERMENT OE A FIVE-THOUSAN D-VEAR-OI.D SYSTEM
Considering the results we have already obtained, we shall make various
attempts to reconcile the totals of the numbers on this tablet, by trying
various different possible values for the numbers labelled N and P, and
making use of the numbers of occurrences of the different signs as shown
in Fig. 11.19 C.
1
$
*.o;
03
)
J
O
O
1
3
N
$
P
to tsi
e s
u
^4- O
0 c
on the
upper side
15
15
24
14
19
26
39
11
7
8
5
Number
each sig
on the
lower side
1
0
2
1
1
2
2
1
1
3
6
Fig. 11 . 19 c. Complete listing of all the numerical signs on the tablet
First attempt:
Following Scheil (1935, see MDP, XXVI), let us assign the value 10,000 to the
wide notch with the circular impression (N), and the value 100,000 to
the circle with the little wings (P). On the upper face of the tablet, we then
obtain the following total for the numbers which appear there (Fig. 11.19 C):
15 x — + 15 x — +24 x — + 14 x — + 19 x -
120 60 30 10 5
+ 26 + 39 x 10 + 11 x 100 + 7 x 300 + 8 x 10,000 + 5 x 100,000
namely 583,622 + —
' 120
On the lower, similarly (Fig. 11.19 C):
1 1 n 1 „ 1 , 1 , 1
1 x + 0 x — + 2x — + 1 x — + 1 x -
120 60 30 10 5
+ 2 + 2 x 10 + 1 x 100 + 1 x 300 + 3 x 10,000 + 6 x 100,000
namely 630,422 +— —
y 120
The difference between these two results is 46,800, far too great to allow
this attempt to be considered correct, if we attribute the discrepancy to an
error on the part of the scribe.
Second attempt:
Now consider the possibilities of assigning the values:
N = 6,000 [V. Scheil (1923)], P = 100,000 [V. Scheil (1935)]
By a similar calculation, we obtain (Fig. 11.19 C):
116
Upper side 551,622 + Lower side B 618,000 +
120 120
This attempt also must be considered to fail, since the discrepancy
between the two faces is again too large.
Third attempt:
Now let us try:
N = 6,000 [V. Scheil (1923)], P = 10,000 [S. Langdon (1925)]
This again fails, since we obtain (Fig. 11.19 C):
Upper side 101,622 + Lower side 78,422 +
120 120
Fourth attempt:
Now let us consider the values proposed by R. de Mecquenem in 1949:
N = 1,000, and P = 10,000
Again from Fig. 11.19 C, we obtain the results
46 46
LIpper side 61,622 + — - Lower side 63,422 + — -
YY 120 120
This possibility seemed to me for a long time to be the most likely
solution. The results it gives are relatively satisfactory, since the discrepancy
between the totals for the two faces of the tablet is only 1,800. On
this belief, 1 had therefore supposed that the scribe had made some error
in calculation, or had omitted to inscribe on the tablet the numbers corre-
sponding to this difference. This, after all, could be likely enough,
considering the many number-signs crowded onto the tablet - errare
humanum est\ Let us not forget that, just as in our own day, the scribes of old
were capable of making mistakes in arithmetic.
Nonetheless, on reflection, it seemed to me that there was something
illogical in attributing the value 1,000 to the number N, for two reasons.
Consider, first of all, the following two numerical entries taken from
proto-Elamite tablets:
Fig. ij. 20 .
On Mecquenem’s hypothesis, these would respectively have values
,c/= l x 1,000 + 6 x 300 =2,800
9 x 300 + 5 x 10 + 1 = 2,751
117
Now, still adopting this hypothesis, the following numbers would be
units of consecutive orders of magnitude:
1 10 100 300 1,000 10,000
Therefore, in the first place, the question arises: if the notch with the
circular impression really corresponded to the value 1,000, why should
the scribes have adopted the above representations of the numbers 2,800
and 2,751, and not the more regular forms in Fig. 11.21 following?
o ogg f d
,?/ = 2 800 = 100 100 300 300 1.000? 1.000?
<
rss o g g f if
JR _ 9 7C, _ 1 10 100 300 300 1,000? 1,000?
Fig. li.zi. «
On the other hand, we know that for the Sumerians the small circular
impression had value 10, the wide notch 60, and the combination of the
latter including the former had value 600:
Fig. 11.22. 10 60 60x10 = 600
in other words, that the last figure follows the multiplicative principle.
But for the Elamites the small circular impression had value 10 while the
wide notch had value 300. By analogy with the Sumerian system, the value
300 x 10 = 3,000 should be assigned to the wide notch compounded with
the small circle:
6 o m
Fig. 11.23. 10 300 300x 10 = 3,000?
For these reasons I was led to reject Mecquenem’s hypothesis.
Fifth attempt:
We are therefore now led to consider the proposed values:
N = 3,000 and P = 10,000
[the latter from S. Langdon (1925) and R. de Mecquenem (1949), the
former from the above reasoning]. Again comparing the totals from the two
faces of the tablet, this hypothesis gives the following results:
Upper side 77,622 + Lower side B 69,422 +
vv 120 120
This hypothesis therefore does not work either. But, if we wish to keep
DETERMINING THE VALUES OF THE PROTO-ELAMITE NUMBERS
the value of 3,000 for the number N, we must seek a different value for the
number P.
Now, close examination of the mathematical structure which can be
inferred from the values so far determined in the proto-Elamite number-
system caused me to suppose that the following three values could be
possible for the number P:
9,000, 18,000 and 36,000
I was led to this supposition by postulating that the proto-Elamite
system of fractions was developed on the same lines as the notation for the
whole numbers, namely that there had to be a certain correspondence
between a scale of increasing values, and a scale of decreasing values,
relative to a given base number.
This, however, is exactly what one observes if one expresses the different
values determined so far in terms of the number M = 300 (Fig. 11.24).
SIGNS VALUES
Fig. 11.24.
Sixth attempt:
This now leads us to contemplate the possibilities based on these three
possible values for P, of which the first is (Fig. 11.19 C):
THE DECIPHERMENT OF A F I V F - T H 0 U S A N D - Y E A R - O I. D SYSTEM
N = 3,000. P = 9,000
But on comparing the totals which result, we find a serious discrepancy:
Upper side 72,622 + Lower side 63,422 +
^ 120 120
Difference 9,200
Therefore this suggestion must be rejected.
Seventh attempt:
The same results from trying the second possibility inferred above, since
the values:
N = 3,000, P = 36,000
also lead to implausible results (Fig. 11.19 C):
Upper side 207,622 + — Lower side 225,422 + — —
120 120
Difference 17,800
Final attempt, and the solution of the problem:
Now consider the final possibility, with the following values:
N = 3,000, P = 18,000
This system, which is compatible with a coherent mathematical struc-
ture, also gives satisfyingly close agreement:
Upper side 117,622 +— (117,622 + -+ — + — + — )
120 5 10 30 120
Lower side 117,422 +— (117,422 + i+ — + — + — )
120 5 10 30 120
Whence, however, comes this discrepancy of 200 which exists between
the two faces if we adopt this hypothesis? Quite simply, I believe, from a
“typographical error”.
Instead of inscribing on the lower side the grand total corresponding to
the inventory on upper side, which should be in the form:
) •.«' p : «*» ) 03 00 CJ d d (J
— + — + — + — + - + 1 + 1 + 10 + 10 + 300 + 300 + 3,000 + 3,000 + 3,000 + 18,000 x 6
120 30 30 10 5
<
117,622 + i + — + — + — + —
5 10 30 30 120
Fig. i l . 25 a .
the scribe in fact made a large circular impression in the place of one of
the two wide notches:
118
30 r a
Error
otj 13 9 3
100 300
<
Fig. 11.25B.
117,422 + i + — + — + — + —
5 10 30 30 120
It is easy to see how this could happen. The scribe held his stylus
with large circular cross-section in the wrong position (See Fig. 8.10 and
8.12 above): instead of pressing the stylus at an angle of 30°-45° to the
surface of the soft clay, which would have given him a wedge, he held it
perpendicular to the surface thereby obtaining the circle.
That is, instead of doing this:
Fig. 11.26A. Result
he did this:
Fig. 11.26B. Result
Therefore, in all probability, we may conclude that the wide notch with
a small circular imprint corresponds to the value 3,000, and the circle with
the little wings corresponds to the value 18,000.
All the numbers in the proto-Elamite system have, therefore, been
definitively deciphered.
We have good reason to suppose that this system is the more ancient
of the two since the following numerals appear on the proto-Elamite
accounting tablets from the archaic epoch onwards.
3 ° O
1 10 100 300 3,000
Fig. 11.27.
The same set of numerals appears on the earliest numerical tablets, as
well as on the outside of the counting balls recently discovered on the site
of the Acropolis of Susa. Finally, the numerals also are those which, accord-
ing to their respective shapes, correspond to the archaic calculi which were
119
formerly enclosed in the counting balls, in fact to the number-tokens of
various shapes and sizes which stood for these numbers (and whose values,
in turn, have themselves now been determined as a result of the decipher-
ment described above; see also Fig. 10.8 and 10.14 above);
B O © A
Rod Ball Disk Cone
1 10 100 300
Fig. 11.28.
As to the second system of writing numbers, I believe that the Elamites
constructed it - maybe in a relatively recent era - for the purpose of record-
ing quantities of objects or of goods, or magnitudes, of a different kind
from those for which the symbols of the first system were used.
I base this hypothesis on an analogy with Sumerian usage. During the
third millennium BCE, the scribes of Lower Mesopotamia in fact used three
different numerical notations:
• the first, the commonest and oldest, which we have studied in
Chapter 8, was used for numbers of men, beasts, or objects, or for
expressing measures of weight and length;
• the second was used for measures of volume;
• the third was used for measures of area.
This hypothesis is in fact confirmed by the tablet shown in Fig. 11.29,
which carries two inventories which have been very clearly differentiated.
UPPER SIDE LOWER SIDE
Fig. n .29 a. Accounting tablet from Susa. Teheran Museum. Ref. MDP, XXVI, diagram 156
FIRST INVENTORY SECOND INVENTORY
Large perforated cone
3,000
Fig.
11.29B.
DETERMINING THE VALUES OF THE PROTO-ELAMITE NUMBERS
The first of these inventories is indicated by a characteristic script char-
acter, and the corresponding quantities are expressed in the numerals of the
first proto-Elamite system (Fig. 11.29 B). The second inventory is indicated
by the signs (which have not yet been deciphered):
and the corresponding quantities are expressed in the numerals of the
second proto-Elamite system (Fig. 11.9).
The numbers given on the reverse of this tablet correspond respectively
to the total of the first inventory and to the total of the second. Using
the values we have already obtained, we can make the totals for the first
inventory:
a) on upper side:
6 x 300 + 2 x 100 + 10 x 10 + 5 + - + — = 2,105 +-+ —
5 10 5 10
b) on lower side:
7 x 300 + 5 + - + T = 2,105 + -+ —
5 10 5 10
(which, by the way, is a further confirmation of the validity of our earlier
result).
Now let us consider the different numerals on the second inventory, and
let us give value 1 to the narrow notch, 10 to the small circular impression,
100 to the double vertical notch and 1,000 to the double horizontal notch.
Then the totals come out as follows:
a) on upper side:
1,000 + 13 x 100 + 12 x 10 + 12 = 2,432
b) on lower side:
2 x 1,000 + 4 x 100 + 3 x 10 + 2 = 2,432
We may therefore fix the values of the following numerals as shown:
Z ° r H
Fig. 11.30.
(where the former of these values, for example, is confirmed by the tablet in
Fig. 11.31, since the totals come to 591 on both sides).
THE DECIPHERMENT OF A FIVE-THOUSAND-YEAR-OLD SYSTEM
120
UPPER SIDE LOWER SIDE
r
^pw*w«
[•WSSSgV :
— ' j
«
Fig. u . 3 1 a . Accounting tablet from Susa. Louvre. Ref. MDP, XVII, diagram 45
Fig. 11 . 31 B.
So there we see pretty well all of the proto-Elamite numerals deciphered.
At the same time, we have discovered that at Susa two different number-
writing systems were in use, probably corresponding to two different
systems of expressing numbers:
• one, strictly decimal* (Fig. 11.32);
• the other, visibly “contaminated” by the base 60 (Fig. 11.33).
! ° z I N sa
1 10 100 100 1,000 1,000
F G I J K L
Fig. u . 3 2 . The values of the number-signs of the second proto-Elamite number-system
We may suppose that the first may have been used for counting such
things as people, animals or things, while the second may have been used
to express different measures in a system of measurement units (volumes
and areas, for example).
SIGNS
X
Y
VALUES
A
fe
— M
iB
1
V
36,000
2
120
B
1 M
B
1
7 V
18,000
60
c
0
‘o’
— — M
2 B
1
0 1
9,000
30
D
03
— M
3,000
6 B
1
10
E
— — M
12 B
1
>
1,500
5
F
n
— M
60 B
1
300
G
o
— M
30
600 B
10
H
O
1 M
3
6,000 B
100
M
a
M
18,000 B = 300 x 60 B
300
N
10 M
180,000 B = 300 x 600 B
?
P
$
60 M
1,800,000 B = 300 x 6,000 B
?
Fig. 11.33. The mathematical structure of the first proto -Elamite number-system
These are of course only hypotheses, but the above results lend
confirmation to the existence of cultural and economic relations between
Elam and Sumer, at any rate from the end of the fourth millennium BCE,
and to the influence exerted by the Sumerians upon Elamite civilisation.
* A question remains for the numeral formed from a double horizontal notch with a small circular
impression in its centre (Fig. 11.32, sign O). Is this the numeral representing 10,000 = 1,000 x 10? It seems
likely. But this could not be stated with certainty, since we lack documents better preserved than those
we have at present, relevant to this numeral.
121
A FOU R-TH OUS AND- Y E A R-Ol.D DIVISION SUM
CHAPTER 12
HOW THE SUMERIANS
DID THEIR SUMS
The arithmetical problems which the Sumerians had to deal with were
quite complicated, as is shown by the many monetary documents which
they have bequeathed to us. The question which we shall now address is to
find out what methods they used in order to carry out additions, multipli-
cations, and divisions. First of all, however, let us have a look at one very
interesting document.
A FOUR-THOUSAND-YEAR-OLD DIVISION SUM
The tablet shown in Fig. 12.1 is from the Iraqi site of Fara (Suruppak), and
it dates from around 2650 BCE.
We shall present its complete decipherment according to A. Deimel’s
Sumerisches Lexikon (1947). This document provides us with the most valu-
able information on Sumerian mathematics in the pre-Sargonic era (the
first half of the third millennium BCE). It shows the high intellectual level
attained by the arithmeticians of Sumer, probably since the most archaic era.
The tablet is divided into two columns, each subdivided into several
boxes.
From top to bottom, in the first box of the left-hand column is a narrow
notch, followed by a cuneiform group ( se-gur 7 ), which signifies “granary
of barley”.
In the box beneath is a representation of the number 7, preceded by
a sign which is to be read sila.
In the third box, the numeral 1 is followed by the sign for “man” ( lu );
below this is a group which is to be read su-ba-ti (the word su means
“hand”) and which might be translated as “given in the hand”.
Finally, at the very bottom of the left-hand column is the sign for "man"
again, above which is the character bi which is simply the indicative “these”.
The literal translation of this column therefore is: “1 granary of barley;
7 sila; each man, given in the hand; these men.”
In the first box of the right-hand column, we can recognise the
representation of 164,571 in the archaic numerals (see Fig. 8.20 above), and
in the box below a succession of signs which represent the phrase “granary
of barley, there remains: 3”.
TRANSCRIPTION
LITERAL TRANSLATION
Left-hand register
Right-hand register
1 “granary of barley”
7 sila (of barley)
164.571
Each man
in his hand receives
Men
these
sila of barley
remain
3
Fig. i2.i. Sumerian tablet from Suruppak (Fara). Date: c. 2650 BCE. Istanbul Museum.
Ref. Jestin (1937), plate XXI, diagram 50 FS
This tablet, which no doubt describes a distribution of grain, shows
all the formal elements of arithmetical division: we have a dividend, a
divisor, a quotient, and even, to an astonishing precision for the time,
a remainder.
The sila and the se-gur ? ("granary of barley”) are units of measurement of
volume. At that time, the former contained the equivalent of 0.842 of our
litre, while the latter came to about 969,984 litres, namely 1,152,000 sila
[see M. A. Powell (1972)]:
1 se-gur 7 (1 granary of barley) = 1,152,000 sila
Thus this distribution involved the division of 1,152,000 sila of barley
between a certain number of people, each of whom is to receive 7 sila.
Now let us do the calculation. 1,152,000 divided by 7 is 164,571, exactly
(36,000) (36,000)
1 se-gur ?
(36,000) (36,000)
(3,600) (3,600) (3,600)
sila 7
(3,600) (3,600)
llu
(600 (600) (60)
su-ba-ti
(600) (600) (60)
lu-
(10) (10) (10) (10) (10) (1)
-bi
se sila
su-kid
3
HOW THE SUMERIANS DID THEIR SUMS
122
the number in the first box of the right-hand column; and the remainder is
3, exactly the information given at the bottom of this column.
There is no doubt about it: you have before your very eyes the written
testimony of the oldest known division sum in history - quite a complex
one; and as old as Noah.
OFFICIAL DOCUMENT OR LEARNER’S EXERCISE?
One may suppose that this tablet was probably an official document in the
archives of the ancient Sumerian city of Suruppak, unless it happened to be
an exercise for apprentice calculators.
On the first supposition, then its translation into plain language is as
follows:
We have divided 1 granary of barley between a certain number of
people, giving 7 sila to each one. These men were 164,571 in number,
and at the end of the distribution there were 3 sila remaining.
On the other hand, if it was really an exercise for learners, then the
appropriate translation would be:
STATEMENT OF THE PROBLEM
Given that a granary of barley has
been divided between several men
so that each man received 7 sila,
find the number of men.
SOLUTION OF THE PROBLEM
The number of men was 164,571
and 3 sila were left over after the
distribution.
For convenience of exposition, we shall adopt the latter interpretation in
what follows.
There is no indication whatever in the document as to the method
of calculation to be used to obtain the result. Nor do we yet know of
any formal description. One thing however is certain, and that is that the
calculation was not carried out by means of Sumerian numerals, which
do not encapsulate an operational capability in the way that our own
numerals do.
Nonetheless the results of the previous chapter give us some basis for
supposition as to what the means of calculation may have been. The
Sumerians most probably made use of the calculi (the very ones shown in
Fig. 10.4), as much prior to the emergence of their numerical notation as
subsequently, since we find these tokens in various archaeological sites of
the third millennium BCE, that is to say, at a time when bullae had almost
entirely been displaced by clay tablets (see Fig. 10.2 above).
We shall now put forward a speculative but entirely plausible recon-
struction of the technique of calculation which was most probably used.
CALCULATION WITH PELLETS,
CONES, AND SPHERES
Let us imagine we are in the year 2650 BCE, in the Sumerian city of
Suruppak. We are in the school where scribes and accountants learn their
skills, and the teacher has given a lesson on how to do a division. Now he
begins the practical class, and sets the problem of dividing one granary of
barley according to the conditions given.
The problem is therefore to divide 1,152,000 sila of barley between a
certain number of persons (to be determined) so that each one gets 7 sila of
barley, which comes down to dividing the first number by 7.
At this time, additions, multiplications, and divisions are carried out by
means of the calculi, those good old imnu of former times which, in their
several shapes and sizes, symbolise the different orders of magnitude of the
units in the Sumerian number-system. Although their use has long disap-
peared from accounting practice, they are still the means that everyone uses
for calculation. This has never worried any of the generations of scribes
since the day when one of them thought of making replicas of the various
calculi on clay tablets, to serve as numerical notations - a narrow notch for
the small cone, a small hole for the pellet, a wide notch for the large cone,
and so on (see Fig. 10.4 above).
Generally, the procedure for performing a division brings in succes-
sively: pierced spheres (= 36,000), plain spheres (= 3,600), large pierced
cones (= 600), large plain cones (= 60), and so on. At each stage, the pieces
are converted into their equivalents as multiples of smaller units whenever
they are fewer than the size of the divisor.
Practically speaking, therefore, the above example proceeds as follows.
In Sumerian, the dividend 1,152.000 is expressed in words (see Fig. 8.5
above) as
sargal-id sar-u-min
which corresponds to the decomposition
216,000 x 5 + 36,000 x 2 = 5 x 60 3 + 2 x (10 x 60 2 )
The largest unit of the written numerals at this time, however, is only
36,000 (see Fig. 10.4 above), which is also the value of the largest of the
calculi. Therefore the dividend must first be expressed in multiples of this
smaller unit, therefore by 32 pierced spheres each of which stands for
36,000 units:
1,152.000 = 32 x 36,000
But we are to divide this by 7, so we arrange these as best we can in
groups of 7:
123
36,000
Fig. 12. 2a.
The number of groups, each with 7 pierced spheres, in this first arrange-
ment is 4, which is the quotient from this first partial division. This, in the
context of the problem, is also equal to the number ( 4 x 36,000) of the first
group of people who will receive 7 sila of barley each. In order not to lose
track of this partial result, we shall put 4 pierced spheres on one side to
represent it.
After this, we have 4 pierced spheres left over. We therefore must divide
these 4 x 36,000 sila. But when it is expressed in pierced spheres, worth
36,000 each, we find that 4 cannot be divided by 7. At this point, therefore,
we convert each one of these into its equivalent number of the next lower
order of magnitude.
Each pierced sphere (36,000) is equivalent to 10 plain spheres, each
worth 3,600. The 4 pierced spheres therefore become 4 x 10 = 40 plain
spheres, which we once again arrange in groups of 7:
3,600
Fig. 12. 2b.
Now we find that there are 5 complete groups of 7 plain spheres, so
we put on one side 5 plain spheres (corresponding to the second group,
5 x 3,600, of people who will each receive 7 sila of barley).
But we find that there are 5 plain spheres left over at the end of this
CALCULATION WITH PELLETS, CONES, AND SPHERES
second division, and 5 is not divisible by 7, so we now replace each plain
sphere by its equivalent number of pieces of the next lower order of
magnitude.
Each “3,600” sphere is equivalent to 6 large pierced cones worth 600
each, so we convert the 5 pierced spheres left over into 5 x 6 = 30 large
pierced cones which we again arrange in groups of 7:
600
Fig. 12. 2c.
Since we have 4 full groups of 7 pierced cones each, we therefore put
aside 4 large pierced cones, corresponding to the third part of the men who
will receive 7 sila of barley each (4 x 600).
However, we now have 2 large pierced cones left over, so we still have to
divide 2 x 600 sila of barley.
Each “600” cone is equivalent to 10 large plain cones worth 60 each, so
we convert the two large pierced cones left over into 2 x 10 = 20 large plain
cones and we arrange these in groups of 7.
60
Fig. 12 . 2 D.
We can form 2 complete groups of 7, with 6 large plain cones left over.
As before, we put aside 2 cones to note the number of complete groups,
ITOW THE SUMERIANS DID THEIR SUMS
124
corresponding to the 2 x 60 men who will each get 7 sila of barley at this
fourth stage of the distribution.
Now we convert the 6 large plain cones left over, worth 60 each, into
their equivalent in pellets worth 10 each, therefore into 6 x 6 = 36 pellets,
and we arrange these into groups of 7, with 1 left over:
Fifth remainder >
Fig. 12. 2E.
Once again, we put aside 5 pellets corresponding to the 5 x 10 men who
will each get 7 sila of barley at this fifth stage of the distribution.
The single pellet left over, worth 10, is now converted into 10 small cones
each worth 1. This makes one complete group of 7, with 3 left over.
l
Sixth remainder >
Fig. 12. 2f.
To note the one complete row, we put aside 1 small cone, and this corre-
sponds to the number (10) of men who will each get 7 sila of barley at this
sixth stage of the distribution. Since the number corresponding to the left-
over cones is 3, and this is less than the divisor, we can proceed no further in
the division of the original number into whole units, and we have finished.
The final quotient can now be easily obtained by totalling the values of the
pieces which we successively set aside in the course of the division, as follows:
4 pierced spheres (quotient from the first division)
5 plain spheres (quotient from the second division)
4 large pierced cones (quotient from the third division)
2 large plain cones (quotient from the fourth division)
5 pellets (quotient from the fifth division)
and
1 small cone (quotient from the sixth division)
Sixth quotient = 1
6
Fifth quotient = 5
gOO © ©
0000060
ooooooo
oodeoeo
odeeooo
ooooooo
o
10
Fig. I2.2G. Result of the division
In other words, the total number of people to whom the barley will be
distributed is
4 x 36,000 + 5 x 3,600 + 4 x 600 + 2 x 60 + 5 x 10 + 1 = 164,571
Back at the school of arithmetic, one student raises his hand and gives
his answer, in Sumerian words pronounced in the following order:
sar-u-limmu = (3,600 x 10) x 4
= 4 pierced spheres
sar-ia = 3,600 x 5
= 5 spheres
ges-u-limmu = (60 x 10) x 4
= 4 large pierced cones
ges-min = 60 x 2
= 2 large cones
ninnu = 50
= 5 pellets
ges = 1
= 1 small cone
Not forgetting to add, of course
se sila su-kid es (“and there are 3 sila of barley left over”)
Another of the students, however, shows up his work to the teacher as he
has traced it onto his clay tablet, which he has divided into boxes and filled
up with Sumerian script. In the top right-hand box, in archaic numerals, he
has written the answer (164,571) exactly as shown in Fig. 8.20 above:
• 4 large circular impressions with small circular impressions within
(a direct representation of 4 pierced spheres, each worth 36,000);
• 5 large plain circular impressions (a direct representation of 5
spheres, each worth 3,600);
• 4 wide notches with small circular impressions within (recalling the
4 large pierced cones, each worth 600);
• 2 plain large notches (for the 2 plain large cones, each worth 60);
• 5 small circular imprints (for the 5 pellets each worth 10); and
• 1 narrow notch (for the small cone representing 1).
And, since the spoken word vanishes into thin air, while what is written
remains, it is thanks to the latter that the division sum from Suruppak has
survived for the thousands of years since the students who solved it
vanished from the face of the earth.
125
THE DISAPPEARANCE OF THE CALCULI
IN MESOPOTAMIA
We can infer that Sumerian arithmetic was done in this kind of way from
the most archaic times down to the pre-Sargonic era. The tablet shown in
Fig. 12.1 is one piece of evidence, and the calculi from this epoch found
in those regions provide another; but the most solid proof is the recon-
struction of the method which we have shown, for it can be easily
demonstrated that the same principles may be applied equally well to
multiplication, addition, and subtraction.
Nonetheless, the historical problems of Mesopotamian arithmetic have
not been completely solved.
At the time at which the tablet we have been examining was made (around
2650 BCE), the calculi were still in use throughout the region, and in
appearance they remained close to the archaic, or curviform, numerals which
had then come into use. These numerals, however, while still present at
the time of Sargon I (around 2350 BCE), gradually disappeared during the
second half of this millennium. Finally, at the time of the dynasty of Ur III
(around 2000 BCE) they had been replaced by the cuneiform numerals.
Correspondingly, the calculi themselves are no longer found in the majority
of the archaeological sites of Mesopotamia dating from this period or later
(Fig. 10.2 above).
While undergoing this transformation from archaic curviform to
cuneiform aspect, the written numerals lost all resemblance to the calculi
which were their concrete ancestors. The Sumerian written number-system,
moreover, was essentially a static tool with respect to arithmetic, since it
was not adapted to manipulation for calculations: the numerals, whether
curviform or cuneiform, instead of having inherent potential to take part
in arithmetical processes, were graphical objects conceived for the purpose
of expressing in writing, and solely for the sake of recall, the results of
calculations which had already been done by other means.
Therefore the calculators of Sumer, at a certain point in time, faced the
necessity of replacing their old methods with new in order to continue to
function. They therefore substituted for the old system of the calculi a new
“instrument" which I shall shortly describe. Meanwhile, we make a detour
to prepare the ground.
FROM PEBBLES TO ABACUS
Only a few generations ago, natives in Madagascar had a very practical way
of counting men, things, or animals. A soldier, for instance, would make his
men pass in single file though a narrow passage. As each one emerged he
FROM PEBBLES TO ABACUS
would drop a pebble into a furrow cut into the earth. After the tenth had
passed, the 10 pebbles would be taken out, and 1 pebble added to a
parallel furrow reserved for tens. Further pebbles were then placed into the
first furrow until the twentieth man had passed, then these 10 would be
taken out and another added to the second furrow. When the second
furrow had accumulated 10 pebbles, these in turn were taken out and 1
pebble was added to a third furrow, reserved for hundreds. And so on until
the last man had emerged. So a troop of 456 men would leave 6 pebbles in
the first furrow, 5 in the second, and 4 in the third.
Each furrow therefore corresponded to a power of 10: the ones, the tens,
the hundreds, and so on. The Malagasies had unwittingly invented the
abacus.
This was not unique to them, however. Very similar means have been
devised since the dawn of time by peoples in every part of the earth, and
the form of the instrument has also varied.
Some African societies used sticks onto which they slid pierced stones,
each stick corresponding to an order of magnitude.
Amongst other peoples (the Apache, Maidu, Miwok, Walapai, or
Havasupai tribes of North America, or the people of Hawaii and many
Pacific islands) the practice was to thread pearls or shells onto threads of
different colours.
Others, like the Incas of South America, placed pebbles or beans or
grains of maize into compartments on a kind of tray made of stone, terra-
cotta, or wood, or even constructed on the ground.
The Greeks, the Etruscans, and the Romans placed little counters of
bone, ivory, or metal onto tables or boards, made of wood or marble, on
which divisions had been ruled.
Other civilisations produced better implementations of the idea, by
using parallel grooves or rods, with buttons or pierced pellets which could
be slid along these. This is how the famous suan pan or Chinese Abacus
came about, a most practical and formidable instrument which is still in
common use throughout the Far East.
But before they used their abacus, the Chinese had for centuries used
little ivory or bamboo sticks, called chou (literally, “calculating sticks”)
which they arranged on the squares of a tiled floor, or on a table made like
a chessboard.
The abacus did not evolve solely in form and construction. Far greater
changes took place in the manner of its use.
The Madagascar natives, who did not profit fully from their great
discovery, no doubt never understood that this way of representing
numbers would give them the means to carry out complex calculations.
So in order to add 456 persons to 328 persons, they would wait out the
HOW THE SUMERIANS DID THEIR SUMS
126
passage of the 456, and then of the 328 others, in order to finally observe
the pebbles which gave the result.
Their use of the abacus was, therefore, purely for counting. Many other
peoples were no doubt in the same state in the beginning. But, in seeking a
practical approach to making calculations which were becoming ever more
complex, they were able to develop procedures for the device by conceiving
of a subtle game in which the pebbles were added or removed, or moved
from one row to another.
To add one number to a number already represented on a decimal
device, all they had to do was to represent the new number also on the
abacus, as before, and then - after performing the relevant reductions - to
read off the result. If there were more than 10 pebbles in a column, then
10 of these would be removed and 1 added to the next, starting with
the lowest-order column. Subtraction can be done in a similar way, but by
taking out pebbles rather than putting them in. Multiplication can be
carried out by adding the results of several partial products.
The “heap of pebbles” approach to arithmetic, indeed the manipulation
of various kinds of object for this purpose, thus once again is central in
the history of arithmetic. These methods are at the very origin of the calcu-
lating devices which people have used throughout history, at times when
the numerals did not lend themselves to the processes of calculation, and
when the written arithmetic which we can achieve with the aid of “Arabic”
numerals did not yet exist.
THE SUMERIAN ABACUS RECONSTRUCTED
It is logical, therefore, to suppose that Sumerian calculators themselves
made use of some sort of abacus, at any rate once their calculi had
disappeared from use.
Archaeological investigation in the land of Sumer has failed so far to
yield anything of this kind, nor has any text been discovered which
precisely describes it as well as its principles and its structure. Nonetheless,
we can with the greatest plausibility reconstruct it precisely.
We may in the first place suppose that the instrument was based on
a large board of wood or clay. It may equally well have been on bricks or
on the floor.
The abacus consists of a table of columns, traced out beforehand,
corresponding to the different orders of magnitude of the sexagesimal
system.
We may likewise suppose that the tokens which were used in the device
were small clay pellets or little sticks of wood or of reed, which each
had a simple unit value (unlike the archaic system of the calculi, whose
pieces stood variously for the different orders of magnitude of the same
number-system).
We may determine the mathematical principles of the Sumerian abacus
by appealing to their number-system itself.
Their number-system, as we have seen, used base 60. This theoretically
requires memorisation of 60 different words or symbols, but the spacing
between successive unit magnitudes was so great that in practice an
intermediate unit was introduced to lighten the load on the memory. In
this way, the unit of tens was introduced as a stepping stone between the
sexagesimal orders of magnitude. The system was therefore based on a
kind of compromise, alternating between 10 and 6, themselves factors of
60. In other words, the successive orders of magnitude of the sexagesimal
system were arranged as follows:
first order
first unit
1
= 1
=
1
of magnitude
second unit
10
= 10
=
10
second order
first unit
60
= 60
=
10.6
of magnitude
second unit
600
= 10.60
=
10.6.10
third order
first unit
3,600
= 60 2
=
10.6.10.6
of magnitude
second unit
36,000
= 10.60 2
=
10.6.10.6.10
fourth order
first unit
216,000
= 60 3
=
10.6.10.6.10.6
of magnitude
second unit
2,160,000
= 10.60 3
=
10.6.10.6.10.6.10
On this basis, therefore, we can lay out the names of the numbers in a
tableau as in Fig. 12.3. There are nine different units, five different tens, nine
different sixties, and so on. From this table, therefore, we can clearly see that
ten units of the first order are equivalent to one unit of the second, that six of
the second are equivalent to one of the third, that ten of the third are equiva-
lent to one of the fourth, and so on, alternating between bases of 10 and 6.
If, therefore, we accept that the Sumerians had an abacus, it must have
been laid out as in Fig. 12.4.
Each column of the abacus therefore corresponded to one of the two
sub-units of a sexagesimal order of magnitude. Since, moreover, the
cuneiform notation of the numerals was written from left to right, in
decreasing order of magnitude starting from the greatest, we may therefore
reconstruct this subdivision in the following manner.
Proceeding from right to left, the first column is for the ones, the second
for the tens, the third for the sixties, the fourth for the multiples of 600, the
fifth for the multiples of 3,600, and so on (Fig. 12.4). To represent a given
number on this abacus, therefore, one simply places in each column the
number of counters (clay pellets, sticks, etc.) equal to the number of units
of the corresponding order of magnitude.
127
THE SUMERIAN ABACUS RECONSTRUCTED
SECOND SEXAGESIMAL ORDER FIRST SEXAGESIMAL ORDER
Sub-order of the
Sub-order of the
Sub-order of
Sub-order of
multiples of 600
multiples of 60
the tens
the units
from 1 x 600
from 1 x 60
from 1 x 10
from 1
to 5 x 600
to 9 x 60
to 5 x 10
to 9
i i i i
i i i i
6 i 10 i 6 i 10 i
'l' 'l' \E
600
ges-u
(= 3 x 600)
60
ges
(= 1 x 60)
10
u
i
ges
1,200
ges-u-min
(= 2 x 600)
120
ges-min
(= 2 x 60)
20
nis
2
min
1,800
ges-u-es
(= 1 x 600)
180
ges-es
(= 3 x 60)
30
usu
3
es
2,400
ges-u-limmu
(= 4 x 600)
240
ge's-limmu
(= 4 x 60)
40
nimin
4
limmu
3,000
ges-u- id
(= 5 x 600)
300
ges-ia
(= 5 x 60)
50
ninnu
5
id
360
ges-as
(= 6 x 60)
6
di
420
ges-imin
(= 7 x 60)
7
imin
480
ges-ussu
(= 8 x 60)
8
ussu
540
ges-ihmmu
(= 9 x 60)
9
ilimmu
Fig. 12 . 3 . Structure of the Sumerian number-system (see also Fig. 8.6 and 10.4)
MULTIPLES OF
1
1
1
1
Nk
1
1
1
1
1
1
1
1
1
*
1
1
1
1
1
1
I
l
1
*
1
1
1
1
1
1
1
I
1
*
1
1
1
1
1
10.60 3
\F
10.60 2
vF
10.60
*
10
N V
60 3
60 2
60
1
CALCULATION ON THE SUMERIAN ABACUS
Suppose one number is already laid out on the abacus, and we wish to add
another number to it. To do this, lay out the second number on the abacus
as well. Then, if there are 10 or more counters in the first column, replace
each 10 by a single counter added to the second. Then replace each 6 in the
second column by 1 added to the third, then each 10 in the third by 1 added
to the fourth, and so on, alternating between 10 and 6. When the left-hand
column has been reached, the result of the addition can be read off.
Subtractions proceed in an analogous way, and multiplication and division
are done by repeated additions or subtractions.
Let us return to the problem in the tablet shown in Fig. 12.1, and try to
solve it on the abacus. We want to divide 1,152,000 by 7. We shall proceed
by means of a series of partial divisions, each one on a single order of
magnitude and beginning with the greatest.
First stage
In Sumerian terms, we are to divide by 7 the number whose expression, in
number-names, is
HOW THE SUMERIANS DID THEIR SUMS
128
sargal-ia sar-u-min,
which breaks down mathematically to:
5 x 60 3 + 2 x (10.60 2 ) = 5 x 216,000 + 2 x 36,000.
In the dividend there are therefore 5 units of order 216,000, and 2 of
order 36,000. But, since the highest is present only five-fold and 5 is not
divisible by 7, these units will be converted into multiples of the next lower
order of magnitude, replacing the 5 counters in the highest order by the
corresponding number of counters on the next.
One unit of order 216,000 is equal to 6 units of order 36,000, so we
take 5 x 6 = 30 counters and add these to the 2 already there. There are,
therefore, 32 counters on the board.
Now, 32 divided by 7 is 4, with remainder 4. 1 therefore place 4 counters
(for the remainder) above the next column down (the 3,600 column) so as
not to forget this remainder. Then the 4 counters (for the quotient) are
placed in the 36,000 column. Then I remove the remaining counters.
Order of the 36,000s 1 i Order
1 | of the
| * 3,600s
'V II II < 1st remainder
—
—
—
—
Fig. 12. 5a.
Second stage
Now I convert the 4 counters of the preceding remainder into units of order
36,000.
One unit of 36,000 is 10 units of 3, 600, so I take 10 x 4 = 40 counters.
But 40 divided by 7 is 5, with remainder 5. Therefore I now place 5 coun-
ters (for this remainder) above the next column down (600) so as not to
forget it.
Then 1 place the 5 counters (for the quotient) in the 3,600 column, and
remove the remaining counters.
Order of the 3,600s , , Order
1 | of the
| ♦ 600s
'h I I I I I < 2nd remainder
Third stage
Now I concert the 5 counters for the preceding remainder into units of
order 600. One unit of 3,600 is 6 units of 600, so I take 5 x 6 = 30 counters.
But 30 divided by 7 is 4, with remainder 2. 1 place 2 counters (for the
remainder) above the next column down (60), as before.
Then 1 place 4 counters (for the preceding quotient) in the 600 column,
and finally I remove the remaining counters (Fig. 12. 5C, opposite).
Fourth stage
Now I convert the 2 counters for the preceding remainder into units of
order 60. One unit of 600 is 10 units of 60, so I take 2 x 10 = 20 counters.
Now 20 divided by 7 is 2, with remainder 6. So I place 6 counters (for the
remainder) above the next column down (10). Then I place 2 counters (for
the preceding quotient) in the 60 column. Then I remove the remaining
counters (Fig. 12.5D, opposite).
129
CALCULATION ON THE SUM F. RIAN ABACUS
Order of the 600s
M'
Order
of the
60s
3rd
remainder
Order
of the
Order of the 60s 1 10s
i
[ ' 4th
1 ^ remainder
; 1 1 ii 1 1 <— i
Fifth stage
I convert the 6 counters for the preceding remainder into units of order 10.
One unit of 60 is 6 units of 10, so I take 6 x 6 = 36 counters.
But 36 divided by 7 is 5 with remainder 1. So I place 1 counter (for
the remainder) above the next column (units) and then 5 counters (for the
preceding quotient) into the tens column. Then 1 remove the remaining
counters. _ ,
Order £
of the 1
Order of the 10s 1 units £
i l I
I Mr
; 1 4-i
Sixth and final stage
Now 1 convert the single counter for the preceding remainder into simple
units. One unit of 10 is 10 simple units, so I take 10 counters.
But 10 divided by 7 is 1, with remainder 3. So 1 place 3 counters (for the
remainder) to the right of the units column. Then 1 place 1 counter (for
the preceding quotient) into the units column, and I remove the remaining
counters.
Since I have now arrived at the final column, of simple units, the
procedure is finished. To obtain the final result, I simply read off from
the abacus to obtain the quotient (Fig. 12.5 F):
4 x 36,000 + 5 x 3,600 + 4 x 600 + 2x 60 + 5x 10 + 1
and the 3 counters which I placed at the right of the last column give me
the remainder.
HOW THE SUMERIANS DID THEIR SUMS
130
210,000 3,600 60 1
36,000 ! 600 ' 10
6th
remainder
^ ^ ^ ^ ^ ^
I I I I I I
Quotient from [ j j | j j
the first stage J [ ' '
i i i i i
Quotient from j i i i i
the second stage 1 i i t i
i 1 I '
i | | |
Quotient from ! i i i
the third stage 1 \ i i
i i i
I I I
Quotient from ] , |
the fourth stage J \ 1
1 i
' i
Quotient from ] i
the fifth stage 1 i
Quotient from the sixth and last stage
1
\k
1
1
1
Nk
FINAL QUOTIENT
4 x 36.000 + 5 x 3,600 + 4 x 600 + 2 x 60 + 5 x 10 + 1
Remainder
3
Fig. 12. 5f.
On the abacus, therefore, the procedures for calculation were much
simpler than for the much more ancient methods using the calculi of old.
Undoubtedly both methods were in use together for a certain time, the
more traditionally-minded tending to stay with the methods of their prede-
cessors. These too were probably the same who continued to use the
curviform notation of times past up until the end of the third millennium
by which time the use of cuneiform notation had spread throughout
Mesopotamia. We may therefore imagine the disputes between “calculists”
and "abacists”, the former standing to the defence of calculating by
means of objects of different sizes and shapes, the latter attempting to
demonstrate the many advantages of the new method.
What I have just said about the quarrel between the specialists is
plausible, but it is merely a figment of my imagination. The rest of what
I have been saying, though, is much more than merely probable.
CONFIRMATION OF THE SUMERIAN ABACUS
AND ABACISTS
The reconstructions described above have in fact received confirmation as
a result of recent discoveries.
I am referring to Sumero-Akkadian texts on cuneiform tablets dating
from the beginning of the second millennium BCE. from various Sumerian
archaeological sites (including Nippur), which have been meticulously
collated, translated and interpreted by Liebermann (in AJA). These texts
are all reports, and detailed analyses, in two languages (Sumerian and
Ancient Babylonian) of various professions exercised at the time in Lower
Mesopotamia. They are, in a way, “yearbooks” for these professions, and
were made in several copies. The reports refer to each profession by giving
a description of its representative, and a brief title of the kind “man of. . .”,
but at the same time in each case they clearly specify the nature of any tools
or devices used in each profession.*
Among all the many sorts of information in these texts, we find precisely
the professions which are of prime interest for us. The lists give with
great precision not only their official designation but also their tools and
* The bilingual texts from which have been taken the names given in Fig. 12.6 A-L occur mainly on tablets
with the following museum references: - 3 NT 297, 3 NT 301 (cf. Field Numbers of Tablets excavated at
Nippur ); - JM 68433, IM 58496 (cf. Tablets in the Collections of the Iraqi Museum of Baghdad)', - NBC 9830
(cf. Tablets in the Babylonian Collection of the Yale University Library, New Haven, Conn.); - MLC 653 and
1856 (cf. Tablets in the Collection of the ]. P. Morgan Library, currently housed in the Babylonian Collection of
the Yale University Library, New Haven). The article by S. J. Liebermann, of which the principal results are
summarised here in a more accessible form and with some supplementary detail, provides the expert with
all necessary philological information and correspondences, and all necessary bibliographical references,
including those referring to the important publication by B. Landsberger (cf. Materialen zum Sumcrischen
I.exikon, Rome, 1937).
131
instruments, down to the very detail of their shape and material, and even
which component goes with which instrument.
This is therefore a sufficiently significant discovery to justify a detailed
philological explanation. The results will be displayed in successive
diagrams each with three columns. On the left we shall place the Sumerian
name (in capital letters); in the centre we shall place the Ancient Babylonian
name (in italics); and, on the right, the equivalent English translation.
First we encounter a word which expresses the verb “to count”:
SID
ma-nu
to count
Fig. 12 .6 a.
Remarkably, the Sumerian graphical etymology of this verb displays
in itself evidence of the existence of the abacus. Originally, this verb was
represented by the following pictogram (Fig. 12.6 B). Here we see a hand, or
at any rate an extreme idealisation of one, doing something with a board in
the shape of a frame or a tray and divided into rows and columns.
Somewhat later on, the same verb was represented by a cuneiform
ideogram, where we seem to see a frame divided into several columns and
intersected by a vertical wedge resembling the figure for unity:
MOST ANCIENT ARCHAIC MORE RECENT
FORM OF CUNEIFORM SIGN SIGN
THE SIGN (Sumerian of the epoch (classical Sumerian)
(archaic Sumerian of of femdet Nasr)
the i'ruk period)
Fig. 12.6 b. Sumerian notations of the verb "to count" ( sidj. Deimel (1947) no. 314
Considering how ancient this sign for unity is (3000-2850 BCE), we are
led to believe that the Sumerian abacus goes back to an even more ancient
period than we had previously supposed. To come back to the “professional
yearbooks”, we find here also a clear reference to the system of calculi,
which are referred to by a word which strictly means “small clay object”:
Fig. 12.6c.
CONFIRMATION OF THE SUMERIAN ABACUS
The “accounting” itself is denoted by a combination of the verb SID (“to
count”) and the word NIG (total, sum):
Fig. 12. 6d.
Here again, the Sumerian etymology traces this to a suggestive origin,
since the signs for the word N1G1 (or NIGIN), meaning “total”, “sum”, “to
collect together”, clearly suggest the successive sections of the abacus:
MOST ANCIENT
FORM OF
THE SIGN
(Archaic Sumerian of
the Uruk period)
ARCHAIC
CUNEIFORM SIGN
(Sumerian of the epoch of
Jemdet Nasr)
□□
dp at
Fig. 12. 6f..
Next we find the word for the expert in weights and measures, in a way
the metrologist of that time and place, the “man of the stones”:
Fig. 12. 6f.
This designation obviously would not be confused with that of the
calculator using the method of the calculi-, this man is distinctly denoted
in these texts by the terms:*
Fig. 12. 6g.
* The texts have been damaged by time at this point, and we cannot make out the corresponding
Babylonian term. We see only its beginning, Sa, which tells us little since sa is simply the Sumerian transla-
tion of the word for ’man'’. But, following the work of A. L Oppenheim ( 1959 ), we know that the Sumerian
word for calculi was abrui (plural: abndti or abac, meaning literally “stone ', “stone object", “kernel" or “hail-
stone”. So w'e may suppose that the complete term was sa abndti-i, where the scribe w'ould use the second
form of the plural in order to avoid confusion with sa abne-e, unless he simply used the Sumerian word
IMNA ( calculi ) in order to coin a term similar to Sa imnaki (or sa imnake).
HOW THE SUMERIANS DID THEIR SUMS
132
What follows next is even more thrilling, since it reveals not only the
name but also the material used for the counter employed by the abacus-
users of the period (the word “GES” means “wood"):
Moreover, it yields not only what the counter is made of; we also learn
its shape, since the profession which made use of it comes under the
heading of “the men of the small wooden sticks”.
As we have supposed above, they indeed made use of rods to perform
their operations on the abacus (Fig. 12.5).
As to the abacus itself, we find that the texts refer to it clearly, using a
figurative expression. To understand what this means, let us first note that,
in Sumerian, “tablet” is said as DAB 4 and, in the absence of any further
details, this word is always understood to mean “clay tablet”, the dominant
medium of the region for writing on. But in this case the material of the
tablet is specified by the word GES, meaning “wood”. The word GESDAB 4
therefore means “wooden tablet” and in this context is therefore quite other
than the “paper of Mesopotamia".
Another word which enters into the makeup of the Sumerian term for
the abacus is DIM. As a verb, this means “to fashion”, “to form”, “to model
in clay”, “to construct" and so on, and hence, by association of ideas, “to
elaborate”, “to perfect”, “to create”, “to invent”. As a noun, it means
“fashion”, “form”, “construction”, and, by extension, “perfecting”, “forma-
tion”, “elaboration", “creation”, or “invention” [A. Deimel (1947), no. 440].
Now we understand that the word DIM frequently, by association of
ideas, referred to the activities associated with Mesopotamian accounting,
not only modelling and moulding clay (to make the calculi and the tablets)
but also, and above all, to perfect, to elaborate results, and consequently to
create and to invent something which nature did not provide in the raw
state. Moreover, calculation is essential for shaping and fashioning objects,
and also to architects for whom it is a vital necessity in their constructions.
Bringing all these terms together into a logical compound, by composing
the expression GESDAB 4 -DIM to designate the instrument in question, the
scribes must have had several simultaneous meanings present in their
minds:
1. “wooden tablet” meaning perfecting;
2. “wooden tablet” meaning elaboration;
3. “wooden tablet” meaning creation;
4. “wooden tablet” meaning invention;
5. “wooden tablet” endowed with form (namely the tablet);
6. “wooden tablet” endowed with forms (the columns);
7. “wooden tablet” meaning the accounts;
8. “wooden tablet” meaning accounting; and so on.
Here we have therefore the characteristics and the many purposes of the
abacus. The word GESDAB 4 -DIM can have but one translation.
GESDAB.-DIM
4
gesdab 4 -dim mu 1
abacus
Fig. 12. 6i.
Even more significant is this other designation of the instrument of
calculation:
GESSU-ME-GE
su-me-ek-ku-u
Fig. )2.6j.
The word SU which is one component of this expression literally means
“hand”. In certain contexts, however, it also means “total”, “totality”, allud-
ing to the hand which assembles and totalises [A. Deimel (1947), no. 354].
The word ME, for its part, means “rite”, “prescription”; in other words
“the determination of that which must be done according to the rules”, or
“an action which is performed according to a precise order as well as a
prescribed order” [A. Deimel (1947), no. 532].
It would not in fact be at all surprising if the practice of calculating
on the abacus corresponded to a genuine ritual, since the knowledge of
abstract numbers and, even more, skill in calculation were not within the
grasp of everyone as they are today. Those who knew how to calculate were
rare indeed.
With all the peoples of the earth, calculation did not merely evoke
admiration for those skilled in the art: they were feared, and regarded
as magicians endowed with supernatural powers. This naturally gave rise to
a certain element of sacred ritual in their activities, not to mention the
numerous privileges which kings and princes often granted them.
In any case, in a context such as the present one, the word ME must be
understood as “the determination of that which must be done in the precise
order prescribed by the rules of calculation”. This is something like what
computer scientists of modern times would mean by “algorithm”.
The term GE (or GI) is the word for a reed and stands for all the names
of objects which can be made from this material [A. Deimel (1947), no. 85].
When we put them all together, these terms give the expression GESSU-
ME-GE, which corresponds to one or other of the following literal
translations:
GES-SID-MA
is- si mi-nu-ti
GES-NIG 2 -SID
is-si nik-kds-si
Fig. 12. 6h.
counting stick;
stick for accounting
133
1. A hand (SU), a reed (GE), the rules (of arithmetic) (ME) and
wood (GES) (“of the tablet” understood);
2. The wood (of the tablet), a reed (GE), the rules (of arithmetic)
(ME) and a total (i.e. provided by the hand) (SU).
In plain language, the expression GESSU-ME-GE clearly refers to the
abacus.
Lastly, for the “professional calculator”, the texts use one or other of the
following expressions:
Fig. 12.6k.
The first of these means literally “the man (LU) of the wooden tablets
for accounting (GES DAB 4 DIM)”, and the second simply means “the man
(LU) of the wooden tablets (GESDAB 4 )“, there being no confusion possible
about the material support.
We also find one or other of the two names
Fig. 12. 6l.
The first of these means literally a man (LU) who manipulates the rules
(MUN) with a reed (GE) on the wood (GES) (“of the tablet" understood),
CONFIRMATION OF THE SUMERIAN ABACUS
while the second corresponds to a symbolic variant of the first which could
be translated as a man (LU) who finds the total (SU) with a reed (GI)
according to the rules (MUN).
There is at this time no doubt that the abacus indeed existed in
Mesopotamia, and even coexisted with the archaic system of calculi, most
probably throughout the third millennium BCE.
This abacus consisted of a tablet of wood, on which were traced before-
hand lines of division which exactly corresponded to the Sumerian
sexagesimal system (Fig. 12.5) and therefore delimited, column by column,
each of the order of unity of this numerical system (1, 10, 10.6, 10.6.10,
10.6.10.6, 10.6.10.6.10, and so on).
The counting tokens themselves were thin rods of wood or of reed, given
the value of a simple unit, such that their subtle arrangement over the
columns of the abacus allowed all the operations of arithmetic to be carried
out. (No doubt it is as a result of their perishable nature that archaeologists
have never brought any of these to light. Another reason may be that, as we
may well suppose, whenever one of these experts did not have a “calculat-
ing board” to hand, he could simply draw the “tablet” on the loose soil.)
Lastly, as with writing but perhaps more so, the use of the abacus gave
rise to a guild, perhaps even with the privileges of a special caste, so much
would its complex rules and practices have been inaccessible to ordinary
mortals: this was the caste of the professional abacists, who no doubt
jealously preserved the secrets of their art.
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
CHAPTER 13
MESOPOTAMIAN NUMBERING
AFTER THE ECLIPSE OF SUMER
THE SURVIVAL OF SUMERIAN NUMBERS
IN BABYLONIAN MESOPOTAMIA
For some time after the decline of Sumerian civilisation, the sexagesimal
system remained in use in Mesopotamia. Just as many French people still
use “old francs” in their everyday reckonings even though “new francs”
replaced them officially as long ago as 1960, so the inhabitants of
Mesopotamia continued to use the “old counting” based on multiples and
powers of 60.
The following examples come from an accounting tablet excavated at
Larsa (near Uruk) and probably dating from the reign of Rim Sin
(1822-1763 BCE). They are characteristic examples of the everyday reckon-
ings that constitute the city archives, and give an
following numerical values:
account of sheep with the
61 (ewes)
| 96 (ewes)
T «<fff
60 1
60 30 6
84 (rams)
105 (rams)
rm
60 20 4
60 40 5
145 (sheep)
IT W 201 (sheep)
rrr« r
120 20 5
180 20 1
F 1 c . 13.1. Birot, tablet 42, p. 85, plate XXIV
The numerals used are indeed those of the old Sumerian cuneiform
system, with its characteristic difficulty in the representation of numbers
such as 61, the vertical wedge signifying 60 being almost indistinguishable
from the wedge used for 1 - and this is certainly the reason why the scribe
leaves a large space between the two symbols, so as to avoid confusion with
the number 2.
There is nothing at all surprising in the persistence of the old system in
Lower Mesopotamia, since that is where the system first arose, in the lands
of Sumer. What is less obvious is why the sexagesimal system survived for
so long in the lands to the north, that is to say in Akkadian areas. However,
134
the evidence is indisputable. This is an example from a tablet written
in Ancient Babylonian, dating from the thirty-first year of the reign
of Ammiditana of Babylon (1683-1647 BCE). It provides an inventory of
calves and cows in the following manner:
w I rrr<«c$ I w STW
240 30 7 180 20 9 8 su-si 6
277 209 486
Fig. 13.2. Finkelstein, tablet 348, plate CXIV, II. 8-10
The number 486 (the sum of 277 and 209) is represented not just by 8
large wedges each standing for 60 and 6 small wedges each standing for 1.
The scribe has chosen to provide an additional phonetic confirmation of
the number by putting the word shu-shi (the name of the number 60 in
Akkadian) after the larger expression, rather like the way in which we write
out cheques with a numerical and a literal expression of the sum involved.
All the same, these are just about the last traces of the unmodified
system in use in Mesopotamia. Sumerian numbering was abandoned
for good around the time that the first Babylonian Dynasty disappeared, in
the fifteenth century BCE. By then, of course, modern Mesopotamian
numerals, of Semitic origin, had been current for some time already.
WHO WERE THE SEMITES?
The term “Semite” derives from the passage in the Old Testament (Genesis
10) where the tribes of Eber (the Hebrews), Elam, Asshur, Aram, Arphaxad,
and Lud are said to be the descendants of Shem, one of Noah’s three sons,
the brother of Ham and Japheth. However, though it may have represented
a real political situation in the first millennium BCE, the biblical map of the
nations of the Middle East makes the Elamites, who spoke an Asianic
language, cousins to the Hebrews, Assyrians, and Aramaeans, whose
languages belong to the Semitic group.
“Asianic” is the term used for the earlier inhabitants of the Asian main-
land whose languages, mostly of the agglutinative kind, were neither
Indo-European nor Semitic. It is generally believed that Mesopotamia was
originally inhabited by Asianic peoples, prior to the arrival of Sumerians. It
is thought that Semitic-speaking populations came in a second wave, and
that Akkadian civilisation constitutes the earliest Semitic nation in the area.
However, significant Semitic elements are to be found in the cultures of
Mari and Kis at the beginning of the third millennium BCE, and it is even
possible that the people of El Obeid were of Semitic origin themselves,
135
A BRIEF HISTORY OF BABYLON
though absorbed and assimilated by the Sumerians. The discovery of
the Ebla tablets revealed the existence of a state speaking a language of the
Semitic family in the mid-third millennium BCE, and so it becomes ever
less certain that the “cradle” of the Semitic languages was the Arabian
peninsula, as was long held to be true. Nonetheless, Arabic is probably the
closest to the proto-Semitic stem-language, which began to differentiate
into numerous branches (Ancient Egyptian, some aspects of the Hamitic
languages of eastern Africa, and possibly even Berber, spoken in Algeria
and Morocco) as early as the Mesolithic era, that is to say (for the Middle
East) in the tenth to eighth millennia BCE. That is too far back in time for it
to be possible to say exactly where Semitic languages first arose or who the
people were who brought them to different civilisations in the Middle East.
Like the term “Indo-European”, “Semitic” does not designate any ethnic
or cultural entity, but serves only to define a broad family of languages.
There was no single “Semitic civilisation”, just as there was never such
a thing as an Aryan or an Indo-European culture. Each of the main Semitic-
speaking civilisations of antiquity developed its own specific culture, even
if there are some features common to several or all of them. It is there-
fore important to distinguish amongst the Semitic cultures those of the
Akkadians, the Babylonians, the Assyrians, the Phoenicians, the Hebrews,
the Nabataeans, the Aramaeans, the various peoples of Arabia, Ethiopia, and
so on. (See Guy Rachet, Dictionnaire de I’archeologie, for further details.)
A BRIEF HISTORY OF BABYLON
At the beginning of the third millennium BCE Sumerians dominated
the southern Mesopotamian basin, both numerically and culturally. To the
north of them, between the Euphrates and the Tigris and on the northern
and eastern edges of the Syro-Arabian desert, lived tribes of semi-nomadic
pastoralists who spoke a Semitic language, called Akkadian. The Akkadian
king, Sargon I The Elder, founded the first Semitic state when he defeated
the Sumerians in c. 2350 BCE. His empire stretched over the whole of
Mesopotamia and parts of Syria and Asia Minor. Its capital was at Agade
(or Akkad), and, for one hundred and fifty years, it was the centre of the
entire Middle East. As a result, Akkadian became the language of
Mesopotamia and gradually pushed aside the unrelated language of Sumer.
Assyrian and Babylonian are both descended from Akkadian and are thus
Semitic, not Asianic, languages.
The Akkadian empire collapsed around 2150 BCE and for a relatively
brief time thereafter Sumerians reasserted their control of the area. But
that was the final period of Sumerian domination, for around 2000 BCE,
the third empire of Ur collapsed under the simultaneous onslaughts of the
Elamites (from the east) and the Amorites (from the west). Sumerian civili-
sation disappeared with it for ever, and in its place arose a new culture, that
of the Assyro- Babylonians.
The Amorites, a Semitic people from the west, settled in Lower
Mesopotamia and founded the city of Babylon, which would become and
remain for many centuries the capital of the country known as Sumer and
Akkad. The famous king and law-maker Hammurabi (1792-1750 BCE) was
one of the outstanding figures of the first Babylonian dynasty established
by the Semites, who became masters of the region. Hammurabi extended
Babylonian territory by conquest over the whole of Mesopotamia and as far
as the eastern parts of Syria.
This huge and powerful kingdom was nonetheless seriously weakened,
from the seventeenth century BCE, by the Kassites, Iranian highlanders
who made frequent raids, and it finally surrendered in 1594 BCE to the
Hittites, who came from Anatolia.
Babylon then remained under foreign domination until the twelfth
century BCE, when another Semitic people, the Assyrians, from the hilly
slopes between the left bank of the Tigris and the Zagros mountain range,
entered the concert of nations. The Assyrians were bearers of a version of
Sumerian culture, which they developed most fully in military conquest,
establishing an empire which stretched out in all directions and which was
one of the most fearsome and feared military powers in the ancient world,
until in 612 BCE, Nineveh, the Assyrians’ capital, was destroyed in its turn.
The Babylonians, although dominated by the Assyrians from the ninth
to the seventh centuries BCE, nonetheless retained their own distinctive
culture throughout this period. However, the fall of Nineveh (and with it of
the whole Assyrian Empire) in 612 BCE allowed a great flowering of
Babylonian culture, which was the prime force in the Middle East for over
a century, most especially under the reign of Nebuchadnezzar II (604-562
BCE). But that was Babylon’s last glory: it was conquered in 539 BCE by
Cyrus of Persia, then in 311 BCE by Alexander the Great, and finally expired
completely shortly before the beginning of the Common Era.
THE AKKADIANS, INHERITORS OF
SUMERIAN CIVILISATION
In the Akkadian period (second half of the third millennium BCE) the
Semites, who were now the masters of Mesopotamia, emerged as
the preponderant cultural influence in the region. They naturally sought to
impose their own language, and also to give it a written form. To do this
they borrowed the cuneiform system of their predecessors, and adapted it
progressively to their language and traditions.
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
136
By the time Sumerian cuneiform was adopted by the Akkadians, the
writing system was already several centuries old. The ideas originally signi-
fied by the ideograms were mostly forgotten, and the signs were now purely
symbolic. What the Akkadians found was a basically ideographic writing
system with an already-established drift towards a phonetic system - a drift
which the Akkadians accelerated, whilst retaining the ideographic meaning
of some of the signs. They did so partly because their own language was
less well suited to ideograms than Sumerian, and also because the signs
which represented words for Sumerians represented only sounds to
Akkadian ears.
The adaptation of cuneiform writing was however not a smooth or easy
process. For one thing, Akkadian had sounds not present in Sumerian,
and vice versa. The two ethnic groups of Akkadians (Babylonians and
Assyrians) proceeded independently in this development, despite the
numerous contacts between them. But by adopting the Sumerian cultural
heritage, the Akkadians gave it its greatest flowering, leading it away from
its origins in mnemotechnics and ultimately towards the creation of a true
literary tradition.
THE NUMBERING TRADITIONS OF
SEMITIC PEOPLES
The spoken numbering system of the Semites was very different from
the way Sumerians expressed numbers orally - not just linguistically, but
also mathematically, since Semitic numbering was, and remains, strictly
decimal.
However, Semitic numbering has one small grammatical oddity, in terms
of the decimal numbering systems to which we are now accustomed.
Hebrew and Arabic numbering (see Fig. 13.3 below) provide characteristic
examples.
In Hebrew as in Arabic, spoken numerals have feminine and masculine
forms, according to the grammatical gender of the noun to which they
are attached. For instance, the name of the number 1, treated as if it
were an adjective, has one form if the noun it qualifies is masculine, and a
different form if the noun is feminine. Similarly, the name of the number
2 agrees in gender with its noun. However, what is unusual is that for
all numbers from 3, the number-adjective is feminine if the noun is mascu-
line, and masculine if the qualified noun is feminine. In Hebrew, for
example, where “men” is anoshim and “three” is shalosh (masculine) or
shloshah (feminine), the expression “three men” is translated by shloshah
anoshim, not, as you might expect from Latin or French grammar, by
shalosh anoshim.
HEBREW
ARABIC
1
2
3
4
5
6
7
8
9
10
Feminine
Masculine
’eh ad
’ah at
shnaim
shtei
shloshah
shalosh
'arba ‘ah
’arba
hamishah
hamesh
shishah
shesh
shib’ah
sheba '
shmonah
shmoneh
tishah
tesha '
‘asarah
‘eser
Feminine
Masculine
ahadun
'ihda
’itnan
'itnatani
talatun
talatatun
’arba’un
’arbaatun
khamsun
khamsatun
situn
sitatun
sab'un
sab ‘atun
tamany
tamanyatun
tis’un
tis'atun
ashrun
‘asharatun
Fig. 13-3-
Numbers from 11 to 19 are formed by the name of the unit followed by
the word for 10, each having masculine and feminine forms, used according
to the previous rule:
11
12
13
14
15
16
HEBREW
Feminine
Masculine
'ahad 'asar
'ahat ‘esreh
shnaim ‘asar
shtei 'esreh
shloshah 'asar
shlosh ‘esreh
'arba' ah ‘asar
'arba ‘esreh
hamishah ‘asar
hamesh ‘esreh
shishah ‘asar
shesh 'esreh
ARABIC
1
Feminine
Masculine
'ahad 'ashara
'itnd 'ashara
talatut 'ashara
'arba'ata ‘ashara
khamsata ‘ashara
sitata 'ashara
'ihda ‘ashrata
’itndta ‘ashrata
talata ‘ashrata
’arba' a ‘ashrata
khamsa ‘ashrata
sita 'ashrata
Fig. 13.4.
Apart from the number 20, which is derived from the dual form of the
word for 10, the tens are derived from the name of the corresponding unit,
with an ending that is derived from the customary mark of the plural:
HEBREW ARABIC
20
30
40
50
60
70
80
90
‘eshrim
’isrun
shloshim
taldtuna
’arba'im
’arba'una
hamishim
khamsuna
shishim
situna
shibim
sib'una
shmonim
tamanuna
tishim
tis'una
derived from dual of 10
plural of name of 3
plural of name of 4
plural of name of 5
plural of name of 6
plural of name of 7
plural of name of 8
plural of name of 9
Fig. 13.5.
The system has special names for 100 and 1,000, and proceeds thereafter
by multiplication for multiples of each of these powers of the base:
137
THF. NUMBERING TRADITIONS OF SEMITIC PEOPLES
HEBREW ARABIC
too
me’ah
mi’dtun
200
ma taim
mi’atany
dual of 100
300
shlosh meot
taldtu midtin
(3 X 100)
1,000
'elef
’alfun
2,000
’alpaim
alfdny
dual of 1,000
3,000
shloshet 'alafim
taldtu alaf
(3 X 1,000)
10,000
‘aseret ’alafim
‘asharat ’alaf
(10 x 1,000)
20,000
‘eshrim 'elef
‘ishrunat 'alaf
(20 x 1,000)
30,000
shloshim ’elef
talatunat 'alaf
(30 x 1,000)
Note: Classical Hebrew also has the word ribo (“multitude”) to designate 10,000, together with
its multiples: shtei ribot for 20,000, shlosh ribot for 30,000, etc. Similar words exist in other
ancient Semitic languages: ribab (Elamite), ribbatum (Mari), r(b)bt (Ugaritic).
Fig. 13.6.
NUMBER-NAMES IN
1 ishten
10 eshru, eshcret
2 sita, sind
20 eshrd
3 shaldshu
30 shaldsha
4 erbettu
40 arba
5 khamshu
50 khamsha
6 sheshshu
60 shushshu, shush i
7 sibu
70 *
8 shamanu
80 *
9 teshu
90 *
* The pronunciation of these
numbers is not known
ASSYRO-BABYLONIAN
100
me’atu, meal
= 10 2
200
sita metin
= 2 X
100
300
shalash meat
= 3 X
100
1,000
lim
= 10 3
2,000
sind lim
= 2 x
1,000
3,000
shalashat limi
= 3 x
1,000
10,000
esheret lim
= 10 X 1,000
20,000
eshrd lim
= 20 X 1,000
100,000
meat lim
= 100
X 1,000
200,000
sita metin lim
= 2 X
100 X 1,000
Fig. 13.7.
For intermediate numbers, addition and multiplication are used in
conjunction. In Arabic, it should be noted, the units are always put before
the tens: 57, for example, is sab'un wa khamsiina (“seven and fifty”), as in
German ( siebenundfiinfiig ).
The same order of expression is found in Ugaritic texts (Ugarit was a
Semitic culture that flourished at Ras Shamra, in northern Syria, around
the fourteenth century BCE) and in biblical Hebrew, most frequently in the
Pentateuch and the Book of Esther. According to Meyer Lambert, this order
of numbers is the archaic form.
However, the inverse order (hundreds followed by tens followed by
units) is also found in the Hebrew Bible, and this is the commonest form in
the first Books of the Prophets, and in most of the books written after the
Exile (Haggai, Zechariah, Daniel, Ezra, Nehemiah, Chronicles). Modern
Hebrew (Ivrit) also uses this order (except for numbers between 11 and 19),
which is also the most frequent structure in Semitic languages as a whole
(Assyro-Babylonian, Phoenician, Aramaic, Ethiopian, etc.).
All these numbering systems therefore demonstrate that they have a
common origin, which gives all Semitic numbering its characteristic mark.
It will now be easier to grasp how the Mesopotamian Semites radically
transformed the cuneiform numerals of the Sumerians, and to under-
stand the method that the western Semites (Phoenicians, Aramaeans,
Nabataeans, Palmyreneans, Syriacs, the people of Khatra, etc.) invented to
put their numbers in writing other than by spelling them out. (See Chapter
18 below, pp.227-32
THE SUMERO-AKKADIAN SYNTHESIS
When the Akkadians took over cuneiform sexagesimal numbering, they
were naturally hampered by a written system whose organisation differed
entirely from the strictly decimal base of their own long-standing oral
number-name system. The cuneiform numerals had a sign for 1 (the
vertical wedge) and for 10 (the chevron) - but, since there was no sign for
100 or for 1,000, it occurred to them to write out the names of these
numbers phonetically. “Hundred” and “thousand” were respectively meat
“6,657" IN ANCIENT & MODERN SEMITIC LANGUAGES
ARABIC
sitalunat 'alaf sitatu midtin sab'un wa khamsiina
six thousand six hundred seven & fifty
6 X 1,000 + 6 X 100 + 7 + 50
UGARITIC
tit’alpin tit mat sab'a I khamishuma
six thousand six hundred seven & fifty
6 X 1,000 + 6 x 100 + 7 + 50
CLASSICAL
HEBREW
sheshet 'alafim sesh meot shib'ah we khamishim
six thousand six hundred seven & fifty
6 X 1,000 + 6 X 100 + 7 + 50
CLASSICAL
& MODERN
HEBREW
sheshet ! alafim sesh meot khamishim we shib'ah
six thousand six hundred fifty & seven
6 x 1,000 + 6 x 100 + 50 + 7
ASSYRO-
BABYLONIAN
sheshshu limi seshshu meat khamsha sibu
six thousand six hundred fifty seven
6 x 1,000 + 6 X 100 + 50 + 7
ETHIOPIAN
sassa ma’at sadastu ma'dt khamsa wa sab'a tu
sixty hundred six hundred fifty & seven
60 x 100 + 6 x 100 + 50 + 7
Fig. 13.8.
MESOPOTAMIAN NUMBERING AFTER IHE ECLIPSE OF SUMER
and lim in Akkadian, so they represented these numbers as words, using
the Sumerian cuneiform signs for ME and AT, on the one hand, and for LI
and IM on the other - rather as if we made puzzle-pictures of “Hun” and
“Dread” to represent the sound and thus the number “hundred”:
for the Akkadian Ml fff- °
ME - AT words for 100 LI-IM LI-IM
100 (me at) and i ,000
1,000 (lim)
Fig. ib. 9 a. Fig. 13 . 9 B.
However, they did not stop at the “writing out” stage, they also created
genuine numerals, even if these were derived from the phonetic notation of
the number-names. The symbols chosen were of course no more than
sound-signs from their point of view, since they had lost the meanings that
they had had in Sumerian. The symbol for 100 was soon shortened to its
first syllable, ME, and for 1,000 the Akkadians used the chevron (= 10)
followed by the sign for ME (1,000 = 10 ME = 10 x 100). And since this was
the sign for the word meaning “thousand”, pronounced lim, the cuneiform
chevron followed by ME came to have the phonetic value of the sound LIM
and to be used in all Akkadian words containing the sound LIM.
y-
ME
100
Akkadian cuneiform numerals for 100 and 1,000 as
used from the second millennium BCE in everyday
accounting documents
LIM
1,000
Fig. 13 . 10 A.
Fig. 13 . 10 B.
Because of the standard Semitic custom of counting orally in hundreds
and thousands, the Akkadians therefore introduced strictly decimal nota-
tions into the sexagesimal numerals that they had adopted from the
Sumerians. The result was a thoroughly mixed Akkadian number-writing
system containing special signs for decimal and sexagesimal units, in the
following manner:
1
10
60
10 2
10x60
10 3
60 2
T
<
Y
*>
ME
LIM
1
10
60
100
600
1,000
3,600
Fic. 13 . 11 .
Let us look at a few characteristic examples. Those shown in Fig. 13.12
(M. J. E. Gautier, 1908, plates XVII, XLII and XLIII) come from clay tablets
found at Dilbat, a small town in Babylonian territory that flourished in the
13 8
nineteenth century BCE. Most of the tablets refer to the main events in the
lives of members of a single family, and constitute as it were the family
record.
60 40
TT h-
2 ME
rhirr
1 ME 3
Th# F
1 MF. 50 4
100
200
103
154
Fig. 13.12. See Gautier, plates XVII, XLII and XLIII
The next figure is a transcription of a tally of cattle found in northern
Babylon (M. Birot, 1970, tab. 33, plate XVIII) dating from the seventeenth
year of the reign of Ami-Shaduqa of Babylon (1646-1626 BCE):
tjPP'tji'
1 SHU-SHI 3
r^-rrr
60 10 3
60 20 5
1 ME 1 SHU-SHI 8
63
73
85
168
Fig. 13.13. See Birot, tablet 33, plate XVIII
These examples show how in this period the Akkadians did not seek to
overturn the sexagesimal system that was deeply rooted in local tradition.
However, for the numbers 60 and 61, and in many cases for multiples
of 60, the Semites coped with the corresponding difficulties of the notation
system rather better than had the Sumerians. It occurred to them to
represent the number 60 by the sound-group shu-shi, which was how
they pronounced the number in Akkadian (see Fig. 13.7 above) or, in
abbreviated form, as shu (see Fig. 13.2, 13.13 and 13.14).
T M<hT
1 SHU-SHI 1
t m^-ir
1 SHU-SHII 2
1 SHU-SIII 6
3 SHU-SHI
5 SHU-SHI
61
62
66
180
300
Fig. 13 . 14 .
In short, up to the middle of the second millennium BCE, Mesopotamian
scribes of public, private, economic, juridical, and administrative tablets
had recourse either to sexagesimal Sumerian numbering, or to decimal
Semitic numbering, or finally to a system constituted by a kind of interfer-
ence between the two bases.
139
MESOPOTAMIAN DECIMALS
MESOPOTAMIAN DECIMALS
When Akkadian speech and writing finally supplanted their Sumerian
counterparts in Mesopotamia, strictly decimal numbering became the
norm in daily use. The ancient signs for 60, 600, 3,600, 36,000, and 216,000
progressively disappeared, and only the symbols ME (= 100) and LIM
(= 1,000) remained, to provide the bases for the entire system of numerals.
As in classical Sumerian, units were represented by vertical wedges,
repeated once for each unit, but whereas the Sumerians had grouped the
wedges on a dyadic principle, the Akkadians put them in three groups:
T
TT
ITT
YT
W
¥
W
1
123456789
Fig. 13-15-
The tens were also usually represented by repetition of the chevron
(= 10), but here again the layout or grouping of the repeated symbols was
quite distinct from older Sumerian patterns:
<
«
60
r<
60+10
T«
60 + 20
60 + 30
10
20
30
40
50
60
70
80
90
Fig. 13.16.
As for the hundreds and thousands, they were symbolised by notations
based on multiplication, that is to say in accordance with the analytical
combinations that existed in the spoken language of the Akkadians:
too
T T-
400
TT-
2,000
TT**-
1 100
4 100"
2 1,000"
200
1.
500
fV
3,000
TIT <F-
2 100
5 wo
3 l.OOo"
300
Tirr-
1,000
T-fls-
4,000
Y<T-
3 100"
1 1.000"
4 1,000
Fig. 13.17.
The following examples show just how radical the transformation of
Sumerian cuneiform numerals was. The numbers shown relate to the booty
taken during Sargon II’s eighth campaign against Urartu (Armenia) in
714 BCE:
iW
60 7
>
67
TF-«( TM&
1 ME 30 1 ME 60
»
130 160
Fig. 13.18. See Thureau-Dangin, lines. 380, 366 and 369
m-t- fffr—
3 LIM 6 ME
^
3,600
As can be seen, 60 is now represented by six chevrons instead of the
vertical wedge that formerly had this numerical value, and numbers such as
130, 160 and 3,600 are given strictly decimal representations.
We can also see that by grounding their written numerals on their
spoken number-names, the Assyrians and Babylonians extended the
arithmetical scope of their numeral system whilst restricting its basic
figures to 100 and 1,000. All they needed to do was to combine these
symbols with the multiplication principle, to produce expressions of
the type 10,000 = 10 x 1,000, 40,000 = 40 x 1,000, 400,000 = 400 x 1,000,
and so on. So Sargon II’s scribe wrote out the number 305,412 in the
following manner:
TTTF-¥f-7F <jr
3 ME 5 LIM 4 ME 10 2
(3 X 100 + 5) x 1,000 + 4 x 100 + 10 +2
Fig. 13.19. See Thureau-Dangin, I. 394
RECONSTRUCTING THE DECIMAL ABACUS
The Akkadians must surely have possessed a calculating device, for they
could not otherwise have performed their complex arithmetical operations
save by the archaic device of calculi, of which barely a handful have been
found in archaeological levels of the second millennium BCE. Indeed, as we
also saw in Chapter 12, the Sumerians themselves must have had a kind of
abacus, which we reconstructed in its most probable form along with the
rules and procedures for its use. Furthermore, the Akkadians, at least in
the Babylonian period, had specific terms for referring not only to the
instrument and the tokens which went with it, but also to the operator of
the abacus.
In Ancient Babylonian (see Fig. 12. 6H above), the arithmetical “token”,
which must have been a stick of wood or a swatch of reed stems, was called
either
• is-si mi-nu-ti ("wood-for-counting"); or
• is-si nik-kas-si (“wood-for-accounts”).
As for the abacus itself, it was referred to by one of the two following
MESOPOTAMIAN NUMBERING AFTER THE ECl.IPSE OF SUMER
loan-words borrowed from the corresponding Sumerian terms (see Fig.
12.61 and 12. 6J above):
• gesdab-dim mu (“wooden-tablet-for-accounts”);
• su-me-ek-ku-u (literally, from the corresponding Sumerian word
GESSUMEGE, “wood (i.e. of the tablet), hand, rule, reed” or alterna-
tively “wood, sum, rule, reed”.
The abacus operator or abacist had two official names (see Fig. 12. 6K
and 12. 6L above):
• sa da-ab-di-mi (“the man for the tablet for accounts”);
• sa su-ma-ki-i (“the man for the abacus”).
Our knowledge of these terms comes from various bilingual tablets
dating from the beginning of the second millennium BCE, which provide a
kind of “Yellow Pages” in both Sumerian and Ancient Babylonian, each
entry consisting of a brief description of a representative of a profession
(“the man for . . .”), followed by the name of the tools associated with the
profession. (See Chapter 12 above, and for references to original sources,
see S. J. Liebermann, in AJA 84.)
In view of all this, we have to suppose that the Akkadians first used the
sexagesimal Sumerian abacus for as long as their arithmetic was dependent
on Sumerian notation, but had to construct sexagesimal-decimal conver-
sion tables for the requirements of their own decimal arithmetic during the
long “transitional period” that lasted until the end of the first Babylonian
dynasty, around the middle of the second millennium BCE. However, when
Akkadian culture itself came to hold sway in Mesopotamia, the situation
changed completely. The mathematical structure of the abacus had to be
radically altered to adapt it to the modified cuneiform notation that was
then used for strictly decimal arithmetic.
Indeed, the Assyro-Babylonian numeral system used base 10 and
allowed all numbers up to one million to be represented by combinations
of just these four signs:
T c b <h
1 10 100 (= ME) 1,000 (=UM)
Fig. 13.20.
For numbers above 1,000, the system used analytical combinations
of the given signs, that is to say it used the principle of multiplication to
designate 10,000, 100,000, and 1,000,000, as follows:
<f~
10.LIM ME.LIM LIM.LIM
(= 10x 1,000) (= 100x 1,000) (= 1,000x1,000)
Fig. 13.21.
140
As we showed for the Sumerian abacus in Chapter 12 above, we can here
show quite easily the most probable form of the Assyro-Babylonian abacus
as it was used by “ordinary” counters (there are good reasons for thinking
that there were two types of arithmeticians - the “ordinaries”, whose arith-
metic was exclusively decimal, and the “learned”, who continued to use the
sexagesimal system for mathematical and astronomical purposes). As for
the way the abacus was used, it must have been very similar to the rules for
the sexagesimal system, simply adapted to base 10:
10 6 10 5 10 J 10 3 10 2 10 1
Units 1
Fig. 13.22. Reconstructed Assyro-Babylonian decimal abacus
It should be noted that a brick marked with rows and columns as in
Fig. 13.22 was discovered in the 1970s by the French Archaeological
Delegation to Iran (DAFI) during the dig at the Acropolis of Susa, and that
a few similar pieces were found in the same area during the Second World
War. Up to now these objects have been taken as game-boards. We suggest
that they should rather be seen as arithmetical abaci. Let us hope that further
archaeological discoveries will provide suffficient evidence to confirm
141
THE LAST TRACES OF SUMERIAN ARITHMETIC
this interpretation. What we can be sure of, all the same, is that Susan
accountants (and the Elamites in general) also used arithmetical tools, of
which the first were of course the calculi. And there are very good reasons
for thinking that the tools they used were similar to those of the
Mesopotamians, for their operations were presumably just as complex as
those being carried out a few hundred miles away by their Sumerian and
Assyro-Babylonian counterparts.
THE LAST TRACES OF SUMERIAN ARITHMETIC
IN THE ASSYRO-BABYLONIAN DECIMAL SYSTEM
In the hands of the Semites, cuneiform numerals and Mesopotamian
arithmetic were gradually adapted and finally transformed into a system
with a different base working on quite different principles. All the same,
base 60 did not disappear entirely, and even continued to play a major role
as “big unit” in “ordinary” Mesopotamian accounting. Although it was
often represented (at least, from the start of the first millennium BCE) by
the decimal expression Assyrians and Babylonians alike continued to
represent the number 60 also by “spelling it out” inside numerical expres-
sions, using either the sign for the sound shu-shi (which is how “sixty” was
said in Semitic languages)
im-
1 shu-shi
or in abbreviated form as shu (the first syllable of the word for “sixty”)
1 shu
Above all, they went on figuring the numbers 70, 80, and 90 in the “old
manner”, that is to say in a way that carries the trace of the obsolete 60-
based arithmetic of the Sumerians (just as, nowadays, the French words
for 80 and 90 ( quatre-vingts , quatre-vingt-dix) carry the trace of a vanished
vigesimal arithmetic):
T <
T«
T«<
60 10
60 20
60 30
70
80 5
90 ^
Fig. 13.23.
The signs for the old base units of 600 and 3,600 never disappeared
entirely either. They continue to crop up in contracts and financial
statements, in auguries and in historical and commemorative texts. In these
later usages, the sign for 3,600 underwent a graphic development in line
with the evolution of Mesopotamian cuneiform writing:
CLASSICAL
SUMERIAN
❖
<>
<0
ASSYRIAN
Ancient
Middle
Late
<<
4
<*
A
&
4
BABYLONIAN
Ancient
Middle
Late
&
A
<T
<>
&
4
$
&
4c
*
Fig. 13.24. Evolution and stability of the Sumerian sign shar (=3,600)
For example, when Sargon II of Assyria inscribed the dimensions of the
walls of his fortress at Khorsabad - 16,280 cubits* - he had the figure
written not in what had by then become the standard notation:
<h
10 6 LIM
Fig. 13.25.
TT Y~ T «
2 ME 60 20
KUS
(cubits)
but in this arithmetically more archaic manner:
<*<*<*-<* R 1 R TMtff HFflft
3,600 . 3,600 . 3,600 . 3,600
. 600 . 600 . 600
.1 us
. 3 QA-NI
. 2 KUS
14,400 cubits
1
1,800 cubits
c. 60 cubits
3x6 cubits
2 cubits
Fig. 13.26. See Lyon, p. 10, 1. 65
But such traces of the old system were mere relics, and had no influence
at all on the strictly decimal arithmetic that the Assyro-Babylonians used
throughout their history for everyday reckoning.
* The cubit ( kus) is a measure of length of approx, 50 cm. Six cubits make a qanum, and sixty cubits
make an us.
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
recapitulation: FROM SUMERIAN to
ASSYRO-BABYLONIAN NUMBERING
There were, in brief, three main stages in Mesopotamian culture after the
establishment of the Akkadian Empire:
• in the first, the Semites assimilated the cultural heritage of their
Sumerian predecessors in the region;
• the second is an intermediate period;
• the third is the period of Semitic predominance in Mesopotamian
culture.
SUMERIAN
SYSTEM
(base 60 with
10 and 6 as
auxiliary bases)
SUMERIAN -AKKAD IAN
SYNTHESIS
(compromise between
base 10 and base 60)
ORDINARY
ASSYRO-
BABYLONIAN
SYSTEM
(Strictly decimal base)
1
T
r
T
10
<
<
<
60
r
r TJ&- or Tg
1 SU-SI 1 SU
70
i<
60 10
T<
T<
80
T«
60 20
T«
T«
90
X4K
60 30
J<«
T<«
100
T#
T# or T T-
r t-
60 40
1 ME
1 ME
120
rr
TTJ^r- 0^ TV«
60 60
2 SU-SI 1 ME 20
1 ME 20
600
*
Tc If-
6 ME
TFT*-
6 ME
1,000
or T<T*-
T <T*-
600 360 40
1 LI-MI 1 LIM
1 LIM
3,600
TTT fff V
3 LIM 6 ME
TTT<r»“ffrr-
3 LIM 6 ME
Fig. 13.27. Evolution of popular Mesopotamian numerals before and after the eclipse of
Sumerian civilisation (see also Fig. 18.9 below)
142
As far as numbers and arithmetic are concerned, these periods
correspond respectively to: pure and simple borrowing of Sumerian sexa-
gesimal numbering; the emergence of a mixed system using a combination
of decimal and sexagesimal signs; and the development of a strictly decimal
system. This profound transformation of cuneiform numbers occurred
under the pressure of oral number-names, whose strictly decimal structure
is a common feature of all Semitic languages (see Fig. 13.7 and 13.19
above). But this is not where the development came to a full stop: as we
shall see, the scribes of the city of Mari evolved their own unique version of
a decimal numeral system.
THE ANCIENT S Y R O - M E S O P O T A M I A N
CITY OF MARI
Various texts refer to the Sumero-Semitic city of Mari as an important place
in the Mesopotamian world, but it was not until 1933 that Andre Parrot,
led on by the suggestions of W. F. Albright and by the chance discovery of
a statue, began to excavate at Tel-Hariri, on the border of Syria and Iraq.
Over the following forty years, Parrot’s team conducted a score of excava-
tions and laid bare a whole civilisation.
The earliest traces of habitation at Mari date from the fourth millennium
BCE, and by the first half of the third millennium it was already highly
urbanised, with a ziggurat and a number of temples decorated with
statuary and painted walls. The art and culture of Mari in this period
resemble those of Sumer, but the facial types represented, as well as the
names and the gods mentioned, are Semitic.
Mari became part of the Akkadian Empire, but regained some
independence around the twenty-second century BCE. From the twentieth
to the eighteenth century BCE Mari flourished as an independent and
expanding city-state, but it was defeated and destroyed by Hammurabi
around 1755 BCE. Though it continued to exist as a town, Mari never again
regained any power or influence.
It was in the early eighteenth century BCE, under Zimri-Lim, that
Mari built its most remarkable structures, including a 300-room palace
occupying a ground area of 200 m x 120 m and in part of which were stored
more than 20,000 cuneiform tablets, giving us a unique insight into the
political, administrative, diplomatic, economic, and juridical affairs of a
Mesopotamian state. The tablets include long lists of the palace’s require-
ments (food, drink, etc.), and many letters written by women, which
suggests that they played an important role in the life of the city.
14 3
THE MARI SYSTEM
WHAT IS THE RULE OF POSITION?
Just as an alphabet allows all the words of a language to be written by
different arrangements of a very limited set of signs, so our current
numerals allow us to represent all the integers by different arrangements
of a set of only ten different signs. From an intellectual point of view, this
system is therefore far superior to most numerical systems of the ancient
world. However, that superiority does not derive from the use of base 10,
since bases such as 2, 8, 12, 20, or 60 can produce the same advantages
and be used in exactly the same way as our current decimal positional
system. As we have already seen, moreover, 10 is by far the most wide-
spread numerical base in virtue not of any mathematical properties, but of
a particularity of human physiology.
What makes our written numeral system ingenious and superior to others
is the principle that the value of a sign depends on the position it occupies in a string
of signs. Any given numeral is associated with units, tens, hundreds, or
thousands depending on whether it occupies the first, second, third, or
fourth place in a numerical expression (counting the places from right to left).
These reminders allow us to understand fully the numbering system of
Mari and of the learned men of Babylon . . .
THE MARI SYSTEM
It has recently come to light that the scribes of Mari used, alongside
“classical” Mesopotamian number-notation, a system of numerals quite
different to all that had preceded it.
As in previous systems, the first nine units were represented by an
equivalent number of vertical wedges:
r
TT I T T f ?
f f
1
2 3 4 5 6 7
8 9
Fig. 13.28.
Similarly, the representation of the tens was in line with previous
traditions, since it was based on the use of an equivalent number of
chevrons. However, unlike the Assyro-Babylonians, the scribes of Mari did
not use the old sexagesimal character for 60, but carried on multiplying
chevrons for the numbers 60, 70, 80, and 90:
<
4&
A
4
4
4 -
4
4 -
10
20
30
40
50
60
70
80
90
For 100, they did not use the old system of a wedge plus the sign for the
word for 100 (ME), with the meaning 1 x “hundred”: what they used was
just the single vertical wedge. The number 200 was figured by two vertical
wedges, 300 by three, and so on.
NOTATIONS OF THE HUNDREDS BY THE SCRIBES OF MARI
Fig. 13.30.
So a wedge represented either a unit or a hundred depending on where
it came in the numerical expression.
For instance, to write “120", “130,” and so on, the scribes of Mari put
down 1 vertical wedge followed by 2, 3, etc. chevrons. And to represent a
number such as 698, all that was needed was a representation of 6 followed
by a representation of 98 (9 chevrons and 8 wedges):
u
U; 101
R
[l; 20]
11; 301
14 f
16; 98]
= 1 x 100 + 10
= 110
= 1 x 100 + 20
= 120
= 1 x 100 + 30
= 130
= 6 x 100 + 98
= 698
Fig. 13.31.
It is clear that the scribes of Mari knew both the classical Mesopotamian
decimal notation and also the positional sexagesimal system of the scholars
(see below). When they drew up their tablets in Akkadian (a language
which they handled with ease), they used the former for “current business”
such as economic and legal documents, and the latter for “scientific”
matters (tables, mathematical problems, and so on). In fact, the system
we are now considering never was the official numeral-system of
the city: for it is only found in quite particular places on the tablets
Fig. 13.29.
MESOPOTAMIAN NUMBERING AETER THE ECLIPSE OF SUMER
(on the edges, on the reverse side, and in the margins) and, in most cases,
the numbers worked out in the new system were written out again in one
or the other of the two standard systems.
In other words, the new system seems to have served only as an
aide-memoire and checking device, to make doubly sure that the results
written out in the traditional way were in fact correct. What we see is a kind
of mathematical bilingualism, in which matching results reached by two
separate notations resolve doubts about the correctness of the sums. And it
is of course only because of the role that the system played, and because of
the position of the new-style numerals on the tablets, that modern scholars
have been able to read and interpret them.
The following examples come from the Royal Archives of Mari, as
quoted, translated and decoded by D. Soubeyran. The first gives the last
column of a tally of people, showing the totals for rows identified by the
words in brackets, which refer to the categories of people counted:
7 <
70
(lii-mes)
m
79
(mi-mes)
i
9
(tur-mes)
6
(mi-tur-mes)
T
1
(tur-gab)
These numbers are written in classical Akkadian manner, so they
represent: 70 + 79 + 9 + 6 + 1 = 165. However, after a space and before the
title of the tablet comes the following expression:
r 4 w
Fig. 13.33A. 1 65
If this were a non-positional expression, its value would either be 1 + 65
= 66, or (allowing the vertical wedge to mean 60, as it often did in Akkadian
arithmetic) 60 + 65 = 125. In neither case could it be a running total for the
column which it follows after a space. However, if the wedge is given
the value 100, then we do indeed get the running total of 165, in the
following manner:
[ 1 ; 65 ]
Fig. 13 - 33 B.
= 1 x 100 + 65 = 165
144
The second example is also a list of people, perhaps of nobles. Each entry
is accompanied by a number, which perhaps indicates the number of
servants owned. There is a running total of 183 brought forward, which is
written in classical decimal form as
1 ME-AT 83 (“1 hundred 83”)
A second subtotal gives the figure of 26 servants, and, as you might
expect, the grand total comes to 209, which is expressed in the same way as:
2 ME-TIM 9 (2 x 100 + 9 = 209)
However, the side-edge of the tablet has the following expression:
Fig. 13.34A. 1 85
If this were taken in the classical (non-positional) way, then it would
mean either 1 + 85 (= 86) or 60 + 85 (= 145), and the totals would not
match at all.
However, if the figures on the edge are taken as a centesimal positional
expression, then the sum is 185, which is roughly the same as the first count
of servants, 183.
Fig. 13 . 34 s. [1; 85] = 1 x 100 + 85 = 185
The last of the three tablets details a sequence of deliveries of copper
scythes, with a running total, written in the standard way, of 471 scythes.
But between the markings for the month and the year, there is this:
ff 41
Fig. 13.35A. 4 76
Once again, this expression would not have much meaning if it were read
in the classical manner, but, taking it as an expression in the positional
system of the scribes of Mari, it would give 476, a good approximation of
the previous running total (471):
f 41
Fig. I3-35B. [4; 76] =4x 100 + 76 = 476
According to Soubeyran, the minor discrepancies between these figures
and the totals, as well as their position on the tablets and the rough and
145
THE MARI SYSTEM
ready way they are written, shows that they are rough drafts or workings-
out, intended to check figures before they were inscribed on the tablets in a
formal way. That makes it all the more interesting to see the scribes of Mari
thinking in a positional, centesimal-decimal system, before converting their
results into sexagesimal notation.
For numbers between 100 and 1,000, the Mari system used the rule of
position, and its base was not 10, but 100: the first “large” unit was the
hundred, with the ten playing the role of auxiliary base. On the other hand,
the system did not have a zero. If it had had such a thing, then it would have
served to mark the absence of units in a given order. In other words, if there
had been a zero in the Mari system, then the multiples of the base would
have been written in the same way as we write multiples of our base (20, 30,
40, etc.), with a zero indicating the absence of units of the first order.
All the same, the scribes of Mari were perfectly aware that the value of
the numerals they wrote down depended on their position in a specific
numerical expression. This is all the more noteworthy because very few
civilisations have ever reached such a degree of simplification in written
numerals, and by the same token discovered the rule of position. This
development took place very early on: the tablets that bear the trace of the
rule of position are not later than the eighteenth century BCE.
However, the system was not strictly or consistently positional. Had that
been the case, then 1,000 (= 10 x 100, or ten units of the second centesimal
order) would have been represented by a chevron, 2,000 by two chevrons,
and so on. As for 10,000, the square of the base of the second centesimal
order, it would have been represented by a vertical wedge (had there been a
zero, it would have been figured in the form [1; 0; 0], the first zero signify-
ing the absence of any units of the first order (numbers between 1 and 99),
the second the absence of units of the second order (multiples of 100 by
a number between 1 and 99)). And since 200 = 2 x 100, represented by
two vertical wedges, so 20,000 = 2 x 10,000 would similarly have been
represented by two vertical wedges.
But it was not so: the Mari system had special signs for 1,000 and for
10.000. However, the “Mari thousand” was rather different from the classi-
cal numeral, and it was combined with a multiplier to make numbers like
2.000, 3,000, etc.:
1£- IT
LI-IM 2 LI-IM
Fig. 13 . 36 . 1,000 2,000
This adds up to a mixed system, using simultaneously all the basic rules,
of addition (for the total), of multiplication (for the thousands), and of
position (for numbers less than 1,000).
The Mari scribes used a figure derived from the thousand overlaid with
a chevron (= 10) to represent 10,000 (which was then combined with units
for multiples of 10,000):
£=f~ * <
Fig. 13 . 37 . 10,000 = 1,000 x 10
This is the only example amongst the decimal numerations of the whole
Mesopotamian region where 10,000 is not written as an analytical combi-
nation of the numerals 10 and 1,000, and it is yet another way in which the
Mari system is quite unique.
The Marian cuneiform sign for 10,000 (found not just in economic
tablets, but in fields as diverse as tallies of bricks, of land areas, and of
livestock) is related to the Sumerian ideogram GAL, which meant “large”,
and was pronounced ribbatum in the language of Mari, with the literal
meaning of “multitude”, whence “large number”. So that was the name of
the number 10,000, and it is clearly the same name as the one found at Ebla
( ri-bab ) in the twenty-fourth century BCE, at Ugarit (r(b)bt) in the fifteenth
century BCE, and then in Syria ( ri-ib-ba-at ), and in Hebrew (ribo, pi. ribot).
The following two examples from tablets found at Mari give a fuller view
of how the system worked:
In etymological and graphological terms, 10,000 was the “biggest
number” in the system of Mari. (The scribes could of course represent
far larger numbers by using the multiplication principle, even if no really
large numbers have yet been found in the tablets.) It was a quite unique
centesimal numeral system, found exclusively in this one city on the
common border of Syria and Mesopotamia, at the time of the patriarch
Abraham. It might have developed into a fully positional system had the
Babylonian king Hammurabi not razed the city to the ground in 1755 BCE,
and buried with it a very large part of Mari’s culture. Ironically, it was
the Babylonians themselves who actually devised the world’s first true
positional system - but it was neither a variant of Akkadian decimal arith-
metic, nor a centesimal system like that of Mari. Used for mathematical and
IT 0 -
2 LI-IM [7; 37]
= 2 x 1,000 + (7 x 100 + 37)
= 2,737
Fig. 13 . 39 . See Soubeyran (1984)
Fig. 13 . 38 . See Durand (1987)
MESOPOTAMIAN NUMBERING AFTER THE E C 1. 1 P S E OF SUMER
astronomical reckonings right down to the dawn of the Common Era, the
“learned” numerals of Babylon were a direct inheritance of Sumer, whose
memory they have perpetuated, directly and indirectly, right down to the
present day.
THE POSITIONAL SEXAGESIMAL SYSTEM OF
THE LEARNED MEN OF MESOPOTAMIA
Although we cannot be sure about the exact date, the first real idea of
a positional numeral system arose amongst the mathematicians and
astronomers of Babylon in or around the nineteenth century BCE.
The Mesopotamian scholars’ abstract numerals were derived from the
ancient Sumerians’ sexagesimal figures, but constituted a system far supe-
rior to anything else in the ancient world, anticipating modern notation in
all respects save for the different base and the actual shapes used for the
numerals.
Unlike the “ordinary” Assyro-Babylonian notation used for everyday
business needs, the learned system used base 60 and was strictly positional.
Thus a group of figures such as
[ 3 ; 1 ; 2 ]
which in modern decimal positional notation would express:
3 x 10 2 + 1 x 10 + 2
signified to Babylonian mathematicians and astronomers:
3 x 60 2 + 1 x 60 + 2
Similarly, the sequence [1; 1; 1; 1] which in our system would mean 1 x 10 3
+ 1 x 10 2 + 1 x 10 + 1 (or 1,000 + 100 + 10 + 1) signified in the Babylonian
system 1 x 60 3 + 1 x 60 2 + 1 x 60 + 1 (or 216,000 + 3,600 + 60 +1).
Instances of this system of numerals have been known since the very
dawn of Assyriology, in the mid-nineteenth century, and, thanks to exca-
vations made throughout Mesopotamia and Iraq at that time, many
examples have come to rest in the great European museums (Louvre, British
Museum, Berlin) and in the university collections at Yale, Columbia,
Pennsylvania, etc. The types of document in which the learned system is
used (and which come from Elam and Mari, as well as from Nineveh, Larsa,
and other Mesopotamian cities) are for the most part as follows: tables
intended to assist numerical calculation (e.g. multiplication tables, division
tables, reciprocals, squares, square roots, cubes, cube roots, etc.); astro-
nomical tables; collections of practical arithmetical and elementary
geometrical exercises; lists of more or less complex mathematical problems.
146
The system is sexagesimal, which is to say that 60 units of one order of
magnitude constitute one unit of the next (higher) order of magnitude. The
numbers 1 to 59 constitute the units of the first order, multiples of 60
constitute the second order, multiples of 3,600 (sixty sixties) constitute the
third order, multiples of 216,000 (the cube of 60) constitute the fourth
order, and so on.
In fact, there were really only two signs in the system: a vertical wedge
representing a unit, and a chevron representing 10:
T <
1 10
Numbers from 1 to 59 inclusive were built on the principle of addition,
by an appropriate number of repetitions of the two signs. Thus the
numbers 19 and 58 were written
A ff ”
(1 chevron + 9 wedges) (5 chevrons + 8 wedges)
So far the system is exactly the same as its predecessors. However,
beyond 60, the learned system became strictly positional. The number 69,
for instance, was not written
MW but t fpf
60 9 [1; 9]
For example, this is how Asarhaddon, king of Assyria from 680 to 669
BCE, justified his decision to rebuild Babylon (wrecked by his father
Sennacherib in 689 BCE) rather sooner than the holy writ prescribed:
After inscribing the number 70 for the years of Babylon’s desertion
on the Tablet of Fate, the God Marduk, in his pity, changed his mind.
He turned the figures round and thus resolved that the city would
be reoccupied after only eleven years. [From The Black Stone, trans.
J. Nougayrol]
The anecdote takes on its full meaning only in the light of Babylonian
sexagesimal numbering. To begin with, Marduk, chief amongst the gods in
the Babylonian pantheon, decides that the city will remain uninhabited
for 70 years, and, to give full force to his decision, inscribes on the Tablet of
Fate the signs:
T <
Fig. 13.40A. [1; 10] ([1; 10] = 1 x 60 + 10)
Thereafter, feeling compassion for the Babylonians, Marduk inverts the
order of the signs in the expression, thus:
147
THE POSITIONAL SEXAGESIMAL SYSTEM
< T
Fig. 13.40B. 10 . 1 (=10 + 1)
T <¥
H ; 15]
(= 1 X 60 + 15 = 75)
Fig. 13.42.
Since the new expression represents the number 11, Marduk decreed that
the city would remain uninhabited only for that length of time, and could be
rebuilt thereafter. The anecdote shows that the Mesopotamian public in
general was at least aware of the rule of position as applied to base 60.
In the Babylonian system, therefore, the value of a sign varied according
to its position in a numerical expression. The figure for 1 could for instance
express
• a unit in first position from the right,
• a sixty in the second position,
• sixty sixties or 60 2 in third position,
and so on.
Fig. 13.41. Representations of the fifty-nine significant units of the learned Mesopotamian
numeral system
For instance, to write the number 75 (one sixty and fifteen units) you put
a “15” in first position and a “1” in second position, thus:
And to write 1,000 (16 sixties and 40 units) you put a “40” in first
position and a “16” in second position, thus:
[16 ; 40]
(= 16 x 60 + 40 = 1,000)
Fig. 13 . 43 -
Conversely, an expression such as
<TT
[48 ; 20 ; 12 ] Fig. 13 . 44 .
expresses the number:
48 x 60 2 + 20 x 60 + 12 = 48 x 3,600 + 20 x 60 + 12 = 174,012
in exactly the same way as we would express “174,012 seconds” as:
48 h 20m 12s
Similarly, an expression such as
[1 ; 50 + 7 ; 30 + 6 ; 10 + 5] or [1 ; 57 ; 36 ; 15]
Fig. 13 . 45 .
symbolises, in the minds of the Babylonian scholars, the number:
1 x 60 3 + 57 x 60 2 + 36 x 60 + 15 (= 423,375)
The next examples come from one of the most ancient Babylonian
mathematical tablets known (British Museum, BM 13901, dating from
the period of the first kings of the Babylonian Dynasty), a collection of
problems relating the solution of the equation of the second degree:
[17 ; 46 ; 40] [1 ; 57 ; 46 ; 40]
(= 17 x 60 2 + 46 X 60 + 40) (= 1 x 60 3 + 57 x 60 2 + 46 x 60 + 40)
> >
64,000 424,000
Fig. 13.46. Fig. 13.47.
The difference between Sumerian numbers and the Babylonian “learned”
system was simply this: the Sumerians relied on addition, the Babylonians
on the rule of position. This can easily be seen by comparing the Sumerian
and Babylonian expressions for the two numbers 1,859 and 4,818:
MESOPOTAMIAN NUMBERING AFTER T II E ECLIPSE OF SUM E R
SUMERIAN SYSTEM BABYLONIAN SYSTEM
600 + 600 + 600 + 50 + 9 [30 ; 59]
>
(= 30 x 60 + 59)
3,600 + 600 + 600+ 18 [1; 20; 18]
->
(= 1 x 60 2 + 20 x 60 + 18)
Fig. 13. 48a. Fig 13.48B.
THE TRANSITION FROM SUMERIAN TO
LEARNED BABYLONIAN NUMERALS
One of the reasons for the “invention" of the learned Babylonian system
is easy to understand - it was the “accident” which gave 1 and 60 the
same written sign in Sumerian, and which originally constituted the main
difficulty of using Sumerian numerals for arithmetical operations.
Moreover, the path to the discovery of positionality had been laid out in
the very earliest traces of Sumerian civilisation. The two basic units were
represented, first of all, by the same name, ges (see Fig. 8.5A and 8.5B
above); then, in the second half of the fourth millennium BCE, they were
represented by objects of the same shape (the small and large cone) (see
Fig. 10.4 above); then, from 3200-3100 BCE to the end of the third millen-
nium, by two figures of the same general shape, the narrow notch and the
thick notch (see Fig. 8.9 above); then, from around the twenty-seventh
century BCE, by cuneiform marks of the same type, distinguished only by
their respective sizes; and, finally, from the third dynasty of Ur onwards
(twenty-second to twentieth century BCE), especially in the writings of
Akkadian scribes, by the same vertical wedge.
In other words, as we can see from Asarhaddon’s story in The Black Stone,
and in the Assyro-Babylonian representations of the numbers 70, 80 and 90
(see Fig. 13. 23 above), the large wedge meaning 60 had evolved in line with
the general evolution of cuneiform writing so as to be indistinguishable
from the small wedge meaning 1.
In everyday usage, that evolution was seen as a problem, which was got
round by “spelling out” 60 as shu-shi in numbers such as 61, 62, 63, where
the confusion was potentially greatest (see Fig. 13.14 above), and eventually
by replacing the sexagesimal unit with a multiple of a decimal one (Fig.
13.18 above).
148
But in the usage of the learned men of Mesopotamia, the graphical
equivalence of the signs for 1 and 60 gave rise (at least for numbers
with two orders of magnitude) to a true rule of position. As the following
notations show:
Fig. 13.49.
Babylonian scholars realised therefore that the rule or principle could
be generalised to represent all integers, provided that the old Sumerian
signs for the multiples and powers of 60 were abandoned. The first to
go was the 600 (= 60 x 10), for which was substituted as many chevrons
(= 10) as there were 60s in the number represented. Then the sign for 3,600
(the square of 60) was dropped, and, since this number was a unit of the
third sexagesimal order, it was henceforth represented by a single vertical
wedge. Subsequently the sign for 36,000 was eliminated, and replaced
by the sign for 10 in the position reserved for the third sexagesimal order,
and so on.
For instance, instead of representing the number 1,859 by three signs for
600 followed by the notation of the number 59 (1,859 = 3 x 600 + 59),
Babylonian scholars now used [30; 59] (= 30 x 60 + 59), as shown in
Fig. 13.48 above, which also gives the example of the “old” and “new”
representations of 4,818.
The vertical wedge thus came to represent not only the unit, but any and
all powers of 60. In other words, 1 was henceforth figured by the same
wedge that signified 60, 3,600, 216,000, and so on, and all 10-multiples of
the base (600, 36,000, 2,160,000, etc.) by the chevron.
The discovery was extremely fruitful in itself, but, because of the very
circumstances in which it arose, it gave rise to many difficulties.
THE DIFFICULTIES OF
THE BABYLONIAN SYSTEM
Despite their strictly positional nature and their sexagesimal base, learned
Babylonian numerals remained decimal and additive within each order of
magnitude. This naturally created many ambiguous expressions and was
thus the source of many errors. For example, in a mathematical text from
Susa, a number [10; 15] (that is to say, 10 x 60 + 15, or 615) is written thus:
[10 ; 15]
Fig. 13.50A.
However, this expression could also just as easily be read as
«YT
[25] or [10; 10; 5] (= 10 x 60 2 + 10 x 60 + 5)
* •>
Fig. 13-sob.
It is rather as if the Romans had adopted the rule of position and base
60, and had then represented expressions such as “10° 3' l"" (= 36,181")
by the Roman numerals X III I, which they could easily have confused
with XI II I (11° 2' 1"), X I III (10° 1' 3") , and so on. Scribes in Babylon
and Susa were well aware of the problem and tried to avoid it by leaving
a clear space between one sexagesimal order and the next. So in the same
text as the one from which Fig. 13.50 is transcribed, we find the number
[10; 10] (= 10 x 60 + 10), represented as:
< <
[10 ; 10 ]
>
Fig. 13.51.
The clear separation of the two chevrons eliminates any ambiguity with
the representation of the number 20.
In another tablet from Susa the number [1; 1; 12] ( = 1 x 60 2 + 1 x 60 +
12) is written
r r<rr
[1:1: 12]
^
Fig. 13.52A.
in which the clear separation of the leftmost wedge serves to distinguish the
expression from
THE DIFFICULTIES OF THE BABYLONIAN SYSTEM
TMT
[2 ; 12] (=2x60 + 12)
>
Fig. 13.52B.
In some instances scribes used special signs to mark the separation of the
orders of magnitude. We find double oblique wedges, or twin chevrons one
on top of the other, fulfilling this role of “order separator”*:
^ or or or
Fig. 13-53-
Here are some examples from a mathematical tablet excavated at Susa:
[1 ; 10 ; [ 18 ; 45] (= 1 x 60 3 + 10 x 60 2 + 18 x 60 + 45)
Separation sign
^
Fig. 13.54A.
120 ; j 3 ; 13 ; 21 ; 331
Separation sign
*
(= 20 x 60“ + 3 x 60 3 + 13 x 60 2 + 21 x 60 + 33)
Fig. 13.54B.
The sign of separation makes the first number above quite distinct from
the representation of [1; 10 + 18; 45] (= 1 x 60 2 + 28 x 60 + 45); and for the
same reason the second number above cannot be mistaken for [20 + 3; 13;
21; 33] (= 23 x 60 3 + 13 x 60 2 + 21 x 60 + 33).
This difficulty actually masked a much more serious deficiency of the
system - the absence of a zero. For more than fifteen centuries, Babylonian
mathematicians and astronomers worked without a concept of or sign for
zero, and that must have hampered them a great deal.
In any numeral system using the rule of position, there comes a point
where a special sign is needed to represent units that are missing from the
number to be represented. For instance, in order to write the number ten
using (as we now do) a decimal positional notation, it is easy enough to
place the sign for 1 in second position, so as to make it signify one unit of
the higher (decimal) order - but how do we signify that this sign is indeed
* In commentaries on literary texts, the same sign was used to separate head words from their explications;
in multilingual texts, the sign was used to mark the switch from one language to another; and in lists of
prophecies, the sign was used to separate formulae and to mark the start of an utterance.
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SDMF.H
150
in second position if we have nothing to write down to mean that there is
nothing in the first position? Twelve is easy - you put “1” in second position,
and “2” in first position, itself the guarantee that the “1” is indeed in second
position. But if all you have for ten is a “1” and then nothing . . . The
1
2
3
4
5
6
7
e
9
10
11
12
13
14
15
16
17
IB
19
20
21
22
23
24
25
26
Fig. 13.55. Important mathematical text from Larsa (Senkereh), dating from the period of the
First Babylonian Dynasty (Louvre, AO 8862, side IV). See Neugebauer, tablet 38. Beneath line 16,
note the representation of the number 18,144,220 as [1; 24; blank space; 3; 40].
problem is obviously acute. Similarly, to write a number like “seven
hundred and two” in a decimal positional system, you can easily put a “7”
in third position and a “2” in first position, but it’s not easy to tell that
there’s an arithmetical “nothing” between them if there is indeed no thing to
put between them.
It became clear in the long run that such a nothing had to be represented
by something if confusion in numerical calculation was to be avoided. The
something that means nothing, or rather the sign that signifies the absence
of units in a given order of magnitude, is, or would one day be represented
by, zero.
The learned men of Babylon had no concept of zero around 1200 BCE.
The proof can be seen on a tablet from Uruk (Louvre AO 17264) which gives
the following solution:
“Calculate the
square of TT«^ and you get
In decimal numbers using the rule of position, the first of these
expressions (2 x 60 + 27) is equal to 147, and the square of 147 is
21,609. This latter number can be expressed in sexagesimal arithmetic as
6 x 3,600 + 0 x 60 + 9, and should therefore be written in learned
Babylonian cuneiform numbers with a "9” in first position, a “6” in third
position, and “nothing” in second position. If the scribe had had a concept
of zero he would surely have avoided writing the square of [2; 27] as the
expression [6; 9] which we see on the tablet - since the simplest resolution
of [6; 9] is 6 x 60 + 9 = 369, which is not the square of 147 at all!
Another example of the same kind can be found on a Babylonian
mathematical tablet from around 1700 BCE (Berlin Archaeological
Museum, VAT 8528), where the numbers [2; 0; 20] (= 2 x 60 2 + 0 x 60 + 20
= 7,220) and [1; 0; 10] (= 1 x 60 2 + 0 x 60 + 10 = 3,610) are represented by
TT* V
2 ; 20 1 ; 10
Fig. 13.56.
These notations are manifestly ambiguous, since they could represent,
respectively, [2; 20] (= 2 x 60 + 20 = 140) and [1; 10] (= 1 x 60 + 10 = 70).
To overcome this difficulty, Babylonian scribes sometimes left a blank
space in the position where there was no unit of a given order of magnitude.
Here are some examples from tablets excavated at Susa (examples A, B, C)
and from Fig. 13.58 below (example D, line 15). Our interpretations are not
speculative, since the values given correspond to mathematical relations
that are unambiguous in context:
151
THE DIFFICULTIES OF THE BABYLONIAN SYSTEM
T |
[1 ; * ; 25]
no units of the
second order
Fig. 13 - 57 A.
T
[1 ; 0 ; 35]
>
Fig. 13 . 57 B.
r A
11 ; 0 ; 40]
>
Fig. 13.57c.
T «W W4f-w
11 ; 27 ; 0 ; 3 ; 45] (= 1 x 60 4 + 27 x 60 3 + 0 x 60 2 + 3 x 60 + 45)
>
Fig. 13.57D.
However, this did not solve the problem entirely. For a start, scribes
often made mistakes or did not bother to leave the space. Secondly, the
device did not allow a distinction to be made between the absence of units
in one order of magnitude, and the absence of units in two or more orders
of magnitude, since two spaces look much the same as one space. And
finally, since the figure for 4, for instance, could mean 4 x 60, 4 x 60 2 ,
4 x 60 3 , or 4 x 60 4 , how could you know which order of magnitude was
meant by a single expression?
These difficulties were compounded by fractions. Whereas their
predecessors had given each fraction a specific sign (see Fig. 10.32 above
for an example from Elam), the Babylonians used the rule of position for
fractions whose denominator was a power of 60. In other words, positional
sexagesimal notation was extended to what we would now call the negative
powers of 60 (60- 1 = 1/60, 60' 2 = 1/60 2 = 1/3,600, 6 O - 3 = 1/60 3 =
1/216,000, etc.). So the vertical wedge came to signify not just 1, 60, 60 2 ,
etc., but also 1/60, 1/3,600, and so on. Two wedges could mean 2 or 120 or
1/30 or 1/1,800; the figure signifying 15 could also signify 1/4 (= 15/60),
and the number 30 might just as easily mean 1/2.
Numerals were written from right to left in ascending order of the
powers of 60, and from left to right in ascending negative powers of 60,
exactly as we now do with our decimal positional numbering - except that
in Babylon there was nothing equivalent to the decimal point that we now
use to separate the integer from the fraction.
(= 1 x 60 2 + 0 x 60 + 25)
(= 1 x 60 2 + 0 x 60 + 35)
(= 1 x 60 2 + 0 x 60 + 40)
1
2
3
4
5
e
7
8
9
10
11
12
13
14
15
18
17
TRANSCRIPTION
LINE 1
2
i
i
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SI-LMP-TIM
^^^^^A*AS-SA-yU«U*(M]A SAG
{b-sa sag
-i ...-U
Ib-SA SI-LMP-TIM
MU -B
-IM
3
4
1 ; 59
2.49
ki
Ki
1
2
6
, 15 •• 33 , 45
56, 7
1,16,41
1 .50,49
KI
3
6
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.29 . 32 . 52 . 16
3.31,49
5 9. 1
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7
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48 . 54 . 1 .40
1 1 l 4
1 i *
- i.V":
8
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I 47 ; ^ j 41 /40
5 .19
K<
9
l'
;43. 11 ,58, 20 . 26.40
38 .11
59 ; 1 _
KL
M
10
ii
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13 . 19
20 . 49
KI
11
E!
: 38 : 33 ,36 .36
9 . 1
12 ; 49
KI
9
12
o
;35 : 10. 2.28.27 ,24.26.40
1 /22 . 41
2 18. 1
KI
10
13
!i
33 , 45
45
1 .15
KI
ii 5
14
n
;29 . 21 , 54,. 2.15
27 . 59
46 . 49
KI
12 •
15
0
: 27 . * •. 3, 45
7 *'12 ; 1
4; 49
KI
16
l 1 .
i 91 * 39 ; 6 ; AO 29 , 31 53 , 49
KI
JMt
17
11
\
* Blank space indicating the absence of units in a given order of magnitude
Fig. 13.58. Mathematical tablet, 1800-1700 BCE, showing that Babylonian mathematicians
were already aware of the properties of right-angled triangles (Pythagoras’ theorem). If we take the
numbers in the leftmost column A, the second column B, and the third column C, we find that the
numbers obey the relationship
A= — ■ B = b; C = c, and a 2 = b 2 + c 2
<r
This expresses the relationship by which in a right-angled triangle (with sides b and c and
hypotenuse a) the square of the hypotenuse is equal to the sum of the squares on the other two sides.
Columbia University, Plimpton 322. Author’s own transcription
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
Naturally enough, this led to enormous difficulties, such as are suggested
by the following three interpretations (out of many others possible) of a
single expression:
Notation:
[25 ; 38]
interpretation 1 interpretation 2 interpretation 3
25 x 60 + 38 25 + | | + ^- Q
Fig. 13.59.
All the same, Babylonian mathematicians and astronomers managed to
perform quite sophisticated operations for over a thousand years despite
the imperfections of their numeral system. Of course, they had the orders
of magnitude present in their minds, and the ambiguities of the notation
were resolved by the context (that is to say, the premises of the problem
being tackled) or by the commentary of the teacher, who must presumably
have indicated the magnitudes involved.
152
THE BIRTH OF THE BABYLONIAN ZERO
At some point, probably prior to the arrival of the Seleucid Turks in 311
BCE, Babylonian astronomers and mathematicians devised a true zero, to
indicate the absence of units of a given order of magnitude. They began to
use, instead of a blank space, an actual sign wherever there was a missing
order of the powers of 60, and the sign they used was a variant of the old
“separator” sign discussed above (see Fig. 13.53):
^ 01 ^
Fig. 13 . 61 .
So, in an astronomical tablet from Uruk (now in the Louvre, AO 6456)
from the Seleucid period, we can read:
[2; 0; 25; 38; 4]
(= 2 x 60 4 + 0 x 60 3 + 25 x 60 2 + 38 x 60 + 4)
written on the back of the tablet in the form:
IT ^ «<^ 'V
[2 ; 0 ; 25 ; 38 ; 4]
^
Fig. 13 . 62 .
The diagonal double wedge thus marks the absence of any sexagesimal
units of the fourth order of magnitude.
On lines 10, 14 and 24 of the tablet reproduced in Fig. 13.60 above, we
can read:
Yf ^46CTTT«
[2 : 0 ; 0 ; 33 ; 20] (= 20 x 60 4 + 0 x 60 3 + 0 x 60 2 + 33 x 60 + 20)
^
Fig. 13 . 63 A.
[1 ; 0 ; 45] (= 1 x 60 2 + 0 x 60 + 45)
Fig. 13 . 63 B.
[1;0 ; 0 ; 16 ; 40] (= 1 x 60 4 + 0 x 60 3 + 0 x 60 2 + 16 x 60 + 40)
MM 11 11 11" '
'HWf it in kM
I TMf ■
Fig. 13 . 60 . Mathematical
tablet from Uruk, late third or
early second century BCE,
containing one of the earliest
known instances of the Babylonian
zero. Louvre, AO 6484 side B.
See Thureau-Dangin, tablet 33,
side B, plate LXII.
Fig. 13 . 63 c.
153
y ^ ^
[1 ; 0 ; 7 ; 30] (= 1 x 60 3 + 0 x 60 2 + 7 x 60 + 30)
*
Fig. 13.63D.
The Babylonian mathematical documents that have been published to
date show the zero only in median positions. For that reason, some
historians of science have inferred that Mesopotamian scholars only ever
used their zero in intermediate positions and that it would therefore be
unwise to treat their zero as identical with ours. They argue that although
Mesopotamians wrote expressions such as [1; 0; 3] or [12; 0; 5; 0; 33], they
would never have thought of expressions of the form [5; 0] or [17; 3; 0; 0].
More recently, however, O. Neugebauer has shown that Babylonian
astronomers used the zero not only in median, but also in initial and
terminal positions. For instance, in an astronomical tablet from Babylon
(Seleucid period), we find 60 written thus:
Fig. 13.64A.
(= 1 x 60 + 0)
Here the double slant chevron is used not as a separator, but to mark the
absence of units of the first order. On the back of the same tablet, we also
find 180 represented in the same way:
ITT*
(3 ; 0] (=3x60 + 0)
Fic. 13.64B. *
And in another astronomical tablet from Babylon of the same period
(British Museum, BM 34581), the number
[2; 11; 46; 0] (= 2 x 60 3 + 11 x 60 2 + 46 x 60 + 0)
is represented as:
, [2 ; 11 ; 46 ; 0]
The final zero in this last example is written in a rather special way, like
a “10” with an elongated tail. Has the upper chevron of the zero just been
omitted? Is it a scribal decoration? Or just a sign of haste? The latter seems
the most likely, since there are other examples in tablets from the same
period and the same astronomical source. The following example comes
from one such tablet on which the zero is also represented several times in
the normal manner:
THE BIRTH OF THE BABYLONIAN ZERO
The double oblique wedge or chevron in initial position also allowed
Babylonian astronomers to represent sexagesimal fractions unambiguously.
Here are some examples from the tablet previously quoted:
Fig. 13.67.
To summarise: the learned men of Mesopotamia perfected an abstract,
strictly positional system of numerals at the latest around the middle of the
second millennium BCE, a system far superior to any other in the Ancient
World. At a much later date, they also invented zero, the oldest zero in
history. Mathematicians seem only to have used it in median position; but
the astronomers used it not only in the middle, but also in the final and
initial positions of numerical expressions.
THE DATING OF THE EARLIEST ZERO IN
HISTORY
As we have seen, there is no zero in scientific texts of the First Babylonian
Dynasty, and the figure is hardly attested in any texts prior to the third
century BCE. Does that mean that the Mesopotamians only invented the
zero in the Seleucid period? That cannot be so easily said, for there are
distinctions to be made between the presumed date of an invention, the
period of its propagation, and the dates of its first occurrence in texts
that have come down to us. It is perfectly possible for an invention to
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
154
have been made several generations before its use became widespread, just
as it is possible for the “oldest documents known to bear a trace” to be
several centuries later than the invention itself - either because the earlier
documents have perished, or because they have not yet been discovered.
It is therefore legitimate to believe that the Babylonian zero arose several
centuries before the third century BCE. This supposition is all the more
plausible because we now know that the literary tablets of the Seleucid
period are actually copies of much earlier documents (see H. Hunger, 1976):
mathematical tablets of the Seleucid period may therefore not all be
contemporary texts.
But these are only suppositions. Only further archaeological discoveries
can provide definite proof.
HOW WAS ZERO CONCEIVED?
The double wedge or double chevron had the meaning of “void”, or rather
of the “empty place” in the middle of a numerical expression, but it does
not appear to have been imagined as “nothing”, that is to say as the result
of the operation 10 - 10.
In a mathematical tablet from Susa a scribe tried to explain the result of
such an operation, thus:
20 minus 20 comes to . . . you see?
Similarly, in another mathematical text from Susa, at the end of an
operation (referring to the distribution of grain) where you would expect
the sum of 0 to occur, the scribe writes simply that “the grain is finished”.
The concepts of “void” and “nothing” both certainly existed. But they
were not yet seen as synonyms.
HOW DID BABYLONIAN SCIENTISTS
DO THEIR SUMS?
There are no known accounts of the computational methods used by
Babylonian mathematicians and astronomers. These methods can nonethe-
less be reconstructed from the numerous mathematical texts that have been
found and deciphered.
Although the rule of position had been adopted, learned Babylonian
numerals remained close to their Sumerian roots in the sense that they
remained sexagesimal, with 10 serving as an auxiliary base within each
order of magnitude. Now, given that we have proved the existence of a
Sumerian abacus and shown what shape it must have had, we can assume
fairly safely that the tool was handed down to Babylonian scholars as part
of their Sumerian heritage, and used by them for the same purposes.
That is very probably how things happened, at least at the beginning of
this story.
But there are very good reasons for believing that the rules and the shape
of the abacus changed very quickly, and that the method became simpler as
the centuries passed.
The simplification of the abacus counting method must have required
as its counterpart the memorisation of “tables” for the numbers between 1
and 60 - these tables constituting the necessary mental “baggage” to be
able to use the abacus for arithmetical operations.
In fact, the Babylonians never bothered to learn such number-tables by
heart: they wrote them out once and for all, and handed the tablets down
from generation to generation. Consequently, the mathematical tablets that
have been discovered include a great number of multiplication tables.
Fig. 13.68 below is a typical example. The transcription can easily be
followed by looking at the face of a clock or watch, and imagining the units
of the first order as minutes, and the units of the second order as hours.
It can then be seen that the tablet on the left gives the numbers from 1 to
20 followed by 30, 40, and 50, and on the right gives the result of multiply-
ing those numbers by 25. It is therefore a
25-times table, completely
analogous to one we could construct using <
)ur current decimal system:
1
(times 25 equals)
25
2
(times 25 equals)
50
3
(times 25 equals)
75
4
(times 25 equals)
100
5
(times 25 equals)
125
6
(times 25 equals)
150
7
etc.
(times 25 equals)
175
Generally speaking, the multiplication tables give the products of a
number n (smaller than 60) of the first twenty integers, then of the
numbers 30, 40, and 50. This clearly suffices to provide the product of n
multiplied by any number between 1 and 60.
With such tables in support, multiplications could be done fairly easily
on an abacus.
The rule of position must have led rather quickly to the realisation that
wooden tablets of the sort shown in Fig. 12.4 above were no longer neces-
sary, and that the divisions of the Sumerian abacus did not have to be
reproduced. All that was now needed was to draw parallel lines to create
vertical columns, one for each of the magnitudes of the sexagesimal system.
Since clay is easier to work than wood, we can surmise that the columns
TRANSCRIPTION
155
HOW DID BABYLONIAN SCIENTISTS DO THEIR SUMS?
were drawn onto the wet clay of a tablet made afresh for each calculation.
Sticks and tokens would not have been needed any longer, since the
numbers involved could be drawn straight onto the clay in the relevant
columns and wiped or scored out as the calculation proceeded. This recon-
struction of the Mesopotamian abacus is of course only a speculation, but
it is in our view a highly plausible one.
Here is an example of how it might have worked, using the multipli-
cation table shown in Fig. 13.68.
The task is to multiply 692 by 25, or, in Babylonian terms, to multiply
[11; 32] (= 11 x 60 + 32) by 25.
Let us begin by scoring the first three columns onto the wet clay tablet,
in which the result will be entered in the three orders of magnitude, start-
ing from the right (numbers from 1 to 59 will be entered in the rightmost
column, multiples of 60 from 1 to 59 in the middle column, and multiples
of 3,600 from 1 to 59 in the leftmost column).
Order of Order of Order of
3,600s 60s units
! | (1 to 59)
I I I
I I I
V * *
Fig. 13.69A.
To the right of the columns, let us inscribe the multiplicand [11; 32]
(= 692) in cuneiform notation:
Fig. 13.69B.
Using the 25 x multiplication table, we look for the product of 2; finding
50, we enter that number in cuneiform notation in the units column of the
abacus tablet:
Fig. 13.69c.
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
We can now rub out the 2 from the multiplicand on the right of the tablet,
and proceed to look up 30 in the 25 x multiplication table. The product
supplied is [12; 30], so we enter 30 in the rightmost column of the units
on our abacus, and 12 in the middle column, reserved for multiples of 60.
Fig. 13.69D.
So we rub out the 30 from the multiplicand on the right of the tablet,
and proceed to look up 11 in the 25 x multiplication table. The product
supplied is [4; 35], so we enter 35 in the middle column (since we have
changed our order of magnitudes) and 4 in the leftmost column, the one
reserved for multiples of 3,600.
So we can now rub out the 11 from the multiplicand, and find that there
is nothing left on the right of the tablet. The first stage of the operation is
complete.
Fig. 13.69E.
The rightmost column now has 8 chevrons in it. Since this is more than
the 6 chevrons which make a unit of the next order, we rub out 6 of them
and “carry" them into a wedge which we enter in the middle column,
leaving 2 chevrons in the units column.
Fig. 13.69F.
So we now have 4 chevrons and 8 wedges altogether in the 60s column.
The sum of these being not greater than 60, we simply rub out the numerals
in the column and replace them with the numeral signifying 48, the sum of
4 chevrons (4 x 10) and 8 wedges (= 8). And as there is only a 4 in the
column of the third order, the result of the multiplication is now fully
entered on the abacus:
[11; 32] x 25 = [4; 48; 20]
(= 4 x 3,600 + 48 x 60 + 20 = 17,300)
156
Fig. 13.69G. 4 48 20
The Babylonians also had tables of squares, square roots (Fig. 13.70),
cube roots, reciprocals, exponentials, etc., for all numbers from 1 to 59,
which enabled far more complex calculations to be performed. For
instance, division was done by using the reciprocal table, i.e. to divide one
number by another, you multiplied it by its reciprocal.
All this goes to show the great intellectual sophistication of the
mathematicians and astronomers of Mesopotamia from the beginning of
the second millennium BCE.
TRANSCRIPTION AND
RECONSTRUCTION
e
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13; 04
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04
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lb-si.
14; 01
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e
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15;
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fb-si.
16
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fb-si.
26
e
5
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ft
32
fb-si.
40
e
7
l&'Sii
ib-si 8
18; off
e
33
fb-si.
19,1*
e
34
fbsi.
1,04
e
8
fb-si.
20X19
e
35
fb-si „
1 '01
e
0
lb-si.
21/39
e
36
fb-si.
1 j40
Go
fb-si.
e
37
fb-si.
2.01
e
^4; 04
e
38
fb-si.
Fig . 13.70. Fragment of a table of square roots, c. 1800 BCE, from Nippur (100 miles SE of
Baghdad). University of Pennsylvania, Babylonian section, CBS 14233 side 2.
THE BABYLONIAN LEGACY
The abstract system of the learned men of Babylon has had a powerful
influence over the scientific world from antiquity down to the present.
From at least the second century BCE, Greek astronomers used the
Babylonian system for expressing the negative powers of 60. However,
instead of using cuneiform numerals, the Greeks used an adapted version
of their own alphabetic numerals. For example, they wrote expressions like
0° 28' 35" and 0° 17' 49" in the following way:
157
THE BABYLONIAN LEGACY
T
KH
AE
0 + 28 +
35 '
[0 ;
28
; 35]
1 60
60 2 >
O
IZ
M0
(=0+ 17 +
49 '
[0 ;
17
; 49]
V 60
60^
TRANSCRIPTION
le
IB
KE]
K
ir
KCJ
KA
• .
IA
KZ J
KB
A
[. ..
IE
[KHJ
KT
B
Mr
B
KA
AC
1C
NB ME
Ke
KA
A
A
r
KC
e
IZ
NA 1C
A
J-
KE
e
KG'
A
KZ
MA
IH
TAYPOY
AIAYM
cr
KG'
c
MZ
E
K»
ir
l»
r Ke nc
«I-K[.]
]•
KZ
H
&
er
A
AC
K
■x- Ne NB
T-MI-J
KH
1
NB
H
AB
IH
KA
*1- KB MH
A
Ke
IB
r
e
AT
N
KB
B NA AC
B
ir
AE
1
AE
KB
KT
A Ke
r
II
IA
NC
IA
AE
NE
KA
E Ne
A
TRANSLATION
12
25]
20
, ,
13
26]
21
' ’
14
27]
1
15
[28]
2
43
2
24
36
16
52
45
29
4
30
3
26
9
17
54
16
30
9
26
4
27
41
18
BULLS
GEMINI
6
26
6
47
5
29
13
19
0 29
56
0 20[.]
]■
27
8
9
6
30
36
20
0 59
52
0 40[.]
28
10
52
8
32
18
21
0 22
48
1
29
12
3
9
34
50
22
2 54
36
2
30
13
35
10
35
22
23
4 29
3
14
56
11
35
55
24
5 59
4
Fig. 13.73. Transcription and translation of a Greek astronomical table, from a third- century
papyrus. University of Michigan Papyrus Collection, Inv. 924. See }. Garctt Winter, pp. 118-20
GREEK PAPYRI
first century
Pap. Aberdeen
No. 128
after + 109
Pap. Lund
Inv. 35A
O
2 nd century
Pap. London
No. 1278
>
467 CE
Pap. Michigan
Inv. 1454
Fig. 1 3 . 7 4 a . The “sexagesimal ” zero of Greek astronomers
ARABO-PERSI AN MANUSCRIPTS
TP t
T
i
<
+ 1082
+1436
+ 1680
+ 1788
Bodleian Library,
Leyden Univ. Lib,
Princeton, Firestone Library,
Ms Or. 516
Cod. Or. 187 B
ELS 147
ELS 1203
Fig. 13 . 72 . Greek astronomical papyrus, second century CE (after 109). Copied from
Neugebauer, plate 2
Fig. 1 3 . 7 4 b . Scribal variants of the “ sexagesimal zero" in Arabic and Jewish astronomical texts
MESOPOTAMIAN NUMBERING AITER THE ECLIPSE OE SUMER
Arab and Jewish astronomers also followed the Greeks’ borrowing of the
Babylonian system, which they “translated” into their own alphabetic
numerals, giving the following forms for the illustrative numbers shown in
Fig. 13.71 above:
n9 ns r J e 5 v
35 ; 28 ; 0 35 ; 28 ; 0
4
tSD P Z LmJj 7
49 ; 17 ; 0 49 ; 17 ; 0
<■ «
Fig. 13.75A. Fig. 13-75B.
Thus the learned Babylonian system has come down to us and is
perpetuated in the way we express measures of time in hours, minutes and
seconds and in the way we count arcs and angles, despite the strictly decimal
nature of the rest of our numerals and metric weights and measures. It is
largely due to the Arabs that the system was transmitted to modern times.
Fig. 13 . 76 . Bilingual (Latin-Pcrsian) astronomical table, transcribed by Thomas Iiyde, 1665.
British Library 757 cc 11 (1), pp. 6-7
CODES AND CIPHERS IN CUNEIFORM NUMERALS
In some periods and in some fields, the scribes of Susa and Babylon were
much given to playing cryptic games with numerals. Some of these games
involved numerical transposition, that is to say the use of numerical expres-
sions in lieu of words or ideograms, generally based on some coherent
system of “coding”, or on complex numerological symbolism.
158
I * — wmrmf
mi jh*** mm ntytf amn\
Am fa jinn
I I V k f ^ 1 N "X I vturnt ■»* !
i
-4
S ' 1
JL_i
i» 1
, *
ft
_£j
tf)
1
•
T
V
T»
T
w
1 M
H
* i !
Til t
Fig. 13.77. Astronomical table by Levi Ben Gerson, a French -Jewish savant, 1288-1344 CE.
British Museum, Add. 26 921, folio 20b. Transcribed by B. R. Goldstein, table 36.1
159
CODES AND CIPHERS IN CUNEIFORM NUMERALS
One of the inscriptions of the name of King Sargon II of Assyria (722-
705 BCE) provides an example of numerical transposition. Recording the
construction of the great fortress of Khorsabad (Dur Sarukin), Sargon says:
I gave its wall the dimensions of (3,600 + 3,600 + 3,600 + 3,600 + 600
+ 600 + 600 + 60 + 3 x 6 + 2) cubits [i.e. 16,280 cubits] corresponding
to the sound of my name. [Cylinder-inscription, line 65]
However, this assertion has not yet yielded all its secrets: we cannot
reconstitute the coding system by which the name was transposed into
numerals from this single example alone.
Another type of number-name game is shown in a tablet from Uruk of
the Seleucid period. At the end of the Exaltation of Ishtar (published by
F. Thureau-Dangin in 1914) the scribe indicates that the tablet belongs to
someone called
«r <«?r «f <r *?ir
" son or
21 35 35 26 44 21 11 20 42
Fig. 13.78.
But who is he? The last line gives his name and the name of his father,
but both names are written in numerals. The scribe gives us a puzzle
without giving us the key [Thureau-Dangin (1914)].
Numerical cryptograms were also widely used for haruspicy, the “secret
science” of divination or fortune-telling. Seers and fortune-tellers used
several different numerical combinations for mystifying the profane and for
ensuring that their magical texts remained impenetrable to the uninitiated
(Fig. 13.79). Commenting on the Esagil tablet, which gives the dimensions
of the great temple of Marduk at Babylon and of the tower of Babel,
G. Contenau wrote:
This difficult text looks on first reading like a bland statement of the
dimensions of yards and terraces - a mere sequence of numbers, as on
a stock list, with all it has to say stated plainly. However, the scribe has
peppered his account with the intercalated formula so often found in
hieratic texts:
May the initiated explain this to the initiated
And the uninitiated see it not!
We should not forget the significant role played by the oral teaching
of the pupil by the master which accompanied the lessons of the
invariably summary texts themselves. Even texts which appear to be
utterly ordinary hid esoteric meanings which we cannot imagine.
The scribes of Susa and Babylon also used cryptograms for word-games,
or rather, writing-games, which are worth some attention. For instance, the
f r r
¥ r
w r
ww r
ITT ¥ T
TTT ¥ TT
TTT ¥ TT
W r YY
W¥ W
Hr ¥ TTT
nr ¥ ttt
0 rw rrr
kywy
iTtr ¥ ¥
i* ¥
Y
...JfcfNSr trr
r
1 T T .jy 1 /r-
rV
r r ^*&**u_.
r ¥ ..
r Y y^t
r
r ¥
r y
r? Tfr
Y ¥ Y^Y
r ,
r ?
rf
r y y^y
X4Y, J
T ¥ Y^r
TY
Fig. 13.79. Astrological table with cryptograms in a code that remains to be deciphered. (Line 5,
for instance, reads: 3; 5; 2; 1; 12; 4; 31). British Museum, 92685 side 1. Copy made by H. Hunger
combination “3; 20” is often found used as an ideogram for the word
meaning “king”, which was pronounced sar or sarru in Akkadian. An
inscription on a brick from the reign of Susinak-Sar-Ilani, king of Susa
(twelfth-eleventh century BCE) bears this formulation of the king’s name:
SUSINAK - TTT-^- ILANI 7TT << ^ SUSI
3 . 20 3 . 20
(“Shushinak-Shar-Ilani, king of Susa”)
MESOPOTAMIAN NUMBERING AFTER THE ECLIPSE OF SUMER
Now, why is the numerical combination “3; 20” a logogram for “king”? In
Akkadian, the word for “king” was sar, pronounced more or less exactly the
same way as sar, the name of the “higher” sexagesimal unit of counting in
the Sumero-Babylonian system, that is to say 3,600. Elamite scribes thus seem
to have made a pun by replacing the word “king” by a numerical combination
[3; 20] which represented 3,600 according to a specific rule of interpretation.
But what was the rule? It clearly has nothing to do with the positional
sexagesimal system of the learned men of Mesopotamia, since in that way
of reckoning [3; 20] = 3 x 60 + 20 = 200, which is not the right answer. On
the other hand, we could decompose sar into a kind of “literal” numerical
expression that would be represented by “sixty sixties”, or, in cuneiform:
mm-
60 SHU-SHI
Punning scribes could have written this out, as a game, in this alternative
way:
-4.
20 20 20 SHU-SHI
or finally as
TTT« B<r
3 x 20 SHU-SHI
So we can see that Susan scribes regarded the sequence [3; 20] as
expressing the product of 3 x 20 (implicitly, x 60), that is to say 3,600,
making a pun on sar, “the king”.
Assyro-Babylonian scribes also used the combination [3; 20] to refer to
the king, but they sometimes added a chevron, making [3; 30]. This latter
variant cannot be accounted for by 3 x 60 + 30 = 210, nor by (3 x 30) x 60
= 5,400. However, if the addition of a chevron (= 10) to the expression
[3; 20] is understood as the mark of a multiplication of [3; 20] by 10, then
the symbol can be understood as:
[3; 30] = [3; 20] x 10 = 3,600 x 10 = 36,000
This gives the number called sar-u in Sumerian, written in that older
system as a 3,600 with an additional chevron in the middle:
<> &
3,600 3,600 x 10
The word sar-u (which means “ten sar” or “the great sar”) is thus what is
meant by the numerical expression
160
TTT<« = TTT« k
3 ; 30 3 ; 20 x 10
SAR-U
- because this Sumerian number-name has exactly the same sound as the
Akkadian word sharru, meaning king. So when a scribe referred to the king
by writing [3; 30], we can deduce that he meant to say “the great king”.
There are many other Babylonian cryptograms which remain unsolved,
however. For instance, we have no idea why the concepts of “right” and
“left” came to be written by the cuneiform numerals [15] and [2; 30] respec-
tively, nor why [1; 20] was used as an ideogram for “throne”, nor, finally,
why the vertical wedge, the sign for the unit in the numerical system, had
the role of the determiner (“the man who . . . ”) in the names of the main
male functions.
ipt
i m
TT m ►
’4x< *wm
»H U * — t
»f- IT
■; Mfrsp? ir*:
17 KR I
\l TT
Eft
70
Fig. i 3 . 8 o a . Cuneiform tablet listing names of gods and their corresponding numbers. Seventh
century BCE, from the “library" of Assurbanipal. British Museum K 170. Trans. J. Bottero
161
CODED CRYPTOGRAMS AND
MYSTICAL NUMEROLOGY
Coded cryptograms were also used in theological speculation, for
Mesopotamian scribes accorded great weight to the numerical transposition
of the names of the gods. Indeed, the religions of the Assyro-Babylonians
assumed that the celestial world was a “numerologically harmonious” one,
in which the numerical value of a name was an essential attribute of the indi-
vidual to which it belonged. For this reason, from the early part of the second
millennium BCE and consistently throughout the first millennium, some of
the Babylonian gods were represented by cuneiform numerals. Fig. 13.80
reproduces a tablet from the seventh century BCE which gives the names of
the gods and for each one a number which could be used as that god’s
ideogram. These are the main points made on the tablet:
• Anu, god of heaven, is attributed the number 60, the higher unit of
the sexagesimal Sumerian and Babylonian system, and considered to
be the number of perfection, because, the scribe says, “Anu is the first
god and the father of all the other gods”;
• Enlil, god of the earth, is represented by 50;
• Ea, god of water, is represented by 40 (elsewhere, she is sometimes
ascribed the number 60);
CODED CRYPTOGRAMS AND MYSTICAL. NUMEROl.OGY
• Sin, the lunar god, corresponds to 30, because, the scribe says (line
9 , column 1, side 1), “he is the lord of the decision of the month”, or,
in other words, Sin is the god who regulates the 30 days of the month
• Shamash, the sun-god, is worth 20;
• ADADherehasthe number 6 (more frequently, he has the number 10);
• Ishtar, daughter of Anu lord of the heavens and held to be the
“queen of the heavens”, has the number 15;
• Ninurta, son of Enlil, has the same number as his father, 50;
• Nergal has the number 14;
• Gibil and Nusku are both represented by 10 because, according
to the scribe (line 16, column 1, side B), “they are the companions of
god 20 ( = Shamash): 2 x 10 = 20”.
The numerological values of the gods of Babylon had all sorts of
consequences. For example, the Babylonian Creation Epic concludes with a
list of the “names” of Marduk, a series of epithets defining his virtues and
powers and intended to demonstrate that he is truly the supreme god and
the most godly of all. First comes a list of ten names, because Marduk’s
“number” is 10, then a second group of forty names, because Marduk is the
son of Ea, whose number is 40; which adds up to fifty names, because 50 is
the number of Enlil, and the main point of the epic is to show how Marduk
replaced Enlil at the head of the universe of gods and men.
TRANSCRIPTION AND TRANSLATION OF
THE TWO RIGHTMOST COLUMNS
LINE 6
r
1 or 60
d A-num
7
9
►tfll WIT
50
d En-lII
UJ
8
H+wirr if
40
d E-a
cn
9
** — <«
30
d Sin (name written as 30)
10
20
d Shamash
11
m
- r
d Adad
" T mL II
LINE 12
<
10
d Bel d Marduk (the Lord Marduk)
«N
13
15
d Ishtar be-Iit ill (Ishtar queen of
the heavens)
UJ
Q
(/}
14
urn
SWSHW-f#
F.n-HI
50 d Nin-urta, mar 50 (50 Nin-urta,
son of the god Enlil) (written as 50)
15
J
14
d U + gur, d Nergal
16
<
10
d Gibil, d Nusku
Fig. 13.80B.
THE NUMBERS OF ANCIENT EGYPT
162
CHAPTER 14
THE NUMBERS OF
ANCIENT EGYPT
Egypt in the time of the Pharaohs had writing and written numerals. They
first arose around 3000 BCE, that is to say, at about the same time that
words and numbers were first written down in Mesopotamia.
We now know that there were regular contacts between Egypt and
Mesopotamia before the end of the third millennium BCE. However, that
does not mean that the Egyptians derived their writing or their counting
from Sumerian models. Egyptian hieroglyphs, Jacques Vercoutter explains,
use signs derived exclusively from the flora and fauna of the Nile basin;
moreover, the tools used for making written signs existed in Egypt from the
fourth millennium BCE.
The pictograms of Egyptian hieroglyphic writing are very different from
Sumerian ideograms, even when we compare signs intended to represent
the same idea or object; the shapes of such signs also seem quite unrelated.
The media of the two systems likewise have little in common. As we saw,
the Sumerians only ever wrote words and numbers by scoring clay tablets
w'ith a stylus, or else by pressing a shaped instrument onto wet clay;
whereas the Egyptians carved their numerals and hieroglyphs in stone, with
hammer and chisel, or else used the bruised tip of a reed to paint them on
shards of stone or earthenware, or onto sheets made by flattening the
dried-out, fibrous, and fragile stems of the papyrus reed.
Egyptian numerals are also quite different from Sumerian ones from a
mathematical point of view. As we have seen, Sumer used a sexagesimal
base; whereas the system of Ancient Egypt was strictly decimal.
So if there was something borrowed by Egypt from Sumer, it could only
have been the idea of writing down numbers in the first place, and not any
part of the way it was done.
Peoples very distant from each other in time and in place but facing
similar situations and needs have discovered quite independently some
of the same paths to follow, and have arrived at similar, if not identical,
results. The Indus civilisation, the Chinese, and the pre-Columbian
populations of Central America (Zapotecs, Maya, etc.), were all faced with
situations probably very similar to those of the Sumerians, and made
much the same mathematical discoveries for themselves. So it seems most
sensible to suppose that around the dawn of the third millennium BCE
the social, psychological, and economic conditions of Ancient Egypt were
such that the invention of writing and of written numerals arose there of its
own accord.
In fact, Egyptian society was already advanced, urbanised, and expand-
ing rapidly long before 3000 BCE. Administrative and commercial logic
led to the slow realisation that human memory could no longer suffice
to fill all the needs of the state without some material support; oral culture
must have come up against its natural limits. We must then suppose that
the Egyptians felt an increasing need to record and thus to retain thoughts
and words, and to fix in a durable form the accounts and inventories of
their commercial activities. And, since necessity is the mother of invention,
the Egyptians overcame the limits of oral culture by devising a system for
writing down words and numbers.
WHAT ARE EGYPTIAN HIEROGLYPHS?
Although they no longer knew how to read them, the Ancient Greeks
recognised the signs carved on the many monuments of the Nile Valley
(temples, obelisks, tombs, and funeral stelae) as “sacred signs" and thus
called them grammata hiera, or, more precisely, grammata hierogluphika
(“carved sacred signs”), whence our word, hieroglyph. It is from these carved
signs, which the Ancient Egyptians considered to be “the expression of the
words of the gods”, that we have derived our knowledge of the spoken
language of Ancient Egypt. The basic writing system for the representation
of this language was designed for and was used for the most part only on
stone monuments, and it is this writing system (rather than the language it
represents) that we commonly call hieroglyphs.
HOW TO READ HIEROGLYPHS
Hieroglyphs are very detailed pictograms representing humans, in various
positions, all kinds of animals, buildings, monuments, sacred and profane
objects, stars, plants, and so on.
to phallus quail bird in fish uraeus
adore flight
fc e? d & r I
woman bull screech- beetle snake flowering
owl reed
fa A <£ %
pregnant hare falcon bee hooded lotus
Fig. 14.1. Some hieroglyphs woman viper
163
HOW TO READ HIEROGLYPHS
Hieroglyphs may be written in lines from left to right or right to left, or
in columns from top to bottom or from bottom to top. The direction of
reading is indicated by the orientation of the animate figures (humans or
animals) - they are “turned” so as to face the start of the line. So they look
left in a text written/read from left to right:
Fig. 14-i-
and they look right in a text written/read from right to left:
j&
Fig. 14.3.
Hieroglyphic signs could be used and understood, first of all, as “integral
picture-signs” or pictograms : pictures that “meant” what they showed. In the
second place, they could also be used as ideograms : that is to say, signs
meaning something more than, or something connected to, what they
showed. For example, an image-sign of a human leg could mean, first, “leg”,
as a pictogram, but also, as an ideogram, the related ideas or actions of
“walking”, “running”, “running away”, etc. Similarly the image of the sun’s
orb could mean “day”, “heat”, “light”, or else refer to the sun-god. The
ideogrammatic interpretation of a sign did not supplant its pictogrammatic
meaning, but coexisted alongside it. The interpretation of a hieroglyphic
sign is therefore open to infinite subjective variation.
Pictograms and ideograms cannot easily cope with every nuance of
language. How can such a system represent actions such as wishing, desiring,
seeking, deserving, and so on? Or abstract notions like thought, luck, fear, or
love ? Moreover, pictograms cannot represent the articulations of spoken
language, and are independent of any particular language spoken.
To overcome these limitations, the Ancient Egyptians used their signs in
a third way, quite at variance with their pictogrammatic and ideogrammatic
values. A sign could also represent the sound of the name of the thing
represented pictographically, and then be used in combination with other
sounds represented by the ideograms of other words, to make a kind of
visual pun or rebus. For instance, let us suppose that in Britain today we
had only a hieroglyphic writing system and in that system had no
pictogram for the things we call “carpets”; but did have a conventional
pictogram for “car”: and, given that we are a nation of dog-lovers,
represented the general idea of “pet” by the ideogram: ^>. Were we to
proceed with the system as the Ancient Egyptians did, we would not invent
a new pictogram for “carpet” but would create a compound picture-pun
^ <S> CAR-PET
Such a system has a built-in propensity towards ambiguity. This is not just
because ideograms by their very nature have variable interpretations, but
also and most especially because a rebus may make a sense in more than
one reading of the phonetic value of the ideograms. To take an equally ficti-
tious example from hieroglyphs to be realised by speakers of English, where
the pictogram 4 has the full pictorial meaning of “fir” and the broader
meaning “wood” when taken as an ideogram, and the ideogram has the
meanings of “house”, “inn” or “home”, the expression (read left to right)
4 &
could be realised phonetically as INN-FIR, with the punning meaning of
the verb infer, or else, read right to left as HOME-WOOD, with the punning
meaning of homeward. In order to reduce the number of total misappre-
hensions of that sort, Egyptian hieroglyphs therefore needed an additional
sign in each compound expression, a kind of ideogrammatic hint or
determiner that showed which way the sound-signs were to be taken. To
continue our example, the determiner when added to 4 would
ensure that it was taken in the directional sense. So
4
would indeed be read as INFER, whereas
— — 4 £b
would be read as HOMEWARD.
That is roughly how Egyptian hieroglyphs evolved from pictorial
evocations of things to phonetic representations of words. For example: the
Ancient Egyptian for “quail chick” was pronounced Wa; the sign depicting
a quail chick signified a quail chick, but also represented the sound Wa.
Similarly, “seat” was pronounced Pe, and the drawing of a seat came to
represent the sound Pe; “mouth” was eR, and a drawing of a mouth meant
the syllable eR; a picture of a hare (WeN) stood for the sound WeN, a
picture of a beetle (KhePeR) made the sound KhePeR, and so on.
| \ |r|
i w p r vvn Hpr Fig. 14.4.
Like Hebrew and other Semitic scripts, Egyptian hieroglyphs are conso-
nantal, that is to say they represented only the consonants, leaving the
vowels to be “understood” by convention and habit. (Where vowels are put
in in modern transcriptions of the language, they are hypothetical and
THF, NUMBERS OF ANCIENT EGYPT
164
conventional: there is in fact no way of knowing how Ancient Egyptian was
actually vocalised.) Since words in Ancient Egyptian contained either one,
two, or three consonants, hieroglyphs used as sound-signs also belonged to
one of three classes: uniliteral (representing a single consonantal sound),
biliteral (representing two sounds), or triliteral (representing a group of
three consonants). With their signs used simultaneously as pictograms, as
ideograms, and with syllabic value, the Ancient Egyptians were thus able to
represent all the words of their language.
An early example is Narmer’s Palette (c. 3000-2850 BCE), which
commemorates the victory of King Narmer over his enemies in Lower
Egypt.
The king can be seen in the centre of the panel, wielding his club over a
captive. The king’s name, written in the cartouche above his regal headgear,
is composed of the hieroglyphs “fish” and “scissor”. The word meaning
“fish” was pronounced N‘R, and the word meaning “scissor” was
pronounced M‘R: the two together thus make N’RMR, or Narmer.
In similar fashion, the word for “woman”, pronounced SeT, was repre-
sented by the image of a bolt (the word for “bolt” being a uniliteral with
value S) and an image of a piece of bread ("piece of bread” being also
a uniliteral, with value T). However, to ensure that S + T was read in the
right way, a pictogram of a woman (unrealised in speech) was added as
a determiner:
S T Determiner
Fig. 14.7.
Likewise the vulture, NeReT in Ancient Egyptian, was represented by N
(“stream of water”), R (“mouth”) and T (“piece of bread”), plus the deter-
miner, “bird”, to ensure that the sound-signs were read as a word belonging
to the class of birds.
Fig. 14.8.
Hieroglyphic writing did not use only these kinds of determiners,
however. In many cases, biliteral and triliteral signs are disambiguated by a
phonetic “complement” which gives a supplementary clue as to how to read
the sign. For instance, the hieroglyph of “hare”, a word pronounced WeN,
would be “confirmed” as meaning the biliteral sound WeN by the addition of
the sign for “stream of water”, a uniliteral sound pronounced N, as follows:
WN
N (Phonetic complement) Fig. 14.9.
It is as if in our imaginary English hieroglyphs we added JK to the sign
to ensure that
M
was recognised as a syllable containing the uniliteral consonant T (as in
"cup of tea") and thus pronounced PET, and not seen as a pictogram
meaning (for example) "Labrador”.
In Ancient Egyptian the name of the god Amon was represented by the
signs whose pronunciation was i (“reed in flower”) and mn (“crenellation”),
supplemented by a determiner (the ideogram signifying the class of gods)
plus a phonetic complement, the sign for "stream of water”, pronounced N,
whose sole function was to confirm that the syllable was to be read in a way
that made it include the sound n.
165
HIEROGLYPHIC NUMERALS
Ik ( 1 )
| ra $
1 si
♦ * Jfr
^ MB* w?
N
Phonetic Ideogram
Fig. 14.10.
complement
HIEROGLYPHIC NUMERALS
Written Egyptian numerals from their first appearance were able to
represent numbers up to and beyond one million, for the system contained
specific hieroglyphs for the unit and for each of the following powers of
10: 10, 100 (= 10 2 ), 1,000 (= 10 3 ), 10,000 (= 10 4 ), 100,000 (= 10 s ), and
1,000,000 (= 10 6 ).
The unit is represented by a small vertical line. Tens are signified by a
sign shaped like a handle or a horseshoe or an upturned letter "U”. The
hundreds are symbolised by a more or less closed spiral, like a rolled-up
piece of string. Thousands are represented by a lotus flower on its stem, and
ten thousands by a slightly bent raised finger. The hundred thousand has
the form of a frog, or a tadpole with a visible tail, and the million is depicted
by a kneeling man raising his arms to the heavens.
READING
RIGHT TO LEFT
READING
LEFT TO RIGHT
1
1
1
10
n
n
100
3>
1,000
i
2
1
1
x
f
t
10,000
t
r
1
1
1
1
100,000
j3
f
0
*
1,000,000
%
i
%
t
ne
$
Fig. 14 . 11 . The basic figures of hieroglyphic numerals with their main variants in stone inscrip-
tions. Note that the signs change orientation depending on which way the line is to be read: the
tadpole (100,000) and the kneeling man (1,000,000) must always face the start of the line.
One of the oldest examples that we have of Egyptian writing and
numerals is the inscription on the handle of the club of King Narmer, who
united Upper and Lower Egypt around 3000-2900 BCE.
Fig. 14 . 12 . Tracing of the knob of King Narmer’s dub, early third millennium BCE
Apart from King Narmer’s name, written phonetically, the inscription on
the club also provides a tally of the booty taken during the king’s victorious
expedition, consisting of so many head of cattle and so many prisoners
brought back. The tally is represented as follows:
Are these real numbers, or are they purely imaginary figures whose sole
aim is to glorify King Narmer? Scholars disagree. But we should note that
the livestock tallies found on the mastabas of the Old Kingdom also often
give very high numbers for individual owners, and that here we are dealing
with the looting of an entire country.
Another example of high numbers can be found on a statue from
Hieraconpolis, dating from c. 2800 BCE, where the number of enemies
slain by a king called KhaSeKhem are shown as 47,209 by the following
signs:
THF. NUMBERS OF ANCIENT EGYPT
200 7,000
G)
i hi i hi
9 3,000 4,000 40,000
*
Fig. 14 . 14 . 47,209
To represent a given number, then, the Egyptians simply repeated the
numeral for a given order of decimal magnitude as many times as neces-
sary, starting with the highest and proceeding along the line to the lowest
order of magnitude (thousands before hundreds before tens, etc.).
Early examples show rather irregular outlines and groupings of the signs.
In Fig. 14.13 above, for example, the number of goats (1,422,000) is written
in a way that is contrary to the rules that were later laid down by Egyptian
stone-cutters, since the figure for the million is placed to the right of the
beast and on the same line, whilst the remainder of the number-signs are
inscribed on the line below. The normal rule was for the signs to go from
right to left in descending order of magnitude on the line below the sign for
the object being counted, thus:
Similarly, Figure 14.14 shows rather primitive features in the representa-
tion of the finger (= 10,000), the grouping of the thousands (lotus flowers)
into two distinct sets, and the relatively poor alignment of the unit signs.
However, from the twenty-seventh century BCE, the execution of hiero-
glyphic numerals became more detailed and more regular. Also, to avoid
making lines of numerals over-long, the custom emerged of grouping signs
for the same order of magnitude onto two or
three lines, which made them
easier to add up by eye:
1 II III |j
III
II
III
III
1111 1111
111 III!
III
III
III
1 2
3
4
5
6
7 8
9
n nn
nnn
nn
nn
nnn
nn
nnn
nnn
nnnn nnnn
nnn nnnn
nnn
nnn
nnn
10 20
30
40
50
60
70 80
90
Fig. 14.16.
166
The evolution of Egyptian numerals can be traced as follows:
1: Old Kingdom period: funerary inscriptions ofSakhu-Re, a Pharaoh of
the Fifth Dynasty, who lived at the time of the building of the pyramids,
around the twenty-fourth century BCE:
10,000 3,000 40 200,000 3,000 30,000 400 10
20,000 400 20,000 400 2,000 3
123,440 + ? 223,400 32,413
Fig. 1418.
Although some parts of them have deteriorated somewhat from age,
the hieroglyphic numerals are entirely recognisable. The tadpoles are all
facing left, and thus these numerical expressions are read from left to right
(see Fig. 14.11 above). In Fig. 14.17, the number 200,000 has been written
along the line, unlike example B in figure 14.18, where the two tadpoles are
put one above the other. The thousands are represented by lotus flowers
connected at the base, a custom which disappeared by the end of the Old
Kingdom period.
2: End of the First Intermediate period (end of third millennium BCE),
from a tomb at Meir:
A
B
C
D
(MAfia
(rDtftftn)
0001
000
9VV
99V
mmt
mr
77
700
7,000
760,000
Fig. 14.19.
3: From the Annals of Thutmosis (1490-1436 BCE), a list of the plunder
of the twenty-ninth year of the Pharaoh's reign (see Fig. 14.21):
The numerals can be transcribed as:
276
4,622
Fig. 14.20.
167
HIEROGLYPHIC NUMERALS
Fig. 14 - 21 . Stone bas-relief from Karnak. Louvre
4: Numerical expression from the stela of Ptolemy V at Pithom, 282-
246 BCE:
Fk;. 14-22.
THE ORIGINS OF EGYPTIAN NUMERALS
The numerical notation of Ancient Egypt was in essence a written-down
trace of a concrete enumeration method that was probably used in earlier
periods. The method was to represent any given number by setting out in a
line or piling up into a heap the corresponding number of standard objects
or tokens (pebbles, shells, pellets, sticks, discs, rings, etc.), each of which
was associated with a unit of a given order of magnitude.
UNITS
TENS
HUNDREDS
THOUSANDS
TENS OF
THOUSANDS
HUNDREDS OF
THOUSANDS
1
D
ft
s
i
I
I
A>
2
OO
ft
ft
99
2!
II
«
A>je
3
DOG
ftft
ft
999
13
in
IIJ
4
0110
m
ftn
9999
nr
iro
li
jeje
5
DO Q
DO
ftfl
999
99
*
HI
n
6
D D 0
ODD
raw
nnn
999
999
111
ifi
i
HI
A&BJB
7
0000
000
mwt
nm
9999
999
nn
111
HU
m
m
i
8
0000
0000
nrem
ravin
9999
9999
131
88
m
m
9
00 0
00 0
0 0 0
non
nnn
nrvi
HI
ill
111 1
1
in
in
nr
J&Q4!
Fig. 14 . 23 . Hieroglyphic representations of the consecutive units in each decimal order
THE NUMBERS OF ANCIENT EGYPT
168
Unlike Sumerian numerals, however, the hieroglyphs give no clue as to
the nature of the tokens used in concrete reckoning prior to the invention
of writing. It seems pretty unlikely that lotus flowers (1,000) or tadpoles
(100,000), were ever practical counting tokens at any period of time. The
spiral, the finger, and the kneeling man with upraised arms pose just as
awkward and still unanswered questions.
It seems most likely to me that the origins of Egyptian numerals are
much more complex than the origins of the written numbers of Sumer and
Elam, and that their inventors used not one but several different principles
at the same time. What follows are no more than plausible hypotheses
about the origins of hieroglyphic numerals, unconfirmed by any hard
evidence.
The origin of the numeral 1 could have been “natural” - the vertical line
is just about the most elementary symbol that humans have ever invented
for representing a single object. It was used by prehistoric peoples from
over 30,000 years ago when they scored notches on bone, and as we have
seen a whole multitude of different civilisations have given the line or notch
the same unitary value over the ages.
In addition, the line (for 1) and the horseshoe (for 10) could well be the
last traces in hieroglyphic numerals of the archaic system of concrete
numeration. The line could have stood for the little sticks used with a value
of 1, and the horseshoe might in fact have been at the start a drawing of the
piece of string with which bundles of ten sticks were tied to make a unit of
the next order.
As for the spiral and the lotus, they most probably arose through
phonetic borrowing. We could imagine that the original Egyptian words
for “hundred” and “thousand” were complete or partial homophones of
the words for “lotus” and “spiral"; and that to represent the numbers, the
Egyptians used the pictograms which represented words which had exactly
or approximately the same sound, irrespective of their semantic meaning,
as they did for many other words in their language and writing.
Parallels for such procedures exist in many other civilisations. In classical
Chinese writing, for instance, the numeral 1,000 was written with the same
character as the word “man”, because “man” and “thousand” are reckoned
to have had the same pronunciation in the archaic form of the language.
On the other hand, the Egyptian hieroglyph for 10,000, the slightly bent
raised finger, seems to be a reminiscence of the old system of finger-
counting which the Egyptians probably used. The system relies on various
finger positions to make tallies up to 9,999.
The hieroglyphic sign for 100,000 may derive from a more strictly
symbolic kind of thinking: the myriads of tadpoles in the waters of the Nile,
the vast multiplication of ffogspawn in the spring . . .
The hieroglyphic numeral for 1,000,000 might more plausibly be
ascribed a psychological origin. The Egyptologists who first interpreted this
sign thought that it expressed the awe of a man confronted with such a
large number. In fact, later research showed that the sign (which also means
“a million years” and hence “eternity”) represented in the eyes of the
Ancient Egyptians a genie holding up the vault of heaven. The pictograms
distant origin lies perhaps in some priest or astronomer looking up to the
night sky and taking stock of the vast multitude of its stars.
SPOKEN NUMBERS IN ANCIENT EGYPTIAN
The spoken numbers of Egyptian have been reconstructed from its modern
descendant, Coptic, together with the phonetic transcriptions of numerical
expressions found in hieroglyphic texts on the pyramids. Here are their
syllabic transcriptions with their approximate phonetic realisations:
1
w‘
[wa‘]
10
md
[medj]
2
snw
[senu]
20
dwty
[dwetye]
3
khmt
[khemet]
30
m‘b’
[m'aba’]
4
fdw
[fedu]
40
khm
[khem]
5
diw
[diwu]
50
diyw
[diyu]
6
srsw
[sersu]
60
si
7
sfkh
[sefekh]
70
sfkh
[sefekh]
8
khmn
[khemen]
80
khmn
[khemen]
9
psd
[pesedj]
90
psd
[pesedj]
St
[shet]
kh’ [kha’] db‘ [djebe‘]
hfn [hefen]
hh [heh]
100
1,000 10,000
100,000
1,000,000
Note that 7, 8, and 9 have the same consonantal structure as 70, 80, and 90
respectively. The Egyptians may well have pronounced them slightly differ-
ently in order to avoid confusion: for instance, sefekh for 7 and sefakh for 70,
khemen for 8 and kheman for 80, etc.
The spoken numerals, as can be seen, were strictly decimal. Compound
numbers were expressed along the lines of the following example:
4,326:
fdw kh khmt sht dwty srsw
“four thousand three hundred twenty six”
FRACTIONS AND THE DISMEMBERED GOD
Fractions were mostly expressed in Ancient Egyptian writing by placing the
hieroglyph “mouth”, pronounced eR and having in this context the specific
169
sense of “part", over the numerical expression of the denominator, thus:
till l
3 5 6 10 100
Fig. 14.24.
When the denominator was too large to go entirely beneath the eR sign,
the remainder of it was placed to the right, thus:
QUO (n)(n) <<==£>
oSo nffl 99
249
Fig. 14.25.
There were special signs for some fractions:
VALUE MEANING
2
3
3
4
“the two parts"
“the three parts
Fig. 14.26.
Save for the last two expressions in Fig. 14.26, the only numerator used
in Egyptian fractions was the unit. So to express (for instance) the equiva-
lent of what we write as 5, they did not write | + 1 + 1 but decomposed the
number into a sum of fractions with numerator 1.
1 + -L = 5 <»><*,<*, = I + I + I = 47
(fi) 2 10 5 Vf im m 3 4 5 60
Fig. 14.27.
Measures of volume (dry and liquid) had their own curious system of
notation which gave fractions of the heqat, generally reckoned to have been
equivalent to 4.785 litres. These volumetric signs used “fractions” of the
hieroglyph representing the painted eye of the falcon-god Horus:
Fig. 14.28.
FRACTIONS AND THE DISMEMBERED GOD
The name of Horus ’s eye was oudjat, written phonetically in hieroglyphs
as follows:
The oudjat was simultaneously a human and a falcon’s eye, and thus
contained both parts of the cornea, the iris and the eyebrow of the human
eye, as well as the two coloured flashes beneath the eye characteristic of the
falcon. Since the most common fractions of the heqat were the half, the
quarter, the eighth, the sixteenth, the thirty-second and the sixty-fourth,
the notation of volumetric fractions attributed to each of the parts or
strokes in the oudjat sign the value of one of these fractions, as laid out in
Fig. 14.30 below.
Fig. 14.30. The fractions of the h eqa t
Horus was the son of Isis and Osiris, the god murdered and cut up into
thirteen pieces by his brother Seth. When he grew up, Horus devoted
himself to avenging his father, and his battles with his uncle Seth were long
and bloody. In one of these combats, Seth ripped out Horus’s eye, tore it
into six pieces and dispersed the pieces around Egypt. Horus gave as good
as he got, and castrated Seth. In the end, according to legend, the assembly
of the gods intervened and put a stop to the fighting. Horus became king of
Egypt and then the tutelary god of the Pharaohs, the guarantor of the legit-
imacy of the throne. Seth became the cursed god of the Barbarians and the
Lord of Evil. The assembly of the gods instructed Thot, the god of learning
and magic, to find and to reassemble Horus's eye and to make it healthy
again. The oudjat thus became a talisman symbolising the wholeness
of the body, physical health, clear vision, abundance and fertility; and so
the scribes (whose tutelary god was Thot) used the oudjat to symbolise the
THE NUMBERS OF ANCIENT EGYPT
170
fractions of the heqat, specifically for measures of grain and of liquids.
An apprentice scribe one day observed to his master that the total of the
fractions of the oudjat came to less than 1:
1111 1 1 63
— + — + — + + + —
2 4 8 16 32 64 64
His master replied that the missing i would be made up by Thot to any
scribe who sought and accepted his protection.
HIERATIC SCRIPT AND CURSIVE NUMERALS
IN ANCIENT EGYPT
With its minutely complex and decorative signs, the hieroglyphic system of
writing words and numbers was only really suitable for memorial inscrip-
tions, and was used mainly, if not quite exclusively, on stone monuments
such as tombs, funeral stelae, obelisks, palace and temple walls, etc. When
Ancient Egyptians needed to note down or record accounts, censuses,
inventories, reports, or wills, for example, or when they penned adminis-
trative, legal, economic, literary, magical, mathematical, or astronomical
works, they had far more frequent recourse to a script that was easier to
handle at speed, namely hieratic script.
Hieratic script uses signs that are simplifications and schematisations of
the corresponding hieroglyphs, with fewer details and with shapes reduced
to skeleton forms. In some cases, the hieratic versions can be recognised as
variants of the original sign; but most often the relationship between the
“cursive” and the “monumental” form is impossible to guess and has to be
learned sign by sign.
OLD
KINO DOM
middle
KINGDOM
NEW
KINGDOM
OLD
KINGDOM
MIDDLE
KINGDOM
NEW
KINGDOM
X
8
X
%
s
r
£
k
jrti
?
S'
JVL
A
?
A
A
?
-A
11
t
A
>
%
1C'
fT
!
\
%
f
S3?
?
?
?
?
A
2
**
*»
P
t
t
V
A
4
j.
1
1
111
)
Fig. 14.31. Some hieroglyphs and their hieratic equivalents
There were also hieratic versions of the hieroglyphic numerals. These are
the numerical signs found in the Harris Papyrus (British Museum), dating
from the Twentieth Dynasty, which gives the possessions of the temples at
the death of Ramses III (1192-1153 BCE):
1 1
10 A
100
1,000
u
2 u
20 A
200
V
2,000
3 U|
30 H
300
3,000
4 ni|
4 U,
400
]»f
4,000
5 1
50 ^
500
UJ
5,000
6 l
60 m
600
6.000
70 *
700
3
7.000
8 SJ
80 ft
800
'$
8,000
dK
9 V
90 a
900
9,000
A
Fig. 14.32.
As can be seen, the hieratic numerals are for the most part visually quite
unrelated to their equivalent hieroglyphs. Although the signs for the first
four units are fairly self-explanatory ideograms, all the other numerals seem
quite devoid of visually intuitive meaning.
So do hieratic numerals constitute a genuinely independent numbering
system? Should we consider the numerals found in the Harris Papyrus as
an arbitrary shorthand, invented by scribes for jotting down numbers
intended to be written quite differently on stone monuments?
In fact, hieratic numerals, like the syllabic signs of this script, are
developed from the corresponding hieroglyphs, and do not constitute an
independent system. However, the changes in the shapes of the signs were
very considerable, imposed in part by the characteristics of the reed-
brushes used for hieratic characters (which, unlike hieroglyphs, were
always written from right to left) and in part by a tendency to use ligatures,
that is to say to run several signs together to produce single compounds.
That is why the groups of five, six, seven, eight, and nine vertical lines
became single signs devoid of any intuitive meaning:
171
Fig. 14 33-
The relationship between hieratic numerals and hieroglyphs is difficult
to see, but it was probably no more difficult for an Ancient Egyptian than it
is for us to see the equivalence between the following ways of writing our
own letters:
ABCDEFKRS
sf & # g> s jr w J
a 6 c </ c f* /r. x 1
Imagine how hard it would be for a speaker of Chinese or Arabic, for
example, with no knowledge of the Latin alphabet, to work out that the
signs on the second and third lines have exactly the same value as the signs
in the corresponding position on the first line!
Fig. 14.34. Detail from the Rhind Mathematical Papyrus (RMP), an important mathematical
document written in hieratic script. From the Hyksos (Shepherd Kings) period (c. seventeenth
century BCE), the RMP is a copy of an earlier document probably going back to the Twelfth
Dynasty ( 1991-1786 BCE). The RMP is in the British Museum.
HIERATIC SCRIPT AND CURSIVE NUMERALS
Hieratic script was therefore not a form of "shorthand”, in the sense that
modern shorthand consists of purely arbitrary signs visually unrelated to
the letters of the alphabet which they represent. Hieratic signs were indeed
derived from hieroglyphs and represent the terminus of a long but specifi-
cally graphical evolution. Hieratic script never replaced the monumental
script used for inscriptions on stone, and never had much impact on the
shape of the hieroglyphs. The two systems were used in parallel for nearly
2,000 years, from the third to the first millennium BCE, and throughout
this period hieratic script, despite its apparent difficulty, provided a
perfectly serviceable tool for all administrative, legal, educational, magical,
literary, scientific, and private purposes.
Hieratic script was gradually displaced from about the twelfth century
BCE by a different cursive writing, called demotic. It survived in specific
uses - notably in religious texts and in sacred funeral books - until the
third century CE, which is why the Greeks called it hieratikos, meaning
"sacred”, whence our term “hieratic”.
FROM HIEROGLYPHIC TO HIERATIC NUMERALS
Hieratic numerals of the third millennium BCE are still fairly close to their
hieroglyphic models; but over the centuries, the use of ligatures and the
introduction of diacritics turn them little by little into apparently quite
different signs with no intuitive resemblance to the original hieroglyphs.
The end result was a set of numerals with distinctive signs for each of the
following numbers:
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
200
300
400
500
600
700
800
900
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
So though they began with a very basic additive numeration, the Egyptians
developed a rapid notation system that was quite strikingly simple, requir-
ing (for example) only four signs to represent the number 3,577, whereas in
hieroglyphs it takes no fewer than 22 signs:
HIEROGLYPHIC NOTATION HIERATIC NOTATION
z w
7 70 500 3,000 7 70 500 3,000
« *
3,577 3,577
Fig. 14-35-
THE NUMBERS OF ANCIENT EGYPT
172
The main disadvantage of the hieratic system was of course that it
required its users to memorise a very large number of distinct signs, and
was thus quite impenetrable to all but the initiate. Here are the shapes that
a hieratic mathematician had to know as well as we know 1 to 9:
HIERATIC NUMERALS: UNITS
OLD
KINGDOM
MIDDLE
KINGDOM
SECOND
INTER-
MEDIATE
PERIOD
NEW
KINGDOM 1
(XVIIITIl &
XIXTH
DYNASTIES)
NEW
KINGDOM II
AND XXIST
DYNASTY
XXIIND
DYNASTY
1
0
1
1
1 f
1
1
1
1
l
1
1
1
1
1
1
f
1
I
2
00
H
M
N
tl
d
11
U
tl
»
11
11
tt
T
H
»I
1
<1
II
u
9
It
U
3
OOQ
<4
M
IU
<u
«i
01
MI
«!
Ul
m
111
“■!
m
a*
til
1
m
^jj
O
U|
N
w
Efl
B
—
«<f
*
n
V
1
V
m
□1
a
E
K
E
a
H
B
a
n
B
*1
1
B
E
U
D
|
*<«
.Ul
u\
B
O
El
11
a
m
B
a
tt
D
□
B
a
D
■Ml
111
a
□
£
E
ID
B
— %
a
B
2
a
IH
— *
8
00 00
0000
E3
B
B
B
3
II
|
a
2 *
■
■
□
H
■111
m\
m
a
a
a
BI
A
Cl
a
a
□
a
m
Bi
HIERATIC NUMERALS: TENS
OI.D
KINGDOM
MIDDLE
KINGDOM
SECOND
INTER-
MEDIATE
PERIOD
NEW
KINGDOM I
(XVIIITH &
XIXTH
DYNASTIES)
NEW
KINGDOM II
AND XXIST
DYNASTY
XXIIND
DYNASTY
10
ft
A
<1
4
A
A
A
A
A
A
/I ^
A
4
A
'I
A
A
A
\
A
A
20
ft
5D
*
a
*
e
A
A
*
A
A
A
*
A
X
ft
A
A
/I
A
A
30
(fSlTD
ra
%
'A
*
A
X
%
X
\
X
>1
A
A
A
A
>
-u.
S
—
>
-1.
t
B
5
JU
-X
-JU
1
1 '
1
A
1
1
11
B
“A
1
1
1
—
Ul
M.
m
JO.
n
Ji
jn.
JL
Ul
UL
4
70
flAn
fl
*
J!
_
A
\
*
*
K
*
VI
A
M
X
B
m
Ul
a.
H
ISL
1L
Jtfl
H
U4
m
3
n
H
a
m
1
4
&
%
Jjt-
A
&,
Fig. 14. 36a.
Fig. 14. 36B.
173
FROM HIEROGLYPHIC TO HIERATIC NUMERALS
HIERATIC NUMERALS: HUNDREDS
HIERATIC NUMERALS: THOUSANDS
OLD
KINGDOM
MIDDLE
KINGDOM
SECOND
INTER-
MEDIATE
PERIOD
NEW
KINGDOM I
(XVII1TH
& XIXTH
DYNASTIES)
NEW KINGDOM II
AND
XXIST DYNASTY
XX I IN D
DYNASTY
1,000
I" I
Jj
I
t
A
l
>
i
J
%
Aj
\
>
A
C
L
L
t
2,000
HU
or
*
_i
*
4
-4
«
A
Jis
C-
— r
A
-Jk
D
B
B
n
H,
— "k
v>
11
A* 3 *
-KJk
is
a
A
A
i
A
a -a
1
J0>
*
A
A -A
1
*
A
A A
Fig. 14.360.
THE NUMBERS OE ANCIENT EGYPT
DOING SUMS IN ANCIENT EGYPT
Let us imagine we’re at a farm near Memphis, in the autumn of the year
2000 BCE. The harvest is in, and an inspector is here to make an assessment
on which the annual tax will be calculated. So he orders some of the
farm workers to measure the grain by the bushel and to put it into sacks
of equal size.
This year’s harvest includes white wheat, einkorn, and barley. So as to
keep track of the different varieties of grain, the workers stack the white
wheat in rows of 12 sacks, the einkorn in rows of 15 sacks, and the barley
in rows of 19 sacks, and for each the total number of rows are respectively
128, 84, and 369.
When this is done, the inspector takes a piece of slate to use as a
“notepad” and starts to do some sums on it in hieroglyphic numerals. For
despite the primitive nature of their numerals, the Egyptians have known
for centuries how to do arithmetic with them.
Adding and subtracting are quite straightforward. To add up, all you do
is to place the numbers to be summed one above the other (or one along-
side the other), then to make mental groups of the identical symbols and to
replace each ten of one set of signs by one sign of the next higher decimal
order.
For instance, to add 1,729 and 696, you first place (as in Fig. 14.37
below) 1,729 above 696. You then make mental groupings respectively of
the vertical lines, the handles, the spirals, and the lotus flowers. By reduc-
ing them in packets of 10 to the sign of the next higher order, you get the
correct result of the addition:
It is also quite easy to multiply and to divide by 10 in Egyptian hiero-
glyphics: to multiply, you replace each sign in the given number by the
sign for the next higher order of decimal magnitude (or the next lower, for
division by 10). So to multiply 1,464 by 10 you take:
174
ii nnn 99 ?
ii nnn 99 i
4 60 400 1,000
Fig. 14.38.
and by following the regular procedure it becomes:
40 600 4,000 10,000
Fig. 14-39.
However, to multiply and to divide by any other factor, the Egyptians
went about it quite differently. They knew only their two times table, and
so they proceeded by a sequence of duplications.
To come back now to the tax-collector who needs to know the total
number of sacks of white wheat in this year’s harvest, and therefore needs
to multiply 12 by 128. He goes about it like this:
1 12
2 24
4 48
8 96
16 192
32 384
64 768
128 1,536
That is to say, he writes the multiplier 12 in the right-hand column of his
slate, and opposite it, in the left-hand column, he writes the number 1.
He then doubles each of the two numbers in successive rows until the
multiplier 128 appears on the left. As the number 1,536 appears on
the right in the row where the left column shows 128, this is the result
of the operation: 12 x 128 = 1,536.
To discover how many sacks of einkorn there are, he now has to multiply
84 by 15. His “doubling table” would look like this:
1 15
2 30
4 60
8 120
16 240
32 480
64 960
As the next doubling would take the multiplier beyond the required
figure of 84, he stops there, and looks down the left-hand column to see
175
DOING SUMS IN ANCIENT EGYPT
which of the multipliers entered would sum to 84. He finds that he can
reach 84 with just three of the multiplications already computed, and he
checks the left-hand column numbers by making a little mark next to them,
and putting an oblique stroke beside their right-hand column products,
thus:
1
15
2
30
-4
60/
8
120
-16
240/
32
480
-64
960/
He can then add up the numbers with the oblique check-mark and arrive at
the result:
84 x 15 = 960 + 240 + 60 = 1,260
To compute the number of sacks of barley, the inspector now has to
multiply 369 by 19. He goes about it in exactly the same way, putting 1 in
the left-hand column of his slate and 19 in the right-hand column, and then
doubling the two terms successively as he goes down the rows. He stops
when the left-hand column reaches 256, since the next step would give a
multiplier of 512, which is higher than the required figure of 369:
-1
19/
2
38
4
76
8
152
-16
304/
-32
608/
-64
1,216/
128
2,432
-256
4,864/
Then he looks down the left-hand column to find those numbers whose
sum is 369, finds that they are 256, 64, 32, 16, and 1, and thus adds up the
corresponding right-hand figures to arrive at his total:
369 x 19 = 4,864 + 1,216 + 608 + 304 + 19 = 7,011
So the harvest adds up to 1,536 sacks of white wheat, 1,260 sacks of
einkorn, and 7011 sacks of barley. And since the Pharaoh’s share of that is
one tenth, the inspector can easily calculate the tax payable as 153 sacks of
white wheat, 126 sacks of einkorn, and 701 sacks of barley.
So multiplication in the Egyptian manner is really quite simple and can
be done without any multiplication tables other than the table of 2.
Division is done similarly by successive duplication, but in reverse, as we
shall see.
Let us suppose that in the time of Ramses II (1290-1224 BCE) robbers
have just stripped the tomb of one of the Pharaohs of the preced-
ing dynasty. They have stolen diadems, ear-rings, daggers, breast-plates,
pendants - a whole mass of precious jewellery decorated with gold leaf and
glass beads. Altogether there are 1,476 items in the robbers’ haul, and the
leader of the gang proposes to divide them equally amongst his eleven men
and himself. So he has to divide 1,476 by 12. He goes about it just as if
he were doing a multiplication, putting 12 in the right-hand column, and
stopping when the right-hand figure reaches 768 since the next step would
take the sequence beyond the total number of items to be shared:
/I
12-
/2
24-
4
48
/8
96-
/16
192-
/32
384-
/64
768-
He now has to find which of the numbers in the right-hand column total
1,476 and after various attempts to make the total he finds that 768, 384,
192, 96, 24, and 12 come out exactly right. So he makes a little mark against
these figures in the right-hand column and puts an oblique against their
corresponding numbers in the left-hand column. So he can now add up the
checked numbers on the left to come out with the exact answer to the ques-
tion: how many twelves go into 1,476?
1,476/12 = 64 + 32 + 16 + 8 + 2 + 1 = 123
So each of the robbers takes 123 pieces from the haul, and off they go with
their fair shares.
This method of division only works when there is no remainder; where
the dividend is not a multiple of the divisor, the Egyptians had a much more
complicated method involving the use of fractions, which will not be
explained here.*
The arithmetical methods of Pharaonic Egypt did not therefore require
any great powers of memorisation, since, to multiply and to divide, all
that you needed to know by heart was your two times table. Compared
*The method is explained in Richard J. Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA:
MIT Press, 1972).
THE NUMBERS OF ANCIENT EGYPT
176
to modern arithmetic, however, Egyptian procedures were slow and very
cumbersome.
Fig. 14 . 40 . The Egyptian Mathematical Leather Roll (known as EMLR) in the British Museum.
It contains, in hieratic notation, and in duplicate, twenty -six additions done in unit fractions and
was probably used as a conversion table , / See Gillings (1972), pp. 89-103 J
ANCIENT EGYPTIAN NUMBER-PUZZLES
Egyptian carvers, especially in the later periods, indulged in all sorts of
puns and learned word-games, most notably in the inscriptions on the
temples of Edfu and Dendara. Some of these word-games involve the names
of the numbers, and the following tables (based on the work of P. Barguet,
H. W. Fairman, J. C. Goyon, and C. de Wit) give a small sample of the innu-
merable curious scribal inventions for the representation of the numbers in
hieroglyphs. The references are to Chassinat’s transcription of the inscrip-
tions on the walls of the temples of Edfu (“E”) and Dendara (“D").
VALUE
SIGN & MEANING
EXPLANATION
REFERENCE
Homophony: ‘'one’’ and
1
harpoon
“harpoon” are both pronounced wa'
E.VII, 18, 10
O
1
sun
Because there is only one sun
E.IV, 6 , 4
1
nO/
moon
Because there is only one moon
E.IV, 6 , 4
«>
Only used in the expression “one
1
fraction 1/30
day" or "the first day”: 1/30 of a
E.IV, 8 , 4;
month is 1 day
E.IV, 7, 1
VALUE
SIGN & MEANING
EXPLANATION
REFERENCE
2
Two X harpoon = 2x1
E.IV, 14, 4
2
0
'O
Sun + moon = 1 + 1
E.VI, 7, 5
3
Three x harpoon = 3x1
E.VII, 248, 10
4
m
jubilaeum
No known explanation
E.IV, 6 , 5;
E.IV, 6 , 6 ;
E.VII, 15, 1
5
5-pointed star
Self-evident
E.IV, 6 , 3;
E.IV, 6 , 5;
E.VII, 6 , 4
6
\iz
Standard sign for 1 + star = 1 + 5
E.IV, 5, 4
7
9k
human head
The head has seven orifices:
two eyes, two nostrils, two ears,
mouth
E.IV, 4, 4;
E.V, 305, 1
7
Standard sign for 2 + star = 2 + 5
E.IV, 6 , 5
7
itm nfiiia
1 1
5 + 30
Only in the expression “seven days”:
1/5 of a month = 6 days + 1/30
= 1 day
E.IV, 8 , 4;
E.IV, 7, 1
8
ibis
The sacred ibis was the incarnation
of the god Thot, the principal
divinity of the city of Hermopolis,
formerly Khmnw or Khemenu,
meaning "the city of eight"
E.III, 77, 17;
E.VII, 13, 4;
E. VII, 14, 2
8
m
A curious "re-formation” in
hieroglyphics of the hieratic
numeral 8
E.VI, 92, 13
8
*
01
Standard notation of 3 + star
= 3 + 5 = 8
E.IV, 5, 2
8
O®
Moon + head = 1 + 7 = 8
E.IV, 6 . 4
8
Standard notation of 1 + head
= 1 + 7 = 8
E.IV, 9, 3
177
VALUE
SIGN & MEANING
EXPLANATION
REFERENCE
9
A
Homophony: “nine” and “shine” are
both pronounced psd
E.IV, 8, 2;
E. VII, 8, 8
9
scythe
Based on the fact that in hieratic
the numeral 9 and the sign for
scythe were identical
E. VII, 15, 3;
E.VI1, 15, 9;
E. VII, 17, 3
9
S*
Standard notation of 4 + star
=4+5=9
E.IV, 6, 1
9
II &
Standard notation of 2 + head
=2+7=9
D.II, 47, 3
10
A
falcon
The falcon-god Horus was the first
to be added to the original nine
divinities of Heliopolis, and thus
represents 10
E.V, 6, 5
14
0M
falcon + jubilaeum = 10 + 4 = 14
E.V, 6, 5
ANCIENT EGYPTIAN NUMBER-PUZZLES
VALUE
SIGN & MEANING
EXPLANATION
REFERENCE
15
fraction l /i
Only in the expression “15 days”
or “fortnight”: 1/2 of a month
= 15 days
E. VII, 7, 6
17
Standard notation of 10 + head
= 10 + 7 = 17
E.VII, 248, 9
18
1 1
2 + 10
Only in the expressions “18 days”
or “the 18th day”: 1/2 month +
1/10 month = 15 + 3 = 18
E.IV, 9, 1;
E.VII, 7, 6;
E.VII, 6, 1
19
Standard notation of 10 + scythe
= 10 + 9 = 19
E.VII, 248, 4
20
AA
Two falcons = 2 x 10 = 20
E.VII, 11, 8
107
Standard notation of 100 + head
= 100 + 7 = 107
E.VII, 248, 11
COUNTING IN THE TIMES OF THE CRETAN AND HITTITE KINGS
178
CHAPTER 15
COUNTING IN THE TIMES OF
THE CRETAN AND
HITTITE KINGS
THE NUMBERS OF CRETE
Between 2200 and 1400 BCE, the island of Crete was the centre of a very
advanced culture: Minoan civilisation, as it is called, from the name of the
legendary priest-king Minos who, according to Greek mythology, was one
of the first rulers of Knossos, the ancient Cretan capital near the modern
port of Heraklion (Candia).
The very existence of Minoan civilisation was almost completely
unknown until the end of the last century, and it is only relatively recently
that archaeologists have uncovered a brilliant and original culture which
was, in many respects, the precursor of Greek civilisation.
When Minoan civilisation fell, around 1400 BCE, probably as a result of
some natural disaster or of the invasion of the island by the Mycenaeans
(of Greek origin), it disappeared almost without trace save for what was
preserved in the fables and legends of the Ancient Greeks.
We owe the most spectacular discoveries - such as the famous Palace of
Knossos - to the indefatigable enthusiasm and energy of the British archae-
ologist Sir Arthur Evans (1851-1941). He was the first to show that the
Greek legends had a historical basis, and constituted a living trace of one of
the oldest known European civilisations.
Since the end of the last century, archaeological investigations carried
out mainly on the sites of Knossos and Mallia have brought to light a large
number of documents whose analysis has revealed the existence of a
“hieroglyphic” script between 2000 and 1660 BCE.
Cretan hieroglyphics have still not been deciphered, and these documents
remain enigmatic. Nevertheless they show evidence of an accounting system
adapted to a “bureaucracy” no doubt born within the earliest palaces of
Minoan civilisation. In proof of this we find clay blocks and tablets covered
with figures and hieroglyphic signs, which are more or less schematic draw-
ings of all kinds of objects. These appear to be accounts giving details of
inventories, supplies, deliveries, or exchanges. The purpose of the symbols
was to note down the quantities of the different kinds of goods.
The numerical notation of Crete was strictly decimal, and was based on
the additive principle. Unity was represented by a short slightly oblique
stroke, or by a small circular arc which could be oriented anyhow. Cretan
FACE II FACE IV
Fig. 15.x. Inscriptions on bars of clay, showing Cretan hieroglyphic signs and numerals. Palace of
Knossos, 2000-1500 BCE. (Evans (1909), Doc. P 100]
hieroglyphic writing went from left to right and from right to left, in
boustrophedon (as a ploughman ploughs a field from side to side). 10 was
represented by a circle (or, on clay, by a small circular imprint as would be
made by the pressure of a round-tipped stylus held perpendicularly to the
surface of the clay). 100 was represented by a large oblique bar (distinctly
different from the small stroke of unity), and 1,000 was represented by a
kind of lozenge.
| or > « / or\
1 10 100 1,000
Fig. 15.2. Cretan hieroglyphic figures
With these as starting points, the Cretans represented other numbers by
repeating each one as many times as required. The hieroglyphic figures
were not, however, the only forms used. Other excavations have revealed a
second script, no doubt derived from the hieroglyphic, in which the picture
symbols give way to schematic drawings which, often, we cannot identify
now. Analysis of these documents led Evans to distinguish two variants of
this kind of writing, which he called “Linear A” and “Linear B”.
The system known as “Linear A” is the older. It was in use from the start
of the second millennium BCE up to around 1400 BCE, that is to say at
about the same time as the hieroglyphic script.
The sites which have yielded documents in Linear A are several,
notably Haghia Triada, Mallia, Phaestos, and Knossos. From Haghia
Triada we have a large collection of accounting tablets, unfortunately
in a somewhat sloppy script [Fig. 15.4]. These are, therefore, invento-
ries, with ideograms and numbers; the tablets are in the format of
small pages. But Linear A can be found as well on a wide variety of
objects: vases (with inscriptions cut, painted, or written in ink), seals,
179
THE NUMBERS OF CRETE
stamps, and labels of clay; ritual objects (libation tables); large copper
ingots; and so on. This writing may therefore be very widely found, not
only in administrative environments but also in holy places and prob-
ably also in people’s homes. [O. Masson (1963)]
^ man
Yf “
/V\ M mountain
shi P
|| tree
eye
::r
jf S° at
p ,„,
$ $ Whe3t
» T grain
p'° u g h
/f crescent
(A. moon
OfO op
IIAAJ1 LLL
J^] \ palace
bee
M tfy* crossed
2a A arms
Fig. 15.3. A selection if Cretan hieroglyphics /after Evans]
Fig. 15 . 4 . Cretan tablet with signs and
numerals from the “Linear A" script. Haghia
Triada, sixteenth century BCE. [GORII.A
(1976), HT 13, p. 26}
The script known as “Linear B” is the
more recent, and the best known, of
the Cretan scripts. It is usually dated
to the period between 1350 BCE and
1200 BCE. At this time, the Mycenaeans
had conquered Crete, and ancient
Minoan civilisation had spread onto
the Greek mainland, especially in the
region of Mycenae and Tyrinth.
The signs of this script were en-
graved on clay tablets, which were first
unearthed in 1900. Since then, 5,000
tablets have been found in Crete (at
Knossos only, but in large numbers)
and on mainland Greece (mainly at
Pylos and Mycenae). Linear B, there-
fore, may be found outside Crete. We
may also note that this script, apparently derived by modification of
Linear A, was used to record an archaic Greek dialect, as demonstrated by
Michael Ventris, the English scholar who first deciphered it. It is the only
Creto-Minoan script to have been deciphered to date (Linear A and the
hieroglyphic script correspond to a language which still remains largely
unknown).
A
Fig. 15.5. Cretan tablets
with signs and numerals from
the “Linear B“ script, fourteenth
or thirteenth century BCE.
[Evans and Myrcs (1952)1
Both Linear A and Linear B used practically the same number-signs
(Fig. 15.6). These were:
• a vertical stroke for unity;
• a horizontal stroke (or, solely in Linear A, sometimes a small circu-
lar imprint) for 10;
• a circle for 100;
• a circular figure with excrescences for 1,000;
• the same, with a small horizontal stroke inside, for 10,000 (found
only in Linear B inscriptions: Fig. 15.6, last line).
Fig. 15.6.
Cretan numerals
i
10
too
1,000
10,000
Hieroglyphic system
c. 2000 to c. 1500 BCE
/
>
<
w
•
/
\
0
?
“Linear A” system
c. 1900 to c. 1400 BCE
1
•
0
0
?
“Linear B’’ system
c. 1350 to c. 1200 BCE
l
—
0
0
0
COUNTING IN THE TIMES OF THE CRETAN AND H ITT 1 T F, KINGS
Fig. 15.7. The principle of the Cretan numerals
To represent a given number, it was enough to repeat each of the above
as many times as needed (Fig. 15.7).
The number-systems used in Crete in the second millennium BCE
(hieroglyphic, Linear A, and Linear B) had, therefore, exactly the same
intellectual basis as the Egyptian hieroglyphic notation and, for the whole
time they were in use, underwent no modification of principle. (Similarly,
the drawing of signs and numbers on clay did not give way to a cuneiform
system, as happened in Mesopotamia). As in the monumental Egyptian
system, these number-systems were founded on base 10 and used the prin-
ciple of juxtaposition to represent addition. Moreover, the only numbers
to which each system gave a special sign were unity, and the successive
powers of 10.
The number 10,000 (found only in Linear B inscriptions) is derived from
the number 1,000 by adding a horizontal bar in the interior of the latter. By
all appearances, therefore, a multiplicative principle has been used (10,000
= 1,000 x 10), since the horizontal bar is simply the symbol for 10 in this
system (Fig. 15.6).
THE HITTITE HIEROGLYPHIC N U M B E R- S Y S T E M
From the beginning of the second millennium BCE the Hittites (a people of
Indo-European origin) settled progressively in Asia Minor, no doubt by a
process of slow immigration. Between the eighteenth and the sixteenth
180
centuries BCE, they there established a great imperial power of which there
were two principal phases: the Ancient Empire (pre-1600 to around 1450
BCE) and the New Empire (1450-1200 BCE).
In the course of the imperial era, the Hittites, with many successes and
failures, pursued a policy of conquest in central Anatolia and northern
Syria. But at the start of the thirteenth century BCE, no doubt under attack
from the “Peoples of the Sea”, this powerful empire abruptly collapsed.
A renaissance, however, ensued from the ninth century BCE in the north
of Syria where several small Hittite states maintained elements of the impe-
rial tradition in the midst of mixed populations. This was the beginning
of what is called the “neo-Hittite” phase of the civilisation. Finally, however,
in the seventh century BCE, all these small states were absorbed by the
Assyrian Empire.
The Hittites had two writing systems. One was a hieroglyphic system
which seems to have been of their own creation, of which the earliest known
evidence is from the fifteenth century BCE. The other was a cuneiform
system borrowed from Assyro-Babylonian civilisation whose introduction
dates from around the seventeenth century BCE.*
rWI
horse
EH house
eatin g
2^ donkey
(3ED god
drinking
{^5) ram
P? cart
J| kin s
dL, bad
mountain
Ml
^ face
j|j|| tower
^ town
anger
£& wail
this
Fig. 15.8. The meanings of some of the Hittite hieroglyphics [after Laroche ( 1 960 ) ]
Thus, for at least three centuries (1500-1200 BCE) the hieroglyphic
lived alongside the cuneiform in Anatolia, and they constituted the
dual medium of expression of the Hittite state. For a people to practise
* The cuneiform system, of Assyro-Babylonian origin, was adapted into at least three Hittite dialects:
Nesitic, spoken in the capital of the empire: Louvitic, employed in southern Anatolia, and Palaitic in
the north. Cuneiform characters were used for the numerous tablets making up the royal archives of the
town of Hattusa, capital of the Hittite Empire, at the place which is now Bogazkoy in Turkey, about
150 km east of Ankara: thanks to these documents, the history and language of the Hittites have been
partially reconstructed.
181
THE HITTITF. HIEROGLYPHIC N U M B E R- S Y S T F. M
two writing systems at the same time is not a frequent phenomenon.
We are now able to perceive the reasons which induced the Hittites
into this paradoxical situation. The scribes of Hattusa, who were the
keepers of the Babylonian tradition, were a small and privileged group
who had sole access to their literature and to the documents on clay.
The establishment of a library answered a need, and the use of the
cuneiform ensured that the kingdom could maintain communication
with its representatives abroad. But the tablet was, in effect, a banned
document: it made no public proclamation of the sublimity of the
god, nor of the grandeur of the king. Without doubt the Hittites felt
that these imprinted cuneiform characters, mechanical and lacking
expression, should take second place to a different writing more visual,
more monumental, more apt for writing of divine effigies and royal
profiles. . . . The hieroglyphs are made to be gazed upon, and contem-
plated upon walls of rock: they give life to a name just as a relief brings
the whole person to life. [E. Laroche (I960)]
All the same, hieroglyphic writing survived the cuneiform after the
destruction of the Hittite Empire around 1200 BCE. It served not only for
religious and dedicatory purposes, but also, and perhaps above all, for lay
purposes in business documents.
BASE NUMBERS
| X X or 4 ° r C ° r <?.
1 10 100 1,000
Examples from lead plates of
the neo-Hittite era (eighth century BCE)
discovered at Kululu [Ozgiif (1971)]
72 A P^* ^
f 13
66 § P‘ LI
II! 11
150 JL pL L
a
80
120 * P' :i L
X
^ pi. LII
X 1.2
600 y
%
A
141 a plu
i 13
200 y p'; 2 l
400 X* pLL
From an inscription of the thirteenth JJ.LL XSSHt , r
century BCE [Hrozny (1939)]
4,400
In the Hittite hieroglyphic number-system, a vertical stroke represented
unity. For the successive integers, small groups of two, three, four or five
strokes were used to allow the eye to grasp the total sum of the units. The
number 10 was represented by a horizontal stroke, a 100 by a kind of
Saint Andrew’s cross, and 1,000 by a sign which looked like a fish-hook
(Figure 15.9). On this basis, the representation of intermediate numbers
presented no difficulty, since it was sufficient to repeat each sign as many
times as required.
The Hittite hieroglyphic number-system was, after the fashion of the
Egyptian, strictly decimal and additive, since the only numbers to have
specific signs were unity and the successive powers of 10.
Fig. 15 . 9 . The Hittite hieroglyphic number-system
GREEK AND ROMAN NUMERALS
182
CHAPTER 16
GREEK AND ROMAN NUMERALS
THE GREEK ACROPHONIC N U M B E R- S Y S T E M
Let us now visit the world of the Ancient Greeks, and look at the number-
systems used in the monumental inscriptions of the first millennium BCE.
The Attic system, which was used by the Athenians, assigns a specific
sign to each of the numbers
The signs for the numbers 50, 500, 5,000, and 50,000 are, as can be seen,
made up by combining the preceding signs according to the multiplicative
principle:
50
f 1 . n.a
5 x 10
500
P * P. H
5 x 100
5,000
P = P. X
5 x 1,000
50,000
P , P.M
5 x 10,000
1 5 10 50 100 500 1,000 5,000 10,000 50,000
and is based above all on the additive principle (Fig. 16.1).
1 1
100 H
10,000 M
2 ll
200 HH
20.000 MM
3 ill
300 HHH
30,000 MMM
4 1111
400 HHHH
40.000 MMMM
5 r
500 01
50,000 P
6 n
600 PH
60,000 P M
7 pii
700 PHH
70,000 PMM
8 Pill
800 phhh
80,000 PMMM
9 Pllll
900 PHHHH
90,000 p MMMM
10 A
1,000 x
20 AA
2,000 XX
30 AAA
3,000 xxx
40 AAAA
4,000 xxxx
50 r
5.000 p
60 PA
6,000 PX
70 PAA
7,000 PXX
80 PAAA
8,000 ^xxx
90 PAAAA
9,000 FXXXX
Fig. 16. i. System of numerical annotation found in Attic inscriptions from around the fifth
century BCE until the start of the Common Era. [Franz (1840); Guarducci (1967); Guild;
Gundermann (1899); Larfeld (1902-7); Reinach (1885); Tod]
The Attic system has an interesting feature: with the exception of the
vertical bar representing 1, the figures are simply the initial letters of
the Greek names of the corresponding number, or are combinations of
these: this is what is meant by an acrophonic number-system.
To show this:
THE SIGN
WHICH IS THE
SAME AS
THE LETTER
WHOSE
VALUE IS
IS THE FIRST
LETTER OF
THE WORD
WHICH IS THE
GREEK NAME OF
THE NUMBER
r
PI (the archaic
form of the
letter El
5
rievTe (Pente)
Five
4
DELTA
10
Aexa (Deka)
Ten
H
ETA
100
Hckchtov (Hekaton)
Hundred
X
KHI
1.000
XiAloc (Khilioi)
Thousand
M
MU
10,000
MiipiOL (Murioi)
Ten thousand
Fig. 16.2.
In other words, in the Attic system, in order to multiply the value of one
of the alphabetic numerals A, H, X and M by 5, it is placed inside the
letter T = 5.
This system, which in fact only recorded cardinal numbers, was used
in metrology (to record weights, measure, etc.) and for sums of money.
We shall later see it used for the Greek abacus.
Originally, ordinal numbers were spelled out in full, but from the fourth
century BCE (probably, indeed, from the fifth) a different system was used
to write these numbers, which we shall study later.
To write down a sum expressed in drachmas, the Athenians made use of
these figures, repeating each one as often as required to add up to the quan-
tity; each occurrence of the vertical bar for “1” was replaced by the symbol
(- which stood for "drachma”:
XXX P H AAA l-H-
3,000 500 100 30 3
->
3,633 drachmas Fig. 16 . 4 .
For multiples of the talent, which was worth 6,000 drachmas, they used
the same number of signs but with T (the first letter of TALANTON)
instead of
pi FTTT
500 50 40 5 3
598 talents Fig. 16 . 5 .
For divisions of the drachma (the obol, the half- and the quarter-oio/, and
the chalkos ) special signs were used:
1 CHALKOS
(or 1/8 of an obol)
X
O or T
X: initial letter of
XAAKOYS
1 QUARTER-OBOL
T: initial letter of
TETAPTHMOPION
1 HALF-OBOL
c
1 OBOL
O: initial letter of
(1/6 of a drachma)
1 or U
OBOAION
183
THE GREEK ACROPHONIC NUMBER-SYSTEM
tttxxxFhhhhaa a Qf
3 3,000 500 400 30 5
TALENTS DRACHMAS
Fig. 16.7. Greek inscription (fragment) from Athens dating from the fifth century BCE. (Museum
of Epigraphy, Athens. Inv. Eml2 355)
By the use of these signs, the Athenians were able to write easily those
sums of money which were of relatively frequent occurrence. The following
examples give the idea. (A quite similar system was also used for weights
and measures such as the drachma, mitia, and stater.)
AA
H+
ill c
T
23 drachmas and (3 + 1/2 + 1/4) obols
20
3
3 'h
*/<
drachmas
obols
l-AAAA
llll
read: 40 drachmas and 4 obols
40
4
XX
P
HAAAII
read: 2,630 drachmas and 2 obols
2,000
500
100 30 2
XXXHHIT QTTT XX FA AAA H-H- Hill
3,000 200 50 10 3 2,000 500 40 4 5
talents
drachmas
obols
3,263 talents 2,544 drachmas and 5 obols
Fig. i 6 . 8 .
In the other states of the Ancient Greek world, the citizens also used
similar acrophonic symbols in their various monumental inscriptions
during the latter half of the first millennium BCE (Fig. 16.9 and 16.10). The
Attic system itself, which is the oldest known of the Greek acrophonic
systems, became more widespread at the time of Pericles, when the city of
Athens was the capital of a number of Greek republics.
However it would be wrong to think that these different number-systems
were all strictly identical to the Athenian one. Each had features which
distinguished it from the others. We should not forget that each Greek state
had its own system of weights and its own system of coinage (by this period
the use of money was widespread throughout the Mediterranean).
Furthermore the very notion of a unified metric system, on the lines of an
international monetary system, was foreign to the Greek spirit.'
1 1- (1 drachma)
10 A
ioo H
1,000 X
2 H-
20 AA
200 HH
2,000 XX
3 FFF
30 A aa
300 HHH
3,000 XXX
4 H-H-
40 A AAA
400 HHHH
4,000 XXXX
5 FFFFF
50 F*
500 IT -
5,000 rr*
6 H-H-H-
60 F'a
600 tT’ 1 H
6,000 P"" X
7 FFFFFFF
70 p^aa
700 IT* HH
7,000 P - " X X
8 H-H-FH-F
80 P’AAA
soo (T* HHH
8,000 P'xxx
9 FFFFFFFFF
9°F*AAAA
900 [T” HHHH
9,000 r r 'xxxx
Example: | 1 J R '' H H A A A A FFFFFFFFF
5,000 500 200 40 9
^
5,749 drachmas
Fig . 16.9. Numerical notation in Greek inscriptions from the island of Cos (third century BCE).
[Tod]
1 drachma
F* or
1 *•
5
n*
11 : first letter of rievre, “five"
10
^ ** or
A*
A: first letter of Ackoi, "ten"
50
PE or
r*
flE: abbreviation of RevTtBeKa, “fifty”
100
F-E
HE: abbreviation of HtKoiTov, “hundred"
300
TE*
T.HE: abbreviation ofTpiaKOCTWR, "three hundred"
500
PE or
PE
n.HE: abbreviation of rUvrotKocrioi, “five hundred"
1,000
y
Ancient Boeotian form of the letter X: first letter of XiXiot,
“thousand”
5,000
r
n.X: abbreviation of ReiraxiAioi., “five thousand”
10,000
M
Letter M: first letter ofMupun, "ten thousand”
* Found only at THESPIAE
** Found only at ORCHOMENOS
Fig. 16.10. Numerical notation in Greek inscriptions from Orchomenos and from Thespiae (third
century BCE) [Tod]
* As P. Devambez (1966) explains: “Money was in the first place defined in terms of weight. Each state chose
from its system of weights one unit to be the standard, and the others were multiples or sub-multiples of
this. For instance, at Aegina in the Peloponnese, the standard unit of weight for commerce was the mina
which weighed 628 gm. The unit of money was chosen to be one hundredth of this, the drachma, which
therefore weighed 6.28 gm. The didrachma or stater was about twice this (12.57 gm). The sub-unit, the obol.
weighing 1 .04 gm, was a sixth of the drachma. At Euboea and in Attica, where the mina weighed 436 gm, the
drachma was 4.36 gm; its multiples, the didrachma and tetradrachma , weighed tw'ice and four times this, or
8.73 gm and 17.46 gm respectively; the obol. a sixth of the unit, weighed 0.73 gm."
GREEK AND ROMAN NUMERALS
184
Drachmas
1
. 1 1- P<
1 2 3 4 5
1 Epidaurus, Argos, Nemea
2 Karystos, Orchomenos
3 Attica, Cos, Naxos, Nesos, Imbros, Thespiae
4 Corcyra (Corfu), Hermione (Kastri)
5 Troezen, Chersonesus Taurica (Korsun), Chalcidice
5
p r p r n
6 7 8 9 10
6 Epidaurus
7 Thera
8 Troezen
9 Attica, Corcyra, Naxos, Karystos, Nesos, Thebes,
Thespiae, Chersonesus Taurica
10 Chalcidice, Imbros
10
O © - $
11 12 13 14
A A A >
15 15 16 17
11 Argos
12 Nemea
13 Epidaurus, Karystos
14 Troezen
15 Corcyra, Hermione
16 Attica, Cos, Naxos, Nesos, Mytilene, Imbros,
Chersonesus Taurica, Chalcidice, Thespiae
17 Orchomenos, Hermione
50
rppppi
18 19 20 21
P t it r 6
22 23 24 24
18 Argos
19 Epidaurus, Troezen, Cos, Naxos, Karystos
20 Nemea, Cos, Nesos, Attica, Thebes
21 Imbros
22 Troezen
23 Chersonesus Taurica
24 Thespiae, Orchomenos
100
B H EE
25 26 27
TEE
28 29 30
25 Epidaurus, Argos, Nemea, Troezen
26 Attica, Thebes, Cos, Epidaurus, Corcyra, Naxos,
Chalcidice, Imbros
27 Thespiae, Orchomenos 28 Karystos
29 Chersonesus Taurica
30 Chersonesus Taurica, Chios, Nesos, Mytilene
500
ni p- it IT*
31 32 33 34
ffl P PflC
35 36 37 38
31 Troezen
32 Epidaurus
33 Karystos
34 Cos
35 Naxos
36 Epidaurus
37 Epidaurus, Troezen, Imbros, Thebes, Attica
38 Thespiae, Orchomenos
1,000
X T
39 40
39 Attica, Thebes, Epidaurus, Argos, Cos, Naxos, Troezen,
Karystos, Nesos, Mytilene, Imbros, Chalcidice,
Chersonesus Taurica
40 Thespiae, Orchomenos
5,000
P f 1
41 42
41 Attica, Cos, Thebes, Epidaurus, Troezen, Chalcidice,
Imbros
42 Thespiae, Orchomenos
10,000
M M X
43 43 44
43 Attica, Epidaurus, Chalcidice, Imbros, Thespiae,
Orchomenos
44 Attica
50,000
P PI
45 46
45 Attica
46 Imbros
Fig. i 6 . i i . Table of the numerical signs found in various Greek inscriptions of the period
1500-1000 BCE, used to express sums of money (in general, the numbers shown here refer to
amounts in drachmas). When they are collected together as here we can see the common origin of all
of the Greek acrophonic numerals which were in use at this time. [Tod}
Bringing together all the different systems, we can observe their
common origin (Fig. 16.11A and B).
Fig. 16.12. The Ancient Greek world
Looking now at Fig. 16.14, 16.15, and 16.16, we can see that the original
number-systems were quite similar to the Egyptian hieroglyphic system
and to the Cretan and Hittite systems.
The inconvenient feature of this kind of notation, in that it required
multiple repetitions of identical symbols, led the Greeks to seek a
simplification by assigning a specific sign to each of the numbers:
i
5
10
50
100
500
1,000
5,000
10,000
1
1
1
1
1
1
1
1
10 2
1
1
1
1
10 3
1
1
1
1
10 4
1
*
1
4
1
4
1
4
auxiliary base
5x10
5 x 10 2
5 x 10 3
Fig. 16.13.
•
1 drachma
10 drachmas
B
100 drachmas
X
1,000 drachmas
2 :
20 Z
200 BB
2,000 *X
3 :•
30 Z-
300 BBB
3,000 XXX
4 ::
40 “
400 BBBB
4,000 XXXX
Fig. 16.14A.
185
THE GREEK ACROPHONIC NUMBER-SYSTEM
IJ*
5 drachmas
50 drachmas
BBB00
500 drachmas
xxxxx
5,000 drachmas
c •••
0 • • •
60 ZZZ
600 BBBBBB
6,000 XXXXXX
7 :::•
70 ZZZ-
700 BABBS BB
7,000 xxxxxxx
Q ••••
O • • « •
80 ZZZZ
800 BBBBBBBB
8,000 xxxxxxxx
9 MM*
90 ZZZZ-
g00 HBHBBHBBB
9,000 xxxxxxxxx
H : ancient form of the letter H; first letter of Hckoitov, “hundred”
X : first letter of XlXioi, “thousand”
Example: X B0BBB000B ZZZ- JJ;;.
1.000 900 70 9
>
1,979 DRACHMAS
Fig. 16.14B. System of numerical notation in Ancient Greek inscriptions from Epidaurus
(beginning of the fourth century BCE). This system, which is based on exactly the same principle as
the Cretan number-systems, and is acrophonicfor the numbers 100 and 1,000 only, has no symbols
for 5, 50, 500, or 5,000, [Tod]
1 •
10 ©
100 B
2 :
20 ©©
200 BB
3 :•
30 ©Q©
300 BBS
4 ::
40 ©©©©
400 BBBB
5
50 pi
500 BBBBB
c m ••
D •••
60 pi®
600 BBBBBB
7 :::•
70 pi©©
700 BBBBBBB
0 • •••
0 ••••
80 (71©©©
800 BBBBBBBB
9
90 (71 ©0©©
900 BBBBBBBBB
F 1 : sign FLA. Abbreviation of rievre AeKOt, “fifty”
bbbb p©0 ©©::::
400 50 40 8
^
498 DRACHMAS
Fig. 16.15. System of numerical notation in Greek inscriptions from Nemaea (fourth century
BCE): a decimal system with a supplementary sign for 50 only ITodl
•
-
B H
X
•
1 drachma
10 drachmas
100 drachmas
1,000 drachmas
10,000 drachmas
2 • •
20 z
200 HH
2,000 XX
20,000 ••
3 • • •
30 Z-
300 0EB
3,000 XXX
30,000 • • •
4 ••••
40 ZZ
400 HHHH
4,000 XXXX
40,000 ••••
n
5 drachmas
|^or |"3
50 drachmas
|""E or p or |™^-
500 drachmas
P
5,000 drachmas
?•
6 1".
60 P-
600 PB
6,000 p x
7 P..
70 PZ
700 P HH
7,000 p XX
8 P...
80 PZ-
800 PBBB
8,000 f? XXX
9
90 PZZ
900 PHHHH
9,000 fx* XXXX
Fig. i 6 . 1 6 b . Numerals in late inscriptions from F.pidaurus (end of fourth to middle of third
centuries BCE.) [ Tod I
They therefore arrived at a mathematical system equivalent to the one
used by the South Arabs and the Romans.
Thereafter, no more than fifteen different signs were required in order to
represent the number 7,699 for example, instead of the thirty-one that were
needed in the Cretan and the archaic Greek system.
I* XX P H F AAAA P INI
5,000 2,000 500 100 50 40 5 4
Fig. 16. 17.
Nevertheless, this advance in notation was a step backwards in the
evolution of arithmetic itself. In the beginning, the Greeks had assigned
specific symbols only to unity and to each power of the base, and they were
able to do written arithmetic after the fashion of the Egyptians. But once
they had introduced supplementary figures into their initial set, the Greeks
deprived it of all operational capability. As result, the Greek calculators
thenceforth had to resort to “counting tables".
THE NUMBERS OF THE KINGDOM OF SHEBA
We now consider the numerical notation used by the ancient people of
South Arabia, especially the Minaeans and the Shebans who shared what
is now Yemen during the first millennium BCE. [M. Cohen (1958);
J. C. Fevrier (1959); M. Hofner (1943)]
The inscriptions which have come down to us from these peoples concern
the most varied subjects: buildings constructed on several floors, irrigation
systems retained by large dikes, offerings to the astral gods, animal sacrifices,
tales of conquest, inventories of booty, and so forth . The writing, in which were
written the neighbouring Semitic Arab languages, was no doubt derived (with
some major changes) from the ancient Phoenician writing and had twenty-
nine consonants represented by characters of geometric form, almost all of
the same size. [M. Cohen (1958); J. G. Fevrier (1959); M. Rodinson (1963)]
Fig. 16.16A.
GREEK AND ROMAN NUMERALS
186
The system used by these people was based on the additive principle.
A distinct symbol was assigned to each power of 10, and also to the number
five and to the number fifty (Fig. 16.19).
Like the Greek systems which we have just analysed, this system was
acrophonic in nature. Except for the signs for 1 and 50, all the others are
letters of the alphabet, and are in fact the initial letters of the Semitic names
of the numbers 5, 10, 100, and 1,000. (Quite possibly the South Arabs were
influenced in this respect by the Greeks. This is conjectural, though we do
in fact know from other studies that there were contacts between the
Greeks, the Shebans, and the Minaeans.)
1
1
y - y
(a) <b)
O
P ° r 4
(a) (b)
fe or 4
(a) (b)
f 1 ! M ft
(a) (b)
Simple vertical bar
5
Letter HA: first letter of HAMSAT, Southern Arabic word
for “five”
10
Letter 'AYIN: first letter of the word 'ASARAT, “ten"
50
Half of the sign for 100
100
Letter MIM: first letter of the word MI’AT, “hundred”
1,000
Letter ALIF, first letter of the word ’ALF, “thousand”
(a) reading from left to right (b) reading from right to left
Fig. 16.18.
In the Minaean and Sheban inscriptions, numerals are usually enclosed
between a pair of signs 1 and I in order to avoid confusion between letters
representing numbers and letters standing for themselves (Fig. 16.22 and
16.24). It often happens, also, that the figures change orientation within
the same inscription, since the South Arab writing was in boustrophedon
(alternately from right to left and from left to right).
H
10 0
100 &
1.000 ri
2 1 1
20 00
200 ££
2,000 rirt
III
30000
300 fcfcP
3.000 rirfri
4 1 1 1 1
400000
400
4,000 ririrtrt
V
so t*’
500 PPI&fe
5.000 rfrfrfrirt
y
60 Po
600
6.000 tWfififiri
v
70 F© O
700
7.000 rirtrtrtrirtrt
■gin
80 P 0 O O
800 fcfcfcfefcfcfcfc
8,000 rirtrtrtrirtrtrt
ymi
90P0OOO
900
9,000 rtrirtrtrtrirtrtrt
Fig. 16.19. The symbols, and the principle, of the Southern Arabian number-system. This system
is known only from the period from the fifth to the second or first centuries BCE. On inscriptions
da ting from after the beginning of the Common Era, it seems, numbers are spelled out in full.
There is one interesting and important difference between the number-
system of the South Arabs - at any rate those of Sheba - and the otherwise
similar Greek system, in that the Arab system incorporated a rudimentary
principle of position.
In fact, when one of the figures
O I" ^ or £
10 50 100 Fig. 16.20.
is placed to the right of the sign for 1,000 (when reading from right to left),
then this figure is (mentally) multiplied by 1,000. In the following,
for example:
plhoool 3 3
2,000 30 50 200
* Fig. 16.21.
we would at first be inclined to read the value
200 + 50 + 30 + 2,000 = 2,280
according to the traditional usage of the additive principle, whereas in fact
it represented, in Sheba, the value
(200 + 50 + 30) x 1,000 + 2,000 = 282,000
CIS IV:
inscr. 924
1 mil 1
<■
5
Notice the irregularity
RES:
inscr. 2740, 1.7
U !
50
RES:
inscr. 2868, 1.4
1.4 1
<r
60
RES:
inscr. 2743, 1.10
8 ill 0 4 I
63
RES:
inscr. 2774, 1.4
1 ll lyfOOOol
47
RES:
inscr. 2965, 1.4
BoX 4 4 I
180
Note the unusual manner
of writing the number 30:
OO instead of OOO
Fig. 16.22. Examples taken from Minaean inscriptions (third to first century BCE). The numbers
shown above refer to the volume capacity of certain recipients offered to the astral gods of ancient
Southern Arabia, or to lists of offerings to these gods, or to animals which have been sacrificed.
1C. Robin (personal communication)!.
Similarly, when reading from left to right, the same effect is produced by
placing the figure to the left of the sign for 1,000. Thus
200 50 30 2,000
^ Fig. 16.23.
gives 282,000 (and not 2,280!).
187
ROMAN NUMERALS
RES:
inscr. 3945, 1.13
144444 1
<•
500
RES:
inscr. 3945, 1.13
1 fifth 1
3,000
RES:
inscr. 3945, 1.19
ahftfthhhhhhfthhi
<■
12,000
CIS IV:
inscr. 413, 1.2
H444 hftfthhftl
<r-
6,350
RES:
inscr. 3945, 1.4
loririrfrirtril
10 6,000
16,000
RES:
inscr. 3943, 1.2
I OOO ri 1
30 1,000
>
31,000
RES:
inscr. 3943, 1.3
1 oooo rfrirfriri 1
40 5,000
>
45,000
Fig. 16.24. Examples from inscriptions from the ancient kingdom of Sheba (fifth century BCE).
These inscriptions, principally from the site of Sirwah, tell of military conquests and give various
inventories: numbers of soldiers, material resources, booty, prisoners, and so on. [C. Robin, personal
communication ]
numbers. This is why Roman accountants, and the calculators of the Middle
Ages after them, always used the abacus with counters for arithmetical
work.
As with the majority of the systems of antiquity, Roman numerals were
primarily governed by the principle of addition. The figures (1= 1, V= 5, X=
10, L = 50, C = 100, D = 500 and M = 1,000) being independent of each other,
placing them side by side implied, generally, addition of their values:
CLXXXVI1 = 100 + 50 + 10 + 10 + 10 + 5 + 1 + 1 = 187
MDCXXVI = 1,000 + 500 + 100 + 10 + 10 + 5 + 1 = 1,626
The Romans proceeded to complicate their system by introducing a rule
according to which every numerical sign placed to the left of a sign of
higher value is to be subtracted from the latter.
Thus the numbers 4, 9, 19, 40, 90, 400, and 900, for example, were often
written in the forms
IV (= 5 - 1) instead of IIII XC (= 100 - 10) instead of LXXXX
IX (= 10 - 1) instead of VIIII CD (= 500 - 100) instead of CCCC
XIX (= 10 + 10 -1) instead of XVIIII CM (= 1,000 - 100) instead of DCCCC
XL (= 50 - 10) instead of XXXX
However, this practice must have surely given rise to confusion among
the readers, and the Sheban stone-cutters therefore took the precaution
of also writing out in words the number represented by the figures.
A lucky precaution, for it has enabled us today to arrive at an
unambiguous interpretation of this number-system!
ROMAN NUMERALS
Like the preceding systems, the Roman numerals allowed arithmetical
calculation only with the greatest difficulty.
To be convinced of this, let us try to do an addition in these figures.
Without translating into our own system, it is very difficult, if not impossi-
ble, to succeed.
The example which is most often cited is the following:
CCXXXII
232
+
CCCCXIII
+
413
+
MCCXXXI
+
1,231
+
MDCCCLII
+
1,852
=
MMMDCCXXVII1
3,728
Roman numerals, in fact, were not signs which supported arithmetic
operations, but simply abbreviations for writing down and recording
It is remarkable that a people who, in the course of a few centuries,
attained a very high technical level, should have preserved throughout that
time a system which was needlessly complicated, unusable, and downright
obsolete in concept.
In fact, the writing of the Roman numerals as well as its simultaneous use
of the contradictory principles of addition and subtraction, are the vestiges
of a distant past before logical thought was fully developed.
Roman numerals as we know them today seem at first sight to have been
modelled on the letters of the Latin alphabet:
I V X L C D M
1 5 10 50 100 500 1,000
Fig. 16.25.
However, as T. Mommsen (1840) and E. Hiibner (1885) have remarked,
these graphic signs are not the first forms of the figures in this system.
They were in fact preceded by much older forms which had nothing to
do with letters of an alphabet. They are late modifications of much older
forms.*
* The oldest known instances of the use of the letters L, D and M as numerals do not go back earlier than the
first century BCE. As far as we know, the earliest Roman inscription which uses the letter L for 50 dates only
from 44 BCE (C1L, I, inscr. 594). The earliest known use of the numerals M and D is in a Latin inscription
which dates from 89 BCE, in which the number 1,500 is written as MD (CII., IV, inscr. 590).
GREEK AND ROMAN NUMERALS
188
Originally, 1 was represented by a vertical line, the number 5 by a
drawing of an acute angle, 10 by a cross, 50 by an acute angle with an addi-
tional vertical line, 500 by a semi-circle at an angle, and 1,000 by a circle
with a superimposed cross (of which the denarii figure for 500 is geometri-
cally one half):
I V X V # ^ $
Fig. 16.26. 1 5 10 50 100 500 1,000
In an obvious way, the original figures for 1, 5 and 10 were assimilated to
the letters I, V and X.
The original figure for 50 (which can still be found as late as the reign of
Augustus, 27 BCE - 14 CE*) evolved progressively as shown below, finally
merging with the letter L around the first century BCE:
V 4^ X ->_L ->1->L
Fig. 16.27. 50
The original figure for 100 initially evolved in a similar way towards a
more rounded form: yc and then, for the sake of abbreviation, was split
into one or other of the forms ) or ( . By similarity of shape, and under
the influence of the initial letter of the Latin word centum (“one hundred”),
it was finally assimilated to the letter C.
The original figure for 500 first of all underwent an anticlockwise
rotation of 45°. It then evolved towards the sign B (these signs can still
be found on texts from the Imperial period 1 ') and finally was assimilated
to the letter D:
Fig. 16.28. 500
The figure for 1,000 first of all evolved towards the form (D. This gave rise
to the many variant forms shown below for which, progressively, the letter
M came to be substituted, from the first century BCE, under the influence
of the first letter of the Latin word milk:
BJ -*• 0
1,000
* CIL, IV, inscr. 9934
+ CIL, VIII, inscr. 2557
1
I
CIL I 638. 1449
2
I)
CIL I 638. 744
3
III
CIL I 1471
4
111]
CIL 1 638, 587, 594
A
CIL 1 1449
5
or
V
CIL 1 590, 809,
1449, 1479, 1853
6
VI
CIL I 618
7
VII
CIL I 638
8
VIII
CIL I 698, 1471
9
villi
CIL 1 594, 590
10
X
CIL I 638, 594, 809, 1449
14
XIIII
CIL I 594
15
XV
CIL 1 1479
19
XVJJII
CIL I 809
20
XX
CIL I 638
24
XXIIJI
CIL 1 1319
40
xxxx
CIL 1 594
\
CIL I 214, 411 and 450
or
X
CIL 1 1471, 638. 1996
50
or
1
CIL 1 617, 1853
j.
CIL I 744, 1853
L
CIL 1594, 1479 and 1492
51
X
CIL I 638
74
>X, xxmi
CIL I 638
95
LXXXXV
CIL I 1479
100
C
CIL I 638, 594, 25, 1853
100
3
CIL VIII 21,701
300
ccc
CIL 1 1853
400
cccc
CIL I 638
500
B
CIL I 638, 1533 and 1853
D
CIL I 590
837
BCCCXXXVH
CIL I 638
X
or
vl/
CIL 1 1533, 1578, 1853
and 2172
CIL X 39
00
or
OO
CIL I 594 and 1853
1,000
CIL X 1019
ExJ
CIL VI 1251a
M
CIL 1 593
M
CIL 1 590
1,200
00 cc
CIL I 594
1,500
MD
CIL I 590
2,000
0000
CIL I 594
2,320
0000 cccxx
CIL I 1853
3,700
®®® BCC
CIL 1 25
CIL X 817
5,000
10
CIL 1 1853 and 1533
b
CIL I 2172
5,000
DD
CIL I 590, 594
7,000
fo»®
CIL I 2172
8,670
1')®** DC1XX
CIL 1 1853
(<l>)
CIL 1 1252, 198
or
CCDD
CIL I 583
10,000
CIL 1 1474
or
CIL I 744
CIL 1 1724
12,000
*®
CIL 1 1578
21,072
AX® iXX "
CIL I 744
30,000
CC DD CCI DD CCDD
CIL 1 1474
30,000
CIL 1 1724
50,000
IXO
CIL I 593
CIL I 801
100,000
CIL I 801
CCCDDD
CIL I 594
Fig. 16.30. Written n umbers from
monumental Latin inscriptions, dating from the Republican and early Imperial periods
189
ROMAN NUMERALS
The various forms associated with the number 1,000 in Fig. 16.29 were
mainly used during the period of the Republic, but they can also be found
in some texts of the Imperial period.* A few of them even survived long
after the fall of the Roman civilisation, since they can be found in quite a
few printed works from the seventeenth century (Fig. 16.69 and 16.70).
CIL 1 1319
CIL 1 1492
CIL 1 1996
268
CJLXIIX
CIL I 617
69
LXIX
CIL I 594
286
CCXXCVI
CIL I 618
CCCX1
CIL 1 1529
78
LXXIIX
CIL I 594
345
CCCX1V
CIL 1 1853
CIL 1 1853
Fig. 16 . 31 . Latin inscriptions from the Republican era showing the use of the principle of
subtraction. Use of this principle (which undoubtedly reflects the influence of the popular system
on the monumental system) was nevertheless unusual on well-styled inscriptions.
VlAMf eCEIA ftRF Gl O AD-CAfVAM E 7
IN EA VIA rONTEISOWNEISMlUARIoS
TABELARIOSa'^rQSEIVEIHINCEISVNI
novceria/wmeiuaxi CAPVAMXXOI
MVRAIMVMXKXIIII COSENTIAMOOflll
VALE NTlAMCiXXX f AOFRETVMAE
STATVAM CXXXlf - REGIVMCCXXXV)
SVMAAACArVAR£CIVAVA 0 UACCC
ETEIDEMTRAE TORIN V^Xll
SICILIAFVGmiVOSfTALICORVM
CONaVAEISIVEI-REDIDEtaVE
HOMINES PCCCCX VII- EIDEMQ.VI
PRI/VW 5 -FECEIVT DEAG»>POn:ICO
AR ATOR IBVKE 0 ERENTPAASTORES
fORVMAEDtsaVEpOPUCASHElCFECEl
Fig. 16.32A. Milestone engravingfound at the Forum Popilii in iucania (southern Italy), and
made by C. Popilius iaenas, Consul in 172 BCE and 158 BCE. Now in the Museo della Civilta
Romana, Rome. ICIL, I, 6381
line 4
XI
51
line 7
CCXXXI
231
line 4
XXCIIII
84
line 7
CCXXXVII
237
line 5
X XXIIII
74
line 8
CCCXXI
321
line 5
CXXIII
123
line 12
BCCCCXVII
917
line 6
cxxxx
180
Fig. 1 6 . 3 2 b . Written numbers on the inscription shown in Fig. 16.32 A
* CIL, IV, inscr. 1251; CIL, X, inscr. 39 and 1019; CIL, IL, inscr. 4397: etc.
Fig. 16.33. Elogiumfl/’
Duilius, who conquered the
Carthaginians at the battle of
Mylae, 260 BCE. The inscription
was re-cut at the start of the
Imperial period, during the reign
of Claudius (41-54 CE), in the
style of the third century BCE.
Found in the Roman Forum at
the place of the rostra (columna
rostrataj, and now in the Palazzo
dei Conservator i in Rome
[CIL, 1. 195].
In lines 15 and 16, the figure
for 100,000 is repeated at least 23
times (and at most 33, according
to the restoration by the Corpus.).
t* 1 op-
t { di enfj D’ IHfMET - LECIOI^gM cmr t m einitn $ it omnit
•^trMOSQVE • M A C I S T R«To3M^e* pmtmm pttlditt
DVEM • CASTREIS EXFOCIONT - MAC^L«Nf«i opidom mi
^/CNANDOD CBPET ENC^E EODEM HA^Wtrro | W< I btmt
CM • NAVEBOS • MARlD • CONSOL • PRlMOS^inl topiatqut
i LASESQVE ■ NAVALES • PRlMOS ■ ORNAVET - t&rmmttqnt
< VMQVE • BIS • NAVEBOS • CLASEIS ■ POENICAS • OMBIt«, iitm mm-
j VM AS • COPIAS * CARTACINIENSlS • FRAESENTEU kmmibmltd
flCTATORED 0>^OM • IN ALTOD • MAKj^ tit"
■ CVM • SOCIEIS • SEPTE\*imoa qui*.
qutrt tm OBA y £~ TRJRZSMOSQVE NAVEIS mtntt tiii
mmrtlM- CAPTOM - NVMEI . OOODCC
«fn«|lOM CAPTOM P R AED A-NVMEI*
- captom AEs oeooooee«
1 OO OQ 00000009 OOOOOfi-
f^OQVB ■ NAVALED • PAAEDAD • POPLOIfrf 0 » « 1 « i pr <-
trimmpod
^ATjyCtirr*«iI8 i«)NVOS^i*<l in
" EIS
On line 13, the number 3,700 is written in the form:
DCC
Note: The capital letters (in upright characters in the figure) correspond to that part of the inscrip-
tion which remains intact. The italic letters correspond to the restoration (by the Corpus) of the
part which has been damaged.
pf Iv/n < c iv*
Ml 1 ^ 1 1
Pi
64 Oc
»\ fiyCviKnV
I W v ' u
f\J t\\\ Nf.
tl - U/M
i\k 1
fr
TRANSCRIPTION
hs n. ccIod ccba ccIdd Iod
00 00 00 LXXVIII*
qu$ pecunia in stipulatum
L. Caecili Iucundi venit ob
auctione (m) M. Lucreti Leri
[mer] cede quinquagesima
minu [s]
* See Fig. 16.29 and 16.30
Fig. 16 . 34 - Second panel of a triptych found at Pompeii, therefore prior to 79 CE (the year the
city was destroyed)
ETRUSCAN NUMERALS
Roman numerals reached their standardised form, identical to letters of the
Latin alphabet, late in the history of Rome; but in reality they began life
many hundreds of years, maybe even thousands, before Roman civilisation,
and they were invented by others.
The Etruscans, a people whose origins and language both remain largely
unknown, dominated the Italian peninsula from the seventh to the fourth
GREEK AND ROMAN NUMERALS
190
century BCE, from the plain of the Po in the north to the Campania region,
near Naples, in the south. They vanished as a distinct people at the time
of the Roman Empire, becoming assimilated into the population of their
conquerors.
Several centuries before Julius Caesar the Etruscans, and the other Italic
peoples (the Oscans, the Aequians, the Umbrians, etc.), had in fact invented
numerals with form and structure identical to those of the archaic Roman
numerals.
1
1
CIE 5710
2
11
C1E 5708
TLE 26
3
111
CIE 5741
4
1111
CIE 5748
A
CIE 5705, 5706, 5683,
5677 and 5741
f \
ACII, Table IV 114
6
JA
CIE 5700
7
JlA
CIE 5635
8
,111.0.
ACII, Table IV 114
9
^niiA
CIE 5673
X
or
X
CIE 5683, 5741. 5710
5748, 5695, 5763, 5797
5707, 5711 and 5834
10
CIE 5689 and 5677
+
TLE 126
19
XIX
CIE 5797
36
,IAXXX
CIE 5683
38
^IIIAXXX
CIE 5741
38
CIE 5707
42
IIX xxx
CIE 5710
44
IIIIXXXX
■4
CIE 5748
CIE 5708, 5695, 5705,
5706, 5677 and 5763
50
A
or
Buonamici, p. 245
T
52
114
4 -* ••
CIE 5708
55
is.
CIE 5705 and 5706
60
X*
■4
CIE 5695
75
/***
CIE 5677
82
Jl+f+t
TLE 26
86
//////xxx 4
4
CIE 5763
X
or
ACII, Table IV 114
100
*
Buonamici, p. 473
106
M*
SE, XXIII, series II
(1965), p. 473
Frc. 16.35. Written numbers from Etruscan inscriptions
Fig. 16.36. Etruscan coins
dating from the fifth century
BCE bearing the numbers
A and X
5 10
Collection of the
Landes museum,
Darmstadt [Menninger
(1957) vol. II, p. 48}
For many centuries, they used these figures according to the principles
of addition and subtraction simultaneously. This is evidenced by several
Etruscan inscriptions of the sixth century BCE, where the numbers 19
and 38 are written on the subtractive principle as 10 + (10 - 1) and
10 + 10 + 10 + (10 - 2) (Fig. 16.35).
Fig. 16 . 37 . Fragments of an Etruscan
inscription bearing the numbers:
X'Mt \nf\xx nx
*r~ <1 *r
160 208 15
[ACII table IV 114]
A QUESTIONABLE HYPOTHESIS
A hypothesis commonly accepted nowadays asserts that all of these
numerals derived from Etruscan numerals, themselves of Greek origin.
We should recall that Latin writing derived from Etruscan writing, and
that this comes directly from Greek writing. The Greek alphabets fall into
two groups: the Western type which (like the Chalcidean alphabet, for
example) assigned the sound “kh” to the letter 'f' or ^ or \y ; the Eastern
type which (like the alphabet of Miletus or Corinth, for example) assigned
to this symbol the sound “ps”, while the sound “kh” is represented by the
letter + or x. Etruscan writing, for several reasons, is associated with the
Western type.
Therefore it has come to be supposed that the Etruscan alphabet "was
borrowed from a Greek alphabet of Western type on the land of Italy itself,
since the oldest of the Greek colonies which had such an alphabet, that of
Kumi, dates from 750 BCE, and its establishment precedes the birth of the
Tuscan civilisation by half a century.” [R. Bloch (1963)]
On this basis, having compared the forms of the letters, many specialists
in the Roman numbering system have therefore inferred that the ancient
Latin signs for the numbers 50, 100 and 1,000 come respectively from
the following letters, which belong to the Chalcidean alphabet (a Greek
alphabet of Western type used, as it happens, in the Greek colonies in
Sicily). These letters represented sounds which did not occur in Etruscan or
in Latin, and later became assimilated to the Latin forms which we know.
191
chi:
t
or
*
or
theta
ffl
or
or
6
o
phi
<t>
or
or
CD
According to this hypothesis, the Greek letter theta 0 (originally ffl or
© ) gradually turned into C, under the influence of the initial letter of the
Latin word centum.
This explanation (which many Hellenists, epigraphers, and historians
of science now hold as dogma) is seductive, but it cannot be accepted.
Why, in fact, should three particular foreign characters be introduced
into the Roman number-system, and three only? And why should they
be letters of the alphabet? No doubt, one may reply, because the Greeks
themselves had often used letters of their alphabet as number-signs.
In antiquity, it is true, the Hellenes used two different systems of written
numerals whose figures were in fact the letters of their alphabet. One of
these used the initial letters of the names of the numbers. The other made
use of all the letters of the alphabet (see Fig. 17.27 below):
A
Alpha
1
I
Iota
10
P
Rho
100
B
Beta
2
K
Kappa
20
2
Sigma
200
r
Gamma
3
A
Lambda
30
T
Tau
300
A
Delta
4
M
Mu
40
Y
Upsilon
400
E
Epsilon
5
N
Nu
50
Phi
500
H
Xi
60
X
Chi
600
H
Eta
8
O
Omicron
70
ip
Psi
700
0
Theta
9
n
Omega
800
Now the letter chi, which was supposed to be borrowed for the number
50 in Latin, has value 1,000 in the first of these systems, and 600 in the
second; the letter theta, “borrowed” for the number 100 in Latin, has value
9 in the second Greek version; and the letter phi, supposed to have been
borrowed for the Roman numeral for 1,000, is worth 500 in the second
system. Why the differences?
If the Romans had borrowed the following Greek signs for the numbers
50 and 100:
chi: 'f' or \J^ or ^
theta © or © or G-
then the same would probably have been borrowed by the Etruscans as
well. How then can we explain that, for the same values, the Etruscans in
fact used quite different figures, namely (see Fig. 16.35):
A QUESTIONABLE HYPOTHESIS
yf or f for 50
and or for 100
One can see that the hypothesis is not very sound. The error is due to
the fact that specialists have believed through many generations that
Roman numerals are the children of Etruscan numerals, whereas in fact
they are cousins.
THE ORIGIN OF ROMAN NUMERALS
Though long obscure, the question is no longer in doubt. The signs I, V and
X are by far the oldest in the series. Older than any kind of writing, older
therefore than any alphabet, these figures, and their corresponding values,
come naturally to the human mind under certain conditions. In other
words, the Roman and Etruscan numerals are real prehistoric fossils: they
are descended directly from the principle of the notched stick for counting,
a primitive arithmetic, performed by cutting notches on a fragment of bone
or on a wooden stick, which anyone can use in order to establish a one-to-
one correspondence between the objects to be counted and the objects used
to count them.
Let us imagine a herdsman who is in the habit of noting the number of
his beasts using this simple prehistoric method.
Up to now, he has always counted as his forebears did, cutting in a
completely regular manner as many notches as there are beasts in his herd.
This is not very useful, however, because whenever he wants to know how
many beasts he has, he has to count every notch on his stick, all over again.
The human eye is not a particularly good measuring instrument.
Its capacity to perceive a number directly does not go beyond the number
4. Just like everyone else, our herdsman can easily recognise at a glance,
without counting, one, two, three, or even four parallel cuts. But his
intuitive perception of number stops there for, beyond four, the separate
notches will be muddled in his mind, and he will have to resort to a
procedure of abstract counting in order to learn the exact number.
Our herdsman, who has perceived the problem, is beginning to look for
a way round it. One day, he has an idea.
As always, he makes his beasts pass by one by one. As each one passes,
he makes a fresh notch on his tally stick. But this time, once he has made
four marks he cuts the next one, the fifth, differently, so that it can be recog-
nised at a glance. So at the number 5, therefore, he creates a new unit of
counting which of course is quite familiar to him since it is the number
of fingers on one hand.
GREEK AND ROMAN NUMERALS
192
For any individual, cutting into wood or bone presents the same
problems, and will lead to the same solutions, whether in Africa or Asia, in
Oceania, in Europe or in America.
Our herdsman only has a limited number of options. To distinguish the
fifth notch from the first four, the first idea he has is simply to change
the direction of cut. He therefore sets this one very oblique to the other
four, and thereby obtains a representation all the more intuitive in that it
reflects the angle that the thumb makes with the other four fingers.
1 4 5 6 10 11
11 10 9 5 4 1
pt uit&m&j
Pt
$
Fig. 16 . 38 .
Another idea is to augment the fifth notch by adding a small supple-
mentary notch (oblique or horizontal), so that the result is a distinctive sign
in the form of a “t”, a “Y” or a “V”, variously oriented:
V A < > y^H l-KKA
He resumes cutting notches in the same way as the first four, counting
his beasts up to the ninth. But, at the tenth, he finds he must once again
modify the notch so that it can be recognised at a glance. Since this is the
total number of fingers on the two hands together, he therefore thinks of a
mark which shall be some kind of double of the first. And so, as in all the
numeral systems, he comes to make a mark in the form of an “X” or a cross:
X X \ +
A
D
■WUl.K.fkYH
1 5 9 10
1 5 10
~B~|
E |
123 4 56 78910 15 20
1 5 10
1 5 10 15
1 5 10
Fig. 16 . 39 . Anyone who counts by cutting notches on sticks will come to represent the numbers 1, 5,
10, 15, and so on in one of the above ways.
So he has now created another numerical unit, the ten, and counting on
the tally stick henceforth agrees with basic finger-counting.
Reverting to his simple notches, the herdsman continues to count beasts
until the fourteenth and then, to help the eye to distinguish the fifteenth
from the preceding ones, he again gives it a different form. But this time he
does not create a new symbol. He simply gives it the same form as the
“figure” 5, since it is like “one hand after the two hands together”.
He carries on as before up to 19, and then he makes the twentieth the
same as the tenth. Then again up to 24 with the ordinary notches, and the
twenty-fifth is marked with the figure 5. And so on up to 9 + 4 x 10 = 49.
This time, however, he must once more imagine a new sign to mark
the number 50, because he is not able to visually recognise more than four
signs representing 10.
This is naturally done by adding a third cut to his notch, so he naturally
chooses one of the following which can be made by adding one notch to one
of the representations of the number 5:
VAVIfAKX A Y A ? ri h
Having done this, he can now proceed in the same way until he has gone
through all the numbers from 50 to 50 + 49 = 99.
At the hundredth, our herdsman once again faces the problem of making
a distinct new mark. So equally naturally he will choose one of the follow-
ing which can be made either by adding a further notch to one of the
representations of 10, or by making a double of one of the representations
of 50:
* m ni m x h hi r
Again as before, he continues counting up to 100 + 49 = 149. For the
next number, he re-uses the sign for 50 and then continues in the same way
up to 150 + 49 = 199.
At 200, he re-uses the figure for 100 and continues up to 200 + 49 = 249.
And so on until he reaches 99 + 4 x 100 = 499.
Now he creates a new sign for 500 and continues as before until
500 + 499 = 999. Then another new sign for 1,000 which will allow him
to continue the numbers up to 4,999 (= 999 + 4 x 1,000), and so on.
And so, despite not being able to perceive visually a series of more than
four similar signs, our herdsman, thanks to some well-thought-out notch-
cutting, can now nonetheless perceive numbers such as 50, 100, 500, or
1,000, without having to count all the notches one by one. And if he runs
out of space on his tally stick and cannot reach one of these numbers, then
all he needs to do is to make as many more tally sticks as he needs.
When the notches are cut in a structured way like this, it is possible to go
up to quite large numbers, as large as are likely to be needed in practice,
193
THE ORIGIN OF ROMAN NUMERALS
without ever having to take account of any series of more than four signs of
the same kind. Such a technique is therefore like a lever, the mechanical
instrument which allows someone to raise loads whose weight far exceeds
his raw physical strength.
The procedure also defines a written number-system which gives a
distinct figure to each of the terms of the series
1
5
10 = 5x2
50 = 5x2x5
100 = 5x2x5x2
500 = 5x2x5x2x5
1.000 = 5x2x5x2x5x2
5.000 = 5x2x5x2x5x2x5
Our herdsman’s approach to cutting notches on sticks therefore gives
rise to a decimal system in which the number 5 is an auxiliary base (and
the numbers 2 and 5 are alternating bases), and its successive orders of
magnitude are exactly the same as in the Roman system; furthermore, it
will naturally give rise to graphical forms for the figures which are closely
comparable with those in the archaic Roman and Etruscan systems.
Again, the use at the same time of both the additive and the subtractive
principles in the Etruscan and Roman systems is yet another relic of this
ancient procedure.
To return to our herdsman. Now that he has counted his various beasts
under various categories, he wants to transcribe the results of this break-
down onto a wooden board. In total 144, his beasts are distributed as:
26 dairy cows
35 sterile cows
39 steers
44 bulls
In order to write down one of these numbers, say the steers, the first idea
which occurs to him is to mark these by simply copying the marks of the
tally stick onto the board:
mi v mi x mi v mi x mi v mi x mi v mi
1 5 10 15 20 25 30 35 39
Fig. 16.40.
But he soon becomes aware that such a cardinal notation is very tedious,
because it brings in all of the successive marks made on the stick. To get
around this difficulty, he therefore thinks of an ordinal kind of representa-
tion, much more abridged and convenient than the preceding one. For the
numbers from 1 to 4, he at first adopts a cardinal notation writing them
successively as
I II III IIII
He can hardly do otherwise for, to indicate that one of the lines is the
third in the series, he must mark two others before it, in order that it shall
be clear that it is indeed the third.
He does not do the same for the number 5, however, since this already
has its own sign (“V”, say), which distinguishes it from the preceding four.
Therefore this “V” is sufficient in itself and dispenses with the need to
transcribe the four notches that precede it on the tally stick. Instead of tran-
scribing this number as III1V, all he needs to do is to write V.
Starting from this point, the number 6 (the next notch after the V) can
be written simply VI, and not IIIIVI; the number 7 can be written as VII, and
so on up to VIIII (= 9).
In turn, the sign in the shape of an “X" can represent the tenth mark in
the series all on its own, and renders the nine preceding signs superfluous.
On the same principle, the numbers 11, 12, 13, and 14 can be written as XI,
XII, XIII, and XIIII (and not IIIIVIIIIXI, etc.). Now the number 15 can be
written simply XV (and not IIIIVIIIIXIIIIV nor XIIIIV): each X can erase
the nine preceding marks, and the last V the four preceding marks. The
numbers from 16 to 19 can be written XVI, XVII, XVIII, XVIIII. Then, for
the number 20, which corresponds to the second “X” in the series, we can
write XX. And so on.
When he has counted his animals by means of the notches on his sticks,
our herdsman can now transcribe the breakdown onto his wooden board:
XXVI
(=26)
for the dairy cows
XXXV
(= 35)
for the sterile cows
XXXVIIII
(= 39)
for the steers
XXXXIIII
(=44)
for the bulls
However, looking for ways of shortening the work, our herdsman comes
up with another idea. Instead of writing the number 4 using four lines (IIII),
he writes it as IV, which is a way of marking the “I” as the fourth in the series
on the stick, since this is the one that comes before the “V”:
IIII ->(III)IV »IV
In this way he cuts down on the number of symbols to write, saving 2. In
the same way, instead of writing the number 9 as VIIII, he writes it as IX
since this likewise marks the “I” as the ninth mark in the series on the
notched stick:
IIIIVIIII » (IIIIVIII)IX » IX
GREEK AND ROMAN NUMERALS
He again cuts down on the number of symbols, saving 3, He does
likewise for the numbers 14, 19, 24, and so on.
This is how one can explain why the Roman and Etruscan number-
systems use forms such as IV, IX, XIV, XIX, etc., as well as IIII, VIIII, XIIII,
XVIIII.
We can now conceive that all of the peoples who, for long ages, had been
using the principle of notches on sticks for the purpose of counting should,
in the course of time, with exactly the same motives as our herdsman
and quite independently of any influence from the Romans or Etruscans
themselves, be led to invent number-systems which are graphically and
mathematically equivalent to the Roman and Etruscan systems.
This hypothesis seems so obvious that it could be accepted even if there
were no concrete evidence for it. But such evidence exists, and in plenty.
A REVEALING ETYMOLOGY
It is hardly an accident that the Latin for “counting” should refer to the
practicalities of this primitive method of doing it.
In Latin, “to count” is rationem putare* As M. Yon (cited by L. Gerschel)
points out, the term ratio not only refers to counting,* but also has a
meaning of “relationship” or “proportion between things”.*
Surely this is because, for the Romans, this word referred originally to
the practice of notching, since counting, in a notch-based system, is a
matter of establishing a correspondence, or one-to-one relationship,
between a set of things and a series of notches. Gerschel has demonstrated
this with a large number of examples.
As for the word putare:
This strictly means to remove, to cut out from something what is
superfluous, what is not indispensable, or what is damaging or foreign
to that thing, leaving only what appears to be useful and without flaw.
In everyday life it was employed above all to refer to cutting back a tree,
to pruning. [L. Gerschel (I960)]
To sum up:
In the method of counting described by the expression rationem putare,
if the term ratio means representing each thing counted by a corre-
sponding mark, then the action denoted by putare consists of cutting
into a stick with a knife in order to create this mark: as many as there
* To one of his contemporaries, Plautus wrote: Postquam comedit rcm, post rationem putat. “It is now that he
has consumed his resources that he counts the cost!” ( Trin . 417).
+ We find an example of this use in Cicero (Flacc. XXVII, 69): Auri ratio constat; aurum in aerario est.
“The count of the gold is correct; the gold is in the public treasury.”
* Cato (in Agr. 1. 5) uses the expression pro ratione to mean “in proportion”, giving ratio an arithmetical
sense; Vitruvius (III, 3, 7) also uses the expression to mean “architectural proportion”.
194
are things to count, so many are the notches on the stick, made by
cutting out from the wood a small superfluous portion, as in the defi-
nition of putare. In a way, ratio is the mind which sees each object in
relation to a mark; putare is the hand which cuts the mark in the wood.
[L. Gerschel (I960)]
FURTHER CONFIRMATION
A different confirmation from the preceding is given by F. Skarpa (1934)
in a detailed study of the different kinds of notches used since time
immemorial by herdsmen of Dalmatia (in the former Yugoslavia). In one of
these, the number 1 is represented by a small line, the number 5 by a
slightly longer one, and the number 10 by a line which is much longer than
the others (which is very reminiscent of the measuring scales on rulers and
on thermometers).
Another type of marking used by the Dalmatian herdsmen represented
the number 1 by a vertical stroke, the number 5 by an oblique stroke, and
the number 10 by a cross. In a third type, for the numbers 1, 5 and 10 we
find:
I k X
Fig. 16.41. Herdsmen's tally sticks from Dalmatia [Skarpa (1934), Table III
Does this not rather closely resemble the Roman and Etruscan numerals?
This is all the more striking in that this same type of tally (Fig. 16.41) shows
that the sign for 100, as below, is identical to the Etruscan figure for 100:
Fig. 16.42.
We may well ask why these people used such a figure for 100, but not the
"half-figure” for 50 as the Etruscans did (Fig. 16.35). But close examination
of the tally stick in question tells us why. We find that the marks for the
tens, from 20 to 90, differ from the others by having very small notches on
the edge of the stick, above and below, and the number of these small
notches gives the corresponding number of tens.
195
This method of marking can dispense with a separate sign for 50.
Suppose that a herdsman wants to keep a record of the fact that he has 83
dairy cows and 77 sterile cows, having already counted them as above. All
he needs to do is write the results as shown below on a separate piece of
wood which he will keep by him:
Fig. 16.43. 80 3 70 5 2
So this herdsman has no need for a special symbol for 50, since he has
executed the idea described above as shown in Fig. 16.44:
1 1 2 2 3 3 4 4
I I tl II lit III tilt Mil
xxxxxxxxx
t II II lit lit lltl III!! Mill
12233 445
Fig. 16,44.
A final type of tally known from Dalmatia gives the following figures:
t V X N •
Fig. 16.45. 1 5 10 50 100
The presence of the sign N for 50 seems fairly natural since it can be
made by adding a vertical bar to the figure for 5, just as the sign for 100 is
made by adding a vertical bar to the sign X (Fig. 16.41).
Fig. 16.46. Herdsmen's tally sticks from Dalmatia lSkarpa( 1934 ), Table IVj
Very similar series are to be found in the Tyrol and in the Swiss Alps.
They are found at Saanen on peasant double-entry tallies, at Ulrichen
on milk-measuring sticks, as well as at Visperterminen on the famous
FURTHER CONFIRMATION
tallies of capital, where sums of money lent by the commune or by religious
foundations to the townsfolk were noted using the figures:
- I- + K X
Fig. 16.47. 15 10 50 100
100
100
50
20
5
2
Fig. 16.48. 277 FS
Further evidence can be found in the calendrical ciphers, strange
numerical signs on the calendrical boards and sticks which were in use
from the end of the Middle Ages up to the seventeenth century in the
Anglo-Saxon and West Germanic world, from Austria to Scandinavia
(Fig. 16.52 to 16.54).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Fig. 16.49.
These wooden almanacs give the Golden Number of the nineteen-year
Metonic cycle in the graphical variants of the numerals shown on Fig. 16.52
to 16.54. See E. Schnippel (1926).
The figures used in the English clog almanacs of the Renaissance (see
Fig. 16.54) are the following:
• • • •
• • •
• •
•
V
HIT
• • ♦ •
• • •
• •
•
+
mi
• • • •
• • •
• •
•
t
tttt
• • « •
• • •
• •
•
t
• •
•
12 3 4
5
6 7 8 9
10
11 12 13 14
15
16 17 18 19
20
21 22
Fig. 16.50.
G R K K K AND ROMAN NUMERALS
and those used in Scandinavian runic almanacs have the prototype:
• • • •
• • •
• •
•
>
>>>>
• • • «
• • •
• •
•
+
+ + + +
• • • •
• • •
• •
•
+
>
♦ + -M-
> > > >
• • • •
• • •
» •
•
4-
+ +
+ +
• •
•
12 3 4
5
6 7 8 9
10
11 12 13 14
15
16 17 18 19
20
21 22
Fig. 16.51.
In all of these notations, which appear dissimilar at first sight but
which on close examination prove to originate with the tally-stick principle,
the signs given to the numbers 1, 5 and 10 are unmistakably similar
to the Roman numerals I, V and X and to the Etruscan numerals I, A, and
+ or X.
Fig. 16.52. “Page”
from a wooden almanac
(Figdorschen Collection,
Vienna, no. 799) [ Riegl
(1888), Table I]
Fig. 16.53. Two
“pages” from a wooden
almanac from the
Tyrol ( 15th century )
(Figdorschen Collection,
Vienna, no. 800) [Riegl
(1888), Table V]
196
Fig. 16.54- F.nglish clog almanac from the Renaissance (Ashmolcan Museum, Oxford, Clog C)
[Schnippel (1926), Table I Hal
Even better, in the nineteenth century the Zunis (Pueblo Indians of
North America living in New Mexico, at the Arizona frontier, whose tradi-
tions go back 2,000 years) still used "irrigation sticks” [F. H. Cushing
(1920)] inscribed with numerals that were just as close to Roman figures:
• a simple notch for the number 1;
• a deeper notch, or an oblique one, for 5;
• a sign in the shape of an X for 10.
20 15 10 5 1
Fig. 16.55. Zuhi irrigation stick (New Mexico). The tally marked towards the right
of the stick totals 24, which is marked as a number at the left-hand end in the form
XXI\, which is reminiscent of the Roman representation XXIV where the principle of
subtraction has been applied. [Cushing (1920)}
There can now be no possible doubt: Roman and Etruscan numerals
derive directly from counting on tally sticks.
We are now in a position to put forward the following explanation of the
genesis of such numerals.
Pastoral peoples, who lived in Italy long before the Etruscans and the
Romans, since earliest antiquity (and possibly even in prehistoric times)
counted by the method of tally sticks, and the Dalmatian herdsmen or the
Zunis, for example, independently discovered the same for their own use.
In a quite natural way, all came to make use of the following signs:
1 I
5 yy °r V or X or X
10 X or Jf or or ^ or
50 or A
100 XC
Fig. 16.56.
Inheriting this ancient tradition, the Etruscans and the Romans who
came after them retained from these only the following:
197
ROMAN NUMERALS FOR LARGE NUMBERS
ETRUSCAN
ROMAN
1 1
1 1
5 A
V 5
10 X or / or +-
X 10
50 A
V 50
100*
*100
Fig. 16.57A.
The Romans then completed the series by adding a sign for 500, and
another for 1,000 (the former was the right-hand half of the latter, itself
generated by drawing a circle on top of the figure for 10 (see Fig. 16.29,
16.30 and 16.35):
Fig. 16.57B. Figures for 1,000 ® or
In their hands, these signs changed form over the centuries until they
were replaced by the alphabetic numerals which we know.
This, therefore, is the most plausible explanation of the origins of
the Roman and Etruscan numerals. The following does not gainsay it.
A. P. Ninni (1899) reports that the Tuscan peasants and herdsmen
were still using, in the last century, in preference to Arabic numerals, the
following signs which they call cifre chioggiotti:
G. Buonamici (1932) saw these as descending from Etruscan or Roman
numerals; may we not with more reason see them as a survival of the
ancient practice of counting by cutting notches, a practice older than any
writing and one which is to be found in every rural community on earth?
ROMAN NUMERALS FOR LARGE NUMBERS
The largest number for which Roman numerals as we know them (and
still sometimes use them) had a separate symbol is 1,000. The simple
application of the additive principle to the seven basic figures of this system
would only take us up to 5,000. Therefore, when we come to make use of
these numerals, we find it effectively impossible to write large numbers.
How do we represent, say, 87,000, except by writing down 87 copies of the
letter M?
The ancients had some trouble getting round the problem, and adopted
a variety of conventions for writing large numbers. The difficulties which
they encountered, as did their successors in the European Middle Ages,
deserve special consideration.
In the Republican period, the Romans had a simple graphical procedure
by which they could assign a special notation to the numbers 5,000, 10,000,
50,000, and 100,000. The principal ones (found sporadically as late as the
Renaissance) are the following:
5,000
Comparing these with each other, and with the various ancient forms
of the symbol for 1,000 (Fig. 16.30), we realise that they have a common
origin. In fact, they are simply stylisations (more or less recognisable) of the
original five signs.
The idea governing the formation of four of these consists of an
extremely simple geometrical procedure. Taking as a starting point the
primitive Roman sign for 1,000 (originally a circle divided in two by a
vertical line), the signs for 10,000 and 100,000 were made by drawing one
GREEK AND ROMAN NUMERALS
or two circles, respectively, around it; and the signs for 5,000 and for 50,000
were made by using the right-hand halves of these (Fig. 16.62):
V'
5,000 50,000
Fig. 16.59E.
Following the same principle, the Romans were able to write the
numbers 500,000, 1,000,000, 5,000,000, etc., in the following forms:
„ 500,000: li or IDDDD 1,000,000: £]£) or CCCCIODOD, etc.
Fig. 16.60. I w
But this kind of graphical representation is complicated, and it is difficult
to recognise numbers above 100,000 at a glance; the Romans do not seem to
have taken it any further. An additional possible reason is that there is no
special word in Latin for numbers greater than 100,000: for example, Pliny
C Natural History, XXXIII, 133) notes that in his time, the Romans were
unable to name the powers of 10 above 100,000. For a million, for example,
they said decks centena milia, “ten hundred thousand”.
Nevertheless such a representation may be found, for numbers up to
one million, in a work published in 1582 by a Swiss writer called Freigius
(Fig. 16.61, 16.62 and 16.70):
I V X L C D CD D'J CCDD DDD CCCDDD DDDD CCCCIDDDD
1 i 10 1 10 2 l 10 3 i 10“ i 10 s i 10 6
5 5x10 5 xlO 2 5 x 10 3 5 x 10 4 5 x 10 s
Fig. 16.61.
Other conventions were frequently used by the Romans, and may be
found in use in the Middle Ages, which simplified the notation of numbers
above 1,000 and allowed considerably larger numbers to be reached.
In one of these, a horizontal bar placed above the representation of a
number meant that that number was to be multiplied by 1,000. In this way
all numbers from 1,000 to 5,000,000 could be easily written.
It should however be noted that this convention could sometimes cause
confusion with another older convention, in which, in order to distinguish
between letters used to denote numbers from those used to write words,
the Romans were in the habit of putting a line above the letters being used
as numerals, as can be found in certain Latin abbreviations such as
IIVIR = duumvir; IIIVIR = triumvir
198
Fig. 16.64.
199
ROMAN NUMERALS FOR LARGE NUMBERS
Probably it is for this reason that at the time of the Emperor Hadrian
(second century CE) the multiplication by 1,000 was indicated by placing a
vertical bar at either side, as well as the horizontal line on top.
Reconstructed examples:
35,000
1 XXXV 1
35 x 1,000
557,274
1 DLVII |
CCLXXIV
557x1,000
+ 274
According to some authors, the logical continuation of the convention of
placing a line above the number was to place a double line to represent
multiplication by 1,000,000, thus allowing the representation of numbers
up to 5,000,000,000:
1.000. 000.000 R = 1,000x1,000,000
2.300.000. 000 MMCCC = 2,300x 1,000,000
However no evidence of this in currently known Roman inscriptions has
been found.
Fig. 16 . 65 .
However, this notation was generally reserved for a quite different
purpose.
Fig. 16 . 66 . The archaic Roman numerals
being used in a work by Petrus Bungus on the
mystical significance of numbers (Mysticae
numerorum significationes opus . . .)
published at Bergamo in 1584-1585.
(Bibliotheque nationale, Paris [R. 7489])
Every Roman numeral enclosed in a kind of incomplete rectangle was, in
fact, usually supposed to be multiplied by 100,000, which allowed the
representation of all numbers between 1,000 and 500,000,000.
Examples from Latin inscriptions
from the Imperial period in Rome:
Fig. 16 . 67 .
1 XII 1 a 1,200,000
12 X 100,000
1 XIII 1 b 1,300,000
13 X 100,000
1 00 ~ 1 c 200,000,000
2,000 x 100,000
a. Cf. CIL, I. 1409
b. Cf. CIL, VIII, 1641
j c. Inscription from Ephesus, 103 CE (Cagnat (1899)1 j
Nuneiatit.
(3D c *“
O O 0003
CCID3
CCI>3
X
X
CCt-CC
DMC
DM3
IM1
U09C>
CCDD
CC-IOD OO
//lot.
CCDD « »
CClOO ■»
IJII*
0CD3"""
CCIDD sss
/jlll-
cax> «
CCfDD Oo
/flM.
CCDO »
//••A.
I MDCLI I LXXVI1I CCCXVI
1,651 x 100,000 + 78 x 1,000 + 316
■>
165,178,316
Fig. 16 . 68 .
Fig. 16 . 69 .
Frontispieces from
works by Descartes
(published 1637) and
Spinoza (published
1677). The dates are
written in the archaic
Roman numerals.
D I SCOUR S
DE LA METHODE
Rir bBKmduirt U ration & dndrr
la «ri* dans Its kitnces
Ktl
LA DIOPTRIQVK
I.ES METRO RES
n
l.A CEOMETRIE
dt cdr tmod*
A LEYDE
Ikl'hfnnvitdrlAN MAIRK
cl i li c invn
Aum 7 >rlm’hgt
B. D.
OPERA
POSTHUM A,
i fir it: T
txeiltfiir
1.CUI fcw'i«r 4 'Wl t r . lVlbr«>
c I 3 I ;> c i.xxvn
Fig. 16 . 70 . Archaic Roman numerals in a
work by Freigius, published in 1582 [Smith,
D.E (1958)]
I 1
V $■
X le
L to
C. io®.
D 1 3- ?®o Qpiflgwt#.
CXj « • Clo X\Mm. M Hit
fcs. 133 5°«»
j^.l333
CCCI3C3- \ •b««o.C*nhmmitf<d.
<333^ QuwigMiftinuIta.
CCCCI333D CCCCl3333 lo®»o®o.XJe««3
ittm \* mill*
RtwniTUtfT«t<ruii prejtWuirdirultw <k>a reiww
»li« 1 Httid pkud fignficdn notu.Uf,
QD GO ICO*
Cl 3- C 1 3 CIO. iooe.
CI3.13. 1500 ® V
GREEK AND ROMAN NUMERALS
200
Fig. 16.71. Detail of a page of a Portuguese manuscript of 1200 CE, referring to the Venerable
Bede's method of calculation (Lisbon Public Library, MS Alcobapa 394 (426), folio 252) [Burnham in
PIB, plate XV] _
Figures on XL XXX = 40,030
the drawings: L XC = 50,090
LX = 60,000
But these kinds of notations could only cause confusion and errors of
interpretation - as a future Roman Emperor learnt to his cost, according to
Seneca ( Galba , 5).
On succeeding to his mother Livia, Emperor Tiberius had to pay large
sums of money to her legatees. Tiberius’s mother had written the amount
of her legacy to young Galba in the form: lCCCl.
But Galba had not taken the precaution of checking that the amount was
written out in words. So when he presented himself to Tiberius, Galba
thought that the five Cs had been enclosed in vertical lines, and that there-
fore the sum due to him was
500 x 100,000 = 50,000,000 sesterces.
But Tiberius took advantage of the fact that the two sides-bars were very
short, and claimed that this representation was a simple line above the five
Cs. “My mother should have written them as C CCCCC'i if you were to
be right,” he said. Since the simple line only represented multiplication by
1,000, Galba only received from Tiberius the sum of
500 x 1,000 = 500,000 sesterces
Which goes to show that an unstable notation system can turn a large
fortune into a mere pittance!
The Romans also devised other conventions. Instead of repeating the
letters C and M for successive multiples of 100 or 1,000, they first wrote
the number of hundreds or thousands they wanted, and then placed the
letter C or M either as a coefficient or as a superscript index:
200: II.C or II C 2,000: II.M or II M
300: III.C or III C 3,000: 1II.M or 1II M
However, instead of simplifying the system, these various conventions
only complicated it, since the principle of addition was completely
subverted by the search for economy of symbols.
We therefore see the complexity and the inadequacy of the Roman number-
system. Ad hoc conventions based on principles of quite different kinds
made it incoherent and inoperable. There is no doubt that Roman numerals
constituted a long step backwards in the history of number-systems.
THE GREEK AND ROMAN ABACUSES
Given such a poor system of numerals, the Greeks, Etruscans and Romans
did not use written numbers when they needed to do sums: they used
abacuses.
The Greek historian Polybius (c. 210-128 BCE) was no doubt referring to
one of these when he put the following words into the mouth of Solon (late
seventh century to early sixth century BCE).
Those who live in the courts of the kings are exactly like counters on
the counting table. It is the will of the calculator which gives them their
value, either a chalkos or a talent. ( History , V, 26)
We can all the better understand the allusion when we know that the
talent and the chalkos were respectively the greatest and the least valuable of
the ancient Greek coins, and they were represented by the leftmost and
rightmost columns of the abacus.
Fig. 16.72. Detail of the Darius Vase from Canossa, c. 350 BCE (Museo Archeologico Nazionale,
Naples)
201
THE GREEK AND ROMAN ABACUSES
Fig. 16.73. The Table of Salamis, originally considered to be a gaming table, which is in fact
a calculating apparatus. Date uncertain (fifth or fourth century BCE). ( National Museum of
Epigraphy, Athens)
The writings of many other Greek authors from Herodotus to Lysias also
bear witness to the existence and use of the abacus.
Descriptions of the Greek abacus are not only to be found in literary
text, but also in images. The “Darius Vase” is the most famous example
(Fig. 16.72). It is a painted vase from Canossa in southern Italy (formerly a
Greek colony) and dates from around 350 BCE. The various scenes painted
on it are supposed to describe the activities of Darius during his military
expeditions.
In one detail of the vase, we can see the King of Persia’s treasurer using
counters on an abacus to calculate the tribute to be levied from a conquered
city. In front of him, a personage hands him the tribute, while another begs
the treasurer to allow a reduction of taxes which are too heavy for the city
he represents.
The Greek calculators stood by one of the sides of the horizontal table
and placed pebbles or counters on it, within a certain number of columns
marked by ruled lines. The counters or pebbles each had the value of 1.
A document from the Heroic Age (fifth century BCE) gives us a more
detailed idea. It is a large slab of white marble, found on the island of
Salamis by Rhangabes, in 1846 (Fig. 16.73).
It consists of a rectangular table 149 cm long, 75 cm wide and 4.5 cm
thick, on which are traced, 25 cm from one of the sides, five parallel lines;
and, 50 cm from the last of these lines, eleven other lines, also parallel, and
divided into two by a line perpendicular to them: the third, sixth and ninth
of these lines are marked with a cross at the point of intersection.
Furthermore, three almost identical series of Greek letters or signs are
arranged in the same order along three of the sides of the table. The most
complete of the series has the following thirteen symbols in it:
TP X F> H PATH ICTX
Fig. 16.74.
As we saw at the beginning of this chapter, these in fact correspond to
the numerical symbols of the acrophonic number system (Fig. 16.1), and
they serve here to represent monetary sums expressed in talents, drachmas,
obols, and chalkoi, that is to say in multiples and sub-multiples of the
drachma.
These symbols represented, from left to right in the order shown, 1 talent
or 6,000 drachmas, then 5,000, 1,000, 50.0, 100, 50, 10, 5 and 1 drachmas,
then 1 obol or one sixth of a drachma, 1 demi-obol or one twelfth of a
drachma, 1 quarter-obol or one twenty-fourth of a drachma, and finally 1
chalkos (one eighth of an obol or one forty-eighth of a drachma). (Fig. 16.75)
GREEK AND ROMAN NUMERALS
202
T
1 talent
First letter of TALANTON, “talent”
1 "
5,000 drachmas
X
1,000 drachmas
First letter of CHILIOI, “thousand” (drachmas)
n
500 drachmas
H
100 drachmas
First letter of HEKATON, “hundred" (drachmas)
l»>
50 drachmas
A
10 drachmas
First letter of DEKA, “ten” (drachmas)
r
5 drachmas
First letter of PENTE, “five” (drachmas)
1-
1 drachma
1
1 obol
Unit mark for counting obols
c
1/2 obol
Half of the letter O. first letter of OBOLION
T
1/4 obol
First letter of TETARTHMORION
X
1 chalkos
First letter of CHALKOUS
1 talent = 6,000 drachmas
1 drachma= 6 obols
1 obol = 8 chalkos
Fig. 16 . 75 .
In the abacus of Salamis, each column was associated with a numerical
order of magnitude.
The pebbles or counters disposed on the abacus changed value accord-
ing to the position they occupied (see Fig. 16.76).
The four columns at the extreme right were reserved for fractions of
a drachma, the one on the extreme right being for the chalkos, the next for
the quarter-obol, the third for the demi-obol, and the last for the obol.
The next five columns (to the right of the central cross on Fig. 16.75)
were associated with multiples of the drachma, the first on the right being
for the units, the next for the tens, the third for the hundreds, and so on. In
the bottom half of each column, one counter represented one unit of the
value of the column. In the upper half, one counter represented five units of
the value of the column.
The last five columns (to the left of the central cross in Fig. 16.76) were
associated with talents, tens of talents, hundreds, and so on. One talent
being worth 6,000 drachmas, the calculator would replace counters
corresponding to 6,000 by one counter in the talents columns (sixth from
the right).
As a result of this method of dividing up the table, additions, subtrac-
tions and multiplications could be done (Fig. 16.77 and 16.78).
TALENTS ' DRACHMAS
0 s
So
!§
1 O 1
1 o' '
O
o' ~
tu O
b *
o4 —1
< <
d a:
o' u
•
•
•
•
•
•
3
1
1
1
Fig. 16 . 76 . The principle of the
Greek abacus from Salamis, showing
the representation of the sum “17
talents, 1,173 drachmas, 3 obols, 1
demi-obol 1 quarter-obol and 1
chalkos”. ( C h a 1 k o i is the plural
ofc h a 1 k o s .)
OPERATION
• 3,646 drachmas, 4 obols,
1/2 obols, and 1 chalkos
O 3,117 drachmas, 1 obol,
1/2 obol, and 1/4 obol
1
1
1
•
•
oi
o
•
O
RESULT
1 talent, 764 drachmas,
1/4 obol, and 1 chalkos
Fig. 16 . 77 . The method of addition on the Salamis abacus, showing the addition of “3,646
drachmas, 4 obols, 1/2 obol and 1 chalkos” (shown in black) and “3,117 drachmas, 1 obol, 1/2
obol and 1/45 obol” (shown in white). By reducing the counters according to the rules, the result is
obtained as “1 talent, 764 drachmas, 1/4 obol, and 1 chalkos".
203
*1.31 -tdVdHmJ *
T fKPHPA r HCTX
• • • • • •
Fig. 16 . 78 . To multiply "121 drachmas, 3 obols, 1/2 obol, and 1 chalkos ” by 42, for example, we
start by placing the multiplier 42 on the abacus, by laying out the corresponding counters under the
appropriate number-signs on the left of the table. Then the multiplicand, the sum of money, is laid
out under the number-signs of one of the two series on the right (black circles). Then by manoeuvring
the counters the result is obtained (see a similar method in Fig. 16.84).
The Etruscans and their Roman successors also employed abacuses
with counters. In Fig. 16.79 we reproduce an Etruscan medallion, a carved
stone which shows a man calculating by means of counters on an abacus,
noting his results on a wooden tablet on which Etruscan numerals can be
seen (Fig. 16.35).
iny Roman texts mention it:
Coponem laniumque balneumque, ton-
sorem tabulamque calculosque et paucos
. . . haec praesta tnihi, Rufe . . .
An innkeeper, a butcher, baths, a
barber, a calculating table (= tabu-
lamque calculosque) with its counters
. . . fetch me all that, Rufus ... *
Computat, et cevet. Ponatur calculus,
adsint cum tabula pueri; numeras sester-
tia quinque omnibus in rebus; numer-
cntur deinde labores.
He calculates, and he wriggles his
rear. Let the counter (= calculus) be
placed, let the slaves bring the (calcu-
Fic. 16 . 79 . The medallion with the Etruscan latin g) table: y ou find five thousand
calculator (date uncertain). ( Coin Room. sesterces in all; now make the total of
Bibhotheque nationale, Paris, handle 1898) m y WO rRs t
At Rome, the abacus with counters was a table, on which parallel lines
* Martial, Epigrams, Vol. 1, book 2, 48
+ Juvenal, Satires, IX, 40-43
THE GREEK AND ROMAN ABACUSES
separated the different numerical orders of magnitude of the Roman
number-system. The Latin word abacus denotes a number of devices with
a flat surface which serve for various games, or for arithmetic (Fig. 16.80).
Each column generally symbolised a power of 10. From right to left, the
first was associated with the number 1, the next with the tens, the third
with the hundreds, the fourth with the thousands, and so on. To represent
a number, as many pebbles or counters were placed as required. The Greeks
called these counters psephoi, ("pebble” or “number”) and the Romans
called them calculi (singular: calculus). Certain authors (notably Cicero,
Philosophica Fragmenta, V, 59) called them aera (“bronze”), alluding to the
material they were often made of after the Imperial epoch (Fig. 16.81).
Fig. i 6 . 8 i . Roman calculating counters.
After the originals in the Stddtisches
Museum, Weis, Germany
To represent the number 6,021 on the columns of the abacus we
therefore place one counter in the first column, two in the second, none
in the third, and six in the fourth.
For 5,673 we place three in the first, seven in the second, six in the third,
and five in the fourth (Fig. 16.82).
Fig. 16.82. The
principle of the Roman
abacus with calculi
Fig. 16.83.
Simplification of the
principle of the Roman
abacus with calculi
GREEK AND ROMAN NUMERALS
To simplify calculation, each column is divided into an upper and a
lower part. A counter in the lower half represents one unit of the value of
the column, and a counter in the upper half represents half of one unit of
the value of the next column (or five times the value of the column it is in).
For the upper halves we therefore have five for the first column, fifty for the
second, 500 for the third, and so on (Fig. 16.83).
By cleverly moving the counters between these divisions (adding to and
taking away from the counters in each division) it is possible to calculate.
To add a number to a number which has already been set up on the
abacus, it is set up in turn, and then the result is read off after the various
manipulations have been performed. In a given column, if ten or more
counters are present at any time then ten of these are removed and one is
placed in the next higher column (to the left) (Fig. 16.82). On the simplified
abacus, this procedure is somewhat modified. If there are five or more
in the lower half, then five are removed and one is placed in the upper
half; while if two or more are present in the upper half then two are
removed, and one is placed in the lower half of the next column, to the left
(Fig. 16.83). Subtraction is carried out in a similar way, and multiplication
is done by addition of partial products.
For example, to multiply 720 by 62, we start setting up the numbers 720
and 62 as shown in Fig. 16.84A. Then the 7 of 720 (worth 700) and the 6
of 62 (worth 60) are multiplied, to give 42 (worth 42,000). Therefore two
counters are placed in the fourth column and four counters are placed in
the fifth.
4 2
First partial product: 6 X 7 = 42
Fig. 16.84A.
62 Multiplier
720 Multiplicand
Then the 7 of 720 (worth 700) and the 2 of 62 (worth 2) are multiplied
to give 14 (worth 1,400), and four counters are placed into the third column
and one is placed into the fourth.
204
Second partial product (shown
white circles): 2 x 7 = 14
Fig. 16.84B.
1 4
62 Multiplier
720 Multiplicand
Now the 7 of 720 has done its work, and can be removed. Next we
multiply to 2 of 720 (worth 20) by the 6 of 62 (worth 60) to get 12 (worth
1,200), and so two counters are placed in the third column and one is
placed in the fourth.
Finally, the 2 of 720 (worth 20) and the 2 of 62 (worth 2) are multiplied
to get 40. Therefore four counters are placed in the second column.
1 2 4
Now the various counters on the table are reduced as explained above to
give the required result of the multiplication:
720 x 62 = 44,640
205
4 4 6 4 0
Fig. 16.84E.
Result
Calculating on the abacus with counters was
therefore a protracted and difficult procedure, and
its practitioners required long and laborious train-
ing. It is obvious why it remained the preserve of a
privileged caste of specialists.
But traditions live on, and for centuries these
methods of calculation remained extant in the
West, deeply attached to Roman numerals and
their attendant arithmetic. They even enjoyed
considerable favour in Christian countries from
the Middle Ages up to relatively recent times.
All the administrations, all the traders and all
the bankers, the lords and the princes, all had their calculating tables* and
struck their counters from base metal, from silver or from gold, according
to their importance, their wealth, or their social standing. “I am brass, not
silver!” was said at the time to express that one was neither rich nor noble.
The clerks of the British Treasury, until the end of the eighteenth century,
used these methods to calculate taxes, employing exchequers, or checker-
boards (because of the way they were divided up). This is why the British
Minister of Finance is still called “Chancellor of the Exchequer”.
Fig. 16. 85 a. The use of abacuses with counters continued in Europe until the Renaissance (and in
some places until the French Revolution). Here we see an expert calculator in a German illustration
from the start of the sixteenth century. [ Treatise on Arithmetic by Kobel, published at Augsburg
in 1514]
* The existence of large numbers of treatises on practical arithmetic which mention these procedures
throughout Europe in the sixteenth, seventeenth, and eighteenth centuries gives an idea of how widespread
these practices were before the French Revolution.
THE GREEK AND ROMAN ABACUSES
Fig. 16.85B. "Madame
Arithmetic " teaching
young noblemen the art
of calculation on the
abacus (sixteenth -
century French tapestry).
(Cluny Museum)
Fig. 16.86. Calculating counter bearing the arms of
Montaigne (and surrounded with the chain of the order of
Saint Michel de Montaigne). This counter was found
earlier, in the ruins of the Chateau de Montaigne, though
its original diestamp was found in the nineteenth century.
Fig. 16 . 88 . Calculating table with three
divisions, sixteenth-seventeenth century, as
formerly used in Switzerland and Germany
to calculate rates and taxes. The letters to be
seen on it are (from the top): d for the
deniers (denarius/- sfor the sols or shillings
(solidus/ lb or lib for the pounds (libras/*
then X, C and M for 10, 100 and 1,000
pounds. (Historical Museum of Basel. Inv.
1892.209. Neg. 1500)
GREEK AND ROMAN NUMERALS
206
At the time of the Renaissance, many writers make reference to this.
Thus Montaigne (1533-1592):
We judge him, not according to his worth, but from the style of his
counters, according to the prerogatives of his rank. ( Essays , Book III,
Bordeaux edition, 192, 1, 17)
Likewise Georges de Brebeuf (1618-1661), adapting the formula of
Polybius:
Courtesans are counters;
Their value depends on their place;
If in favour, why then it’s millions,
But zero if they’re in disgrace.
Again Fenelon (1651 - 1715), who makes Solon say:
The people of the Court are like the counters used for reckoning: they
are worth more or less depending on the whim of the Prince.
And Boursault (1638-1701):
Never forget, if I may have your grace,
Whatever more power either of us might have had
We are still but counters stamped with value by the King.
Finally, Madame de Sevigne, who sent these words to her daughter
in 1671:
We have found, thanks to these excellent counters, that I would have
had five hundred and thirty thousand pounds if I counted all my little
successions.
The abacus of this period also consisted of a table marked out into
divisions corresponding to the different orders of magnitude (Fig. 16.87
and 16.88). Numbers were set up on the table with counters (made of the
most diverse materials), whose values depended on where they were placed.
Placed on successive lines, from bottom to top, a counter would be worth 1,
10, 100, 1,000, and so on. Between successive lines, a counter was worth five
of the value of the line below it (Fig. 16.89 and 16.90).
POUND
SOL
DENIER
•
•
•
•
80 (= 50 + 30)
9 (= 5 + 4)
5
7 (= 5 + 2)
10 6
10 s
to 4
10 3
10 2
10
1
GULDEN
GROSCHEN
PFENNIG
•
•
•
6.M8 GULDEN
BGROSCHEN 3 PFENNIG
Fig. 16 . 90 . Representation of the sum of 6,148 gulden. 18 groschen and 3 pfennigs on the
German calculating table (sixteenth-eighteenth century).
The counting tables facilitated addition or subtraction, but lent
themselves with difficulty to multiplication or division and even less well
to more complex operations.
Arithmetical operations practised by this means had little in common
with the operations of modern arithmetic with the same names.
Multiplication, for example, was reduced to a sum of partial products or to
a series of duplications. Division was reduced to a succession of separation
into equal parts.
Such difficulties were at the origin of the fierce polemic which, from the
beginning of the sixteenth century, ranged the abacists on one side, clinging
to their counters and to archaic number-systems like the Greek and the
Roman, against the algorists on the other, who vigorously defended calcula-
tion with pen and paper, the ancestor of modern methods.
Here, for example, is what Simon Jacob (who died in Frankfurt in 1564)
had to say about the abacus:
It is true that it seems to have some use in domestic calculations, where
it is often necessary to total, subtract, or add, but in serious calcula-
tions, which are more complicated, it is often an embarrassment. I do
not say that it is impossible to do these on the lines of the abacus, but
every advantage that a man walking free and unladen has over he who
stumbles under a heavy load, the figures have over the lines.
Pen and paper soon gained the day amongst mathematicians and
astronomers. The abacus was in any case used almost exclusively in finance
and in commerce. Only with the French Revolution would the use of the
abacus finally be banished from schools and government offices.
Fig. 16 . 89 . The layout of the sum "89 pounds, 5 sols and 7 deniers ” on the French calculating
table (sixteenth-eighteenth century).
207
ABACUS IN WAX AND ABACUS IN SAND
ABACUS IN WAX AND ABACUS IN SAND
The Latin word abacus derives from the Greek abax or abakon signifying
“tray”, “table” or "tablet”, which possibly in turn derives from the Semitic
word abq, “sand”, “dust”.
It is true that the “abacus in sand” is part of these oriental traditions, but
it is mentioned also in the Graeco-Roman West, along with the abacus with
counters, especially by Plutarch and by Apuleus. It consisted of a table
with a raised border which was filled with fine sand on which the sections
were marked off by tracing the dividing lines with the fingers or with a
point. (Fig. 16.91).
F i c . 16 . 91 . Mosaic showing Archimedes (2877-212 BCF.) calculating on an abacus with
numerals (sand or wax), at the moment when a Roman soldier was about to assassinate him
(eighteenth century). (Stddtischc Galerie/Ilebieghaus, Frankfurt)
Another type of calculating instrument used in Rome was the abacus in
wax. It was a true portable calculator which was carried hanging from the
shoulder, and it consisted of a small board of wood or of bone coated with
a thin layer of black wax; the columns were marked by tracing in the wax
with a pointed iron stylus (whose other end, being flat, was used to erase
marks by pressing on the surface of the wax).
A specimen from Rome, dating from the sixth century, has been
described by D. E. Smith. It is in the collections of the John Rylands Library
in Manchester. It is made of bone, and consists of two rectangular iron
plates joined by an iron hinge, with three iron styluses.
Horace (65-8 BCE) was perhaps alluding to this instrument in this
passage from the first book of Satires *
. . . causa fuit pater his, qui macro pauper agello noluit in Flavi ludum me
mittere, magni quo pueri magnis e centurionibus orti laevo suspensi loculos
tabulanque lacerte ibant octonos referentes Idibus aeris . . .
I owe this to my father who, poor and with meagre possessions, did
not wish to send me to the school of Flavius, where the noble sons
of noble centurions went, their box and their board ( tabulanque )
hanging from their left shoulder, paying at the Ides their eight bronze
coins. . . .
The Europeans of the Middle Ages probably also used one or other of
these, as well as the abacus with counters.
In his Vocabularium (1053), Papias (who may be considered one of the
authorities on the knowledge of his time) also talks of the abacus as “a table
covered with green sand”, which is exactly what can be found in Remy
d’Auxerre in his commentary on the Arithmetic of Martianus Capella
(c. 420^90 CE) where he describes it as “a table sprinkled with a blue or
green sand, where the figures [the numbers] are drawn with a rod”.
As for the abacus in wax, Adelard of Bath (c. 1095-c. 1160) alludes to it
as follows [B. Boncompagni (1857)]:
Vocatur (Abacus) etiam radius geometricus, quia cunt ad multa pertineat,
maxime per hoc geometricae subtilitates nobilis illuminantur.
(The abacus) is also called the “geometrical radius” since it permits
so many operations. In particular, thanks to it the subtleties of geo-
metry become perfectly clear and comprehensible.
Finally, it is perfectly possible that Radulph de Laon (c. 1125) was think-
ing of one or other of these in writing [D. E. Smith & L. C. Karpinski
(1911)]:
... ad arithmaticae speculationis investigandas rationes, et ad eos qui
musices modulationibus deserviunt numeros, necnon et ad ea quae astrolo-
gorum sollerti industria de variis errantium siderum cursibus . . . Abacus
valde necessarius inveniatur.
For the examination of the rules of mathematical thought and of the
numbers which are at the base of musical modulations, and for the
calculations which, thanks to the skilful industry of the astrologers,
explain the various trajectories of the moving stars, the abacus shows
itself absolutely indispensable.
* Satires, I, VI, 70-75
GREEK AND ROMAN NUMERALS
These authors do not however say what kinds of numeral were used with
the abacuses of these two types, though especially at the time of Papias,
Adelard and Radulph the Arab numerals were used and were already
well known in Europe. But the Greek numerals were used also (from a = 1
to 0 =9) which had been much better known before this time, as well
as the Roman numerals which were in a way the “official" numerals of
mediaeval Europe.
In any case, which figures were used is not of great importance with
instruments of this type for, by reason of its structure (which assigns
variable values to the symbols according to their positions), the columns of
the abacus in sand or the abacus in wax can render even the most primitive
figures operational. Of which the proof follows, for the Roman numerals.
Let us again take up the multiplication of 720 by 62, and try to do it with
Roman numerals on a tablet covered with sand or with wax.
The technique works for any decimal number-system whatever, provided
the figures greater than or equal to 10 are not used. We start by writing the
720 and the 62 in the bottom lines. (Fig. 16.92A)
Fig. 16.92A.
Now we multiply the 7 (700) of 720 by the 6 (60) of 62 and get 42
(42,000). Therefore we write this result at the top, the 2 in the fourth
column and the 4 in the fifth. (Fig. 16.92B)
Then we multiply the 7 (700) of 720 by the 2 of 62 and get 14 (1,400),
and we write this result at the top below the last one, with a 4 in the third
column and a 1 in the fourth. (Fig. 16.92C)
Now we can forget the 7 of 762, and multiply the 2 (20) of 720 by the
6 (60) of 62, and get 12 (1,200) which we again write at the top below
the last result: 2 in the third column and 1 in the fourth. (Fig. 16.92D)
209
Fig. 16.92D.
Finally we multiply the 2 (20) of 720 by the simple 2 of 62 and get 4 (40),
so we write a 4 in the second column.
Fig. 16.92E.
We can now erase the 720 and the 62, and proceed to reduce the figures
which remain. Here we can start with the second column.
Since this figure is less than 10, we pass immediately to the next column,
the third. We add 4 and 2 and get 6, which is less than 10, we erase the two
figures 4 and 2 and write in 6.
ABACUS IN WAX AND ABACUS IN SAND
Then we pass to the fourth column, where we add 2, 1 and 1 to get 4,
which is less than 10, so we erase the three figures and write a 4 in the
fourth column.
The fifth column will remain unchanged since the single figure in it is
less than 10.
It only remains to read the result directly off the columns:
720 x 62 = 44,640
THE FIRST POCKET CALCULATOR
As well as the “desk-top models for school and office”, some of the Roman
accountants used a real “pocket calculator” whose invention undoubtedly
predates our era. The proof of this is a bas-relief on a Roman sarcophagus
of the first century, which shows a young calculator* standing before his
master, doing arithmetic with the aid of an instrument of this type (Fig.
16.96). This instrument consisted of a small metal plate, with a certain
number of parallel slots (usually nine). Each slot was associated with an
order of magnitude, and mobile beads could slide along themT
Ignoring for the moment the two rightmost slots, the remaining seven
are divided into two distinct segments, a lower and an upper. The lower one
* Among the Romans, the word calculator meant, on the one hand, a “master of calculation” whose
principal task was to teach the art of calculation to young people using a portable abacus or an abacus
with counters; and, on the other hand, the keeper of the accounts or the intendant in the important houses
of the patricians, where he was also called di spa} sat or. If these were slaves, they were called cahulonix but
if they were free men then they were called ailculalores or numcrarii.
+ As well as the abacus shown in Fig. 16.94, we know of at least two other examples. One is in the British
Museum in London, and the other in the Museum of the Thermae in Rome.
GREEK AND ROMAN NUMERALS
210
contains four sliding beads, and the upper one, which is the shorter,
contains only one.
In the space between these two rows of slots a series of signs is inscribed,
one for each slot. These are figures expressing the different powers of 10
according to the classical Roman number-system which the bankers and
the publicans used to count by as, by sestertii, and by denarii* (Fig. 16.62
and 16.67 above):
fxl
io 6
Fig. 16 . 93 .
$>)) frld (|) C X I
10 5 10 4 10 3 10 2 10 1
Fig. 16 . 94 . Roman "pocket
abacus" (in bronze), beginning
of Common Era. (Cabinet des
medailles, Bibliotheque
nationale, Paris.) (br. 1925)
IX VIII VII VI V IV III II 1
Fig. 16 . 95 . The principle of
the portable Roman abacus.
This specimen belonged to the
German Jesuit Athanasius
Kircher (1601-1680). (Museum
of the Thermae, Rome)
* The unit of the Roman monetary system was the as of bronze. Its weight continually diminished, from the
origin of the monetary system around the fourth century BCLi until the Empire. It successively weighed
273 gm, 109 gm, 27 gm, 9gm and finally 2.3 gm. Its multiples were the sestertius (first silver. later bronze,
then brass), the denarius (silver), and, from the time of Caesar, the aureus (gold). In the third century BCE,
1 denarius was 2.5 as, 1 sestertius was 4 denarii or 10 as. From the second century BCE, after a general mone-
tary reform, 1 sestertius was 4 as, 1 denarius was 4 sestertii or 16 as. and 1 aureus was 25 denarii or 400 as.
Each of these seven slots was therefore associated with a power of 10.
From right to left, the third corresponded to the number 1, the fourth to the
tens, the fifth to the hundreds and so on (Fig. 16.95).
If the number of units of a power of 10 did not exceed 4, it was indicated
in the lower slot by pushing the same number of beads upwards. When it
exceeded 4, the beads in the upper slot was pulled down towards the centre,
and 5 units were removed from the number and this was represented in the
lower slot.
If we are considering a calculation in denarii, the number represented on
the abacus in Fig. 16.95 corresponds (leaving aside the first two slots on the
right) to the sum of 5,284 denarii'. 4 beads up in the lower slot III means
4 ones or 4 denarii', the upper bead down and 3 beads up in the lower slot
IV means (5 + 3) tens or 80 denarii', two beads up in the lower slot V means
two hundreds or 200 denarii', and finally the upper bead down in slot VI
means five thousands or 5,000 denarii.
Fig. 16.96. Bas-relief on a sarcophagus from a tinman tomb dating from the first century CE.
(Capit 'aline Museum, Rome.)
The first two slots on the right were used to note divisions of the as.*
The second slot, marked with a single O, has an upper part with a single
bead, and a lower which has not four, but five beads: it was used to repre-
sent multiples of the uncia (ounce) or twelfths of the as, each lower bead
being worth one ounce and the upper bead being worth six ounces, which
* In Roman commercial arithmetic, fractions of a monetary unit were always expressed in terms of the as,
the basic unit of money which was divided into twelve equal parts called unciae ('ounces’) - which gave its
name to the corresponding unit of value whatever its nature. Each multiple or sub-multiple of the as (or of
the unit which the as represented) was then given a particular name. For example, for the sub-multiples we
have 1/2: as semis ; 1/3: as triens ; 1/4: as quad r any, 1/5: as quincunx: 1/6: as sextans ; 1/7: as septunx: 1/8: as
octans ; 1/9: asdodrans ; 1/10: as dextans ; 1/11: as deunx ; 1/12: as uncia: 1/24: as semuncia: 1/48: as sicilicus:
1/72: as sextula.
211
THE FIRST POCKET CALCULATOR
allows counting up to 11/12 of an as. The first slot, divided into three and
carrying four sliding beads, was used for the half ounce, the quarter ounce
and the duella, or third part of the ounce. The upper bead was was worth
1/2 ounce or 1/24 as if it was placed at the level of the sign: S or i or
i. : the sign of as semuncia, 1/24 of an as.
The middle bead was worth 1/4 ounce or 1/48 as if it was placed at the
level of the sign: 0 or ) or 7 : the sign of as sicilicus, 1/48 of an as.
Finally, either of the two beads at the bottom of the slot was worth 1/3
ounce or 2/72 as if it was placed at the level of the sign: Z or 2 or 2. :
the sign of as duae sextulae, 2/72 of an as or duella.
The four beads of the first slot probably had different colours (one for
the half ounce, one for the quarter ounce, and one for the third of an ounce)
in case the three should find themselves on top of each other (as in
Fig. 16.95). In certain abacuses these three beads ran in three separate slots.
Therefore we have here a calculating instrument very much the same as
the famous Chinese abacus which still occupies an important place in the
Far East and in certain East European countries.
With a highly elaborate finger technique executed according to precise
rules, this “pocket calculator” (one of the first in all history) allowed those
who knew how to use it to rapidly and easily carry out many arithmetic
calculations.
Why did Western Europeans of the Middle Ages - the direct heirs of
Roman civilisation - carry on using ancient calculating tables in preference
to this more refined, better conceived, and far more useful instrument? We
still do not know. Perhaps the invention belonged to one particular school
of arithmeticians, which disappeared along with its tools at the fall of the
Roman Empire.
L E TT E R S AND NUMBERS
CHAPTER 17
LETTERS AND NUMBERS
THE INVENTION OF THE ALPHABET
The invention of the alphabet was a huge step in the history of human
civilisation. It constituted a far better way of representing speech in
any articulated language, for it allowed all the words of a given language
to be fixed in written form with only a small set of phonetic signs called
letters.
This fundamental development was made by northwestern Semites
living near the Syrian-Palestinian coast around the fifteenth century
BCE. The Phoenicians were bold sailors and intrepid traders: once
they had broken with the complex writing systems of the Egyptians and
Babylonians by inventing their simpler method for recording speech, they
took it with them to the four corners of the Mediterranean world. In the
Middle East, they brought the idea of an alphabet to their immediate
neighbours, the Moabites, the Edomites, the Ammonites, the Hebrews,
and the Aramaeans. These latter were nomads and traders too, and thus
spread alphabetic writing to all the cultures of the Middle East, from
Egypt to Syria and the Arabian peninsula, from Mesopotamia to the
confines of the Indian subcontinent. From the ninth century BCE, alpha-
betic writing of the Phoenician type also began to spread around the
Mediterranean shores, and was gradually adopted by speakers of Western
languages, who adapted it to their particular needs by modifying or adding
some characters.
The twenty-two letters of the Phoenician alphabet thus gave rise directly
to Palaeo-Hebraic writing (in the era of the Kings of Israel and Judaea),
whence came the modern alphabet of the Samaritans, who have main-
tained ancient Jewish traditions. Aramaic script developed a little later,
whence came the “square” or black-letter Hebrew alphabet, as well as
Palmyrenean, Nabataean, Syriac, Arabic, and Indian writing systems. At
the same time, Phoenician letters gave birth to the Greek alphabet, the first
one to include full and rigorous representation of the vowel sounds. From
Greek came the Italic alphabets (Oscan, Umbrian and Etruscan as well
as Latin), and at a later stage the alphabets used for Gothic, Armenian,
Georgian, and Russian (the Cyrillic alphabet). In brief, almost all alphabets
in use in the world today are descended directly or indirectly from what the
Phoenicians first invented.
212
mm
\«
V
rt-EViM 3 U.++Q «4HW A
* w 1 ° t- i 6 \
.i * 0 1° WM mrWW+M tilt*,
if « ■ *
x wmik. */. A
iyi W# \
Fig. 17 . 1 . Stela of the Moabite king Mesha, a contemporary oj the Jewish kings Ahab (874-853 BCE)
and Joram (851-842 BCE), in the Louvre (M. Lidzbarski, vol II, tab. 1). This is one of the oldest
examples of palaeo-Hebrew script (used here to write in Moabitic, a dialect of Canaan close to Hebrew
and Phoenician). This stela, put up in 842 BCE at Diban-Gad, the Moabite capital, gives several clues
to the relations that existed between Moab and Israel at that time; it is also the only document of the
period found so far outside of Palestine in which the name of the God Yahweh is explicitly mentioned.
213
LETTERS AND ALPHABETIC NUMBERING
LETTERS AND ALPHABETIC NUMBERING
It is a remarkable fact that the names and the order of the twenty-two
letters of the original Phoenician alphabet have been maintained more or
less intact by almost all derivative alphabets, from Hebrew to Aramaic,
from Etruscan to ancient Arabic, from Greek to Syriac. According to J. G.
Fevrier (1959), we can be sure of the order of the Phoenician letters because
there are alphabetic primers in Etruscan dating from 700 BCE the order of
whose letters is the same as the one encoded in many acrostics in biblical
Hebrew (the lines of Psalms 9, 10, 25, 34, 111, 112, etc. begin with each of
the letters of the Hebrew alphabet, in alphabetic order). In fact, the same
order of the letters is even found in Ugaritic primers, dating from the
fourteenth century BCE. These primers contain thirty letters written in
cuneiform: however, as M. Sznycer has shown, the eight “extra” Ugaritic
signs, intercalated or appended to the original twenty-two, do not alter the
fundamental Phoenician order of the letters.
It is because the order of the ABC ... is so ancient and so fixed that
letters were able to play an important role in numbering systems.
ARCHAIC
PHOENICIAN
PALAEO-HEBREW SCRIPT
Aramaic cursive. Elephantine,
5th century BCE
HEBREW
Inscription of Akhiram,
11th century BCE
Inscription of Yehimilk,
10th century BCE
Stela of Mesha,
842 BCE
Samarian Ostraca,
8 th century BCE
Arad Ostraca.
7th century BCE
Lakhish Ostraca,
6 th century BCE
Dead Sea Scrolls
Rabbinical cursive
Black-letler Hebrew
aleph
K
4C
¥
♦
•r
<
H
e>
N
bet
$
9
3
9
j
/
>
3
i
a
gimmel
n
A
7
>N
1
A
A
A
i
dalet
a
0
A
4
4
s
T
1
1
he
*
4
'A
fl
9
n
vov
V
y
Y
K
J
1
j
1
zayin
I
13
=£
1
1
t
r
het
06
N
a
R
n
it
r>
n
tet
®
19
&
)»
O
B
yod
£.
£1
H'
'V
f
t
*
4
>
%
kof
\k
y-
y
9
y
J
3
9
S
lamed
<C
L
/
e
/
i
V
5
?
mem
A
y
9
9
>
i
a
W
&
nun
h
>
y
a
>
>
;
S
3
3
samekh
¥
*
T
f
->
V
a
D
ayin
O
0
0
0
*
«
V
9
V
pe
y>
i
s
7
)
A
B
B
tsade
'A 1
h.
a
Y”
■S'
r
r
3
X
quf
(9]
f
T
y
V
t
i*
i»
P
P
resh
4
9
*
<\
4
1
•>
■5
*1
shin
vt/
w
w
W
•V
v
V
y v
C
t?
tav
Y X-
X
X
X
%
A
J)
p
n
Fig. 17.2. Western
Semitic alphabets
Fig. 17 . 3 *
Phoenician and
Hebrew alphabets
compared to Greek
and Italic
PHOENICIAN
II EH RE W
ANCIENT GREEK
ITALIC ALPHABETS
lOtll 10 (ill
centuries BCE
Early
Modem
5th century BCE
Oscan
Umbrian
Etruscan
aleph
*
n
M
A alpha
A
A
A
a
bet
9
a
3
F beta
0
a
0
b
gimmel
A
A
J
r gamma
>
'I
g
dalet
a
•\
n
a delta
8
a
d
he
n
a
^ epsilon
3
3
\
e
vov
Y
1
1
f digamma
O
O
A
V
zayin
i
i
T
zeta
I
*
I
z
het
0
n
n
0 eta
0
a
B
h
tet
©
A
D
e theta
®
th
yod
z
4
*
iota
1
1
1
i
kof
V
»
3
It kappa
A
pi
X
k
lamed
L
\
?
^ lambda
4
<1
1
mem
y
3
O
fA mu
m
m
nun
>
J
:
A/ nu
N
*
n
samekh
¥
V
0
t x‘
e
s?
ayin
0
/
?
0 omicron
V
0
0
pe
9
J
0
P pi
n
1
n
p
tsade
h.
r
3
san
M
s
quf
?
p
?
cf koppa
9
q
resh
9
•»
a
p rho
<3
a
A
r
shin
W
•
0
^ sigma
e
Z
b
s
tav
XT
n
n
T tau
T
■r
■r
t
Y upsilon
y
V
u
phi
%
8
<t>
f
chi
t
kh
psi
S
dh
omega
d
c
Fig. 17 . 4 - Theorder
of the twenty-two
Phoenician letters has
in most cases been
preserved unaltered.
The names here given
to the Phoenician
letters are only
confirmed from the
sixth century BCE,
but their order and
phonetic values go
back much further, to
at least the fourteenth
century BCE.
PHOENICIAN
et
ARAMAEAN
HEBREW
SYRIAC
ANCIENT
ARABIC
GREEK
aleph
0
aleph
0
olap
0
alif
0
alpha
(a)
bet
(b)
bet
(b, v)
bet
(b)
ba
(b)
beta
(b)
gimmel
W
gimmel
<g>
gomal
(g)
jim
gamma
(g)
dalet
(d)
dalet
(d)
dolat
(d)
dal
(d)
delta
(d)
he
(h)
he
(h)
he
(h)
ha
(h)
epsilon
(e)
waw
(w)
vov
(v)
waw
(w)
wa
(w)
faw*
(O
zayin
W
zayin
(z)
zayin
(z)
zay
(z)
zeta
(z)
het
(h)
het
(h)
het
(h)
ha
(h)
eta
(e)
tet
(t)
tet
(0
tel
(t)
ta
<0
theta
(th)
yod
(y)
yod
(y)
yud
(y)
ya
(y)
iota
(i)
kaf
(k)
kof (k, kh)
kop
(k)
kaf
(k)
kappa
(k)
lamed
(i)
lamed
a)
lomad
(i)
lam
(1)
lambda
(1)
mem
(m)
mem
(m)
mim
(rn)
mim
(m)
mu
(m)
nun
(n)
nun
M
nun
(n)
nun
(n)
nu
(n)
samekh
(s)
samekh
(s)
semkat
(s)
sin
(S)
ksi
(ks)
ayin
0
ayin
0
e
0
ayin
0
omicron
(O)
pe
(p)
pe
(p. 0
pe
(p.O
fa
(0
P‘
(P)
sade
(s)
tsade
(ts)
sode
(s)
sad
(S)
san
0)
qof
(d)
quf
(q)
quf
(q) 1
qaf
(q)
qoppa
(?)
resh
(r)
resh
(0
rish
(r) |
ra
(r)
rho
(r)
shin (s
. sh)
shin (
[a, sh)
shin
(sh)
shin
(sh)
sigma
(s)
taw
(t)
tav
( 1 )
taw
(0
ta
CO
lau
(0
tha
m
upsilon
(tf)
kha
(kh)
phi
<pk)
* or digamma, subsequently dropped
LETTERS AND NUMBERS
214
Fig. 17.5. Ugaritic
alphabet primer, fourteenth
century BCE, found in 1948
at Ras Shamra. Damascus
Museum. Transcription
made by the author from
a cast. Sec PRU II (1957),
p. 199, document 184 A
SILENT NUMBERS
North African shepherds used to count their flock by reciting a text that
they knew by heart: “Praise be to Allah, the merciful, the kind Instead
of using the fixed order of the number-names (one, two, three . . . ), they
would use the fixed order of the words of the prayer as a “counting
machine”. When the last of the sheep was in the pen, the shepherd would
simply retain the last word that he had said of the prayer as the name of
the number of his flock.
This custom corresponds to an ancient superstition in this and many
other cultures that counting aloud is, if not a sin, then a hostage to the
forces of evil. In this view, numbers do not just express arithmetical quan-
tities, but are endowed with ideas and forces that are sometimes benign
and sometimes malign, flowing under the surface of mortal things like an
underground river. People who hold such a belief may count things that are
not close (such as people or possessions belonging to others), but must not
count aloud their own loved ones or possessions, for to name an entity is to
limit it. So you must never say how many brothers, wives or children you
have, never name the number of your cattle, sheep or dwellings, or state
your age or your total wealth. For the forces of evil could capture the hidden
power of the number if it were stated aloud, and thus dispose of the people
or things numbered.
The North African shepherd using the prayer as a counting device was
therefore doing so not only to invoke the protection of Allah, but also to
avoid using the actual names of numbers. In that sense his custom is similar
to the use of counting-rhymes by children - fixed rhythmic sequences of
words which when recited determine whose go it is at a game. In Britain,
for instance, children chant as they point round at each other in a circle:
eeny meeny miny mo - catch a blackman by his toe - if he hollers let him go - eeny
meeny miny mo! The child whose “go” it is is the one to whom the finger is
pointing when the reciter reaches mo!
The use of a fixed sequence like this is reminiscent of the archaic
counting methods of pre-numerate peoples, for whom points of the body
functioned much like a counting rhyme. Similarly, disturbed children
(and sometimes quite normal ones) invent their own counting sequences:
one boy I got to know counted Andre, Jacques, Paul, Alain, Georges, Jean,
Frangois, Gerard, Robert, (for 1,2. . . 9) in virtue of the position of his dorm-
mates’ bunks with respect to his own; and G. Guitel (1975) reports the case
of a girl who counted things as January, February, March . . . etc.
The girl could of course have used instead the invariable order of the
letters of the alphabet (A, B, C, D, E . . . ), for any sequence of symbols
can be used as a counting model - provided that the order of its elements
is immutable, as it is with the alphabet. And for that reason many
civilisations have thought of representing numbers with the letters of their
alphabet, still set in the order given them by the Phoenicians.
From the sixth century BCE, the Greeks developed a written numbering
system from 1 to 24 by means of alphabetic letters, known as acrophonics:
I 9
K 10
A 11
M 12
N 13
3 14
O 15
n i6
Fig. 17.6.
The tablets of Heliastes, like the twenty-four songs of the Iliad and the
Odyssey, used this kind of numbering, which is also found on funerary
inscriptions of the Lower Period. However, what we have here is really
only a simple substitution of letters for numbers, not a proper alpha-
betic number-system which, as we will now see, calls for a much more
elaborate structure.
p
17
2
18
T
19
Y
20
3>
21
X
22
'P
23
n
24
A
1
B
2
r
3
A
4
E
5
Z
6
H
7
0
8
215
HEBREW NUMERALS
HEBREW NUMERALS
Jews still use a numbering system whose signs are the letters of the
alphabet, for expressing the date by the Hebrew calendar, for chapters and
verses of the Torah, and sometimes for the page numbers of books printed
in Hebrew.
Hebrew characters, in common with most Semitic scripts, are written
right to left and, a little like capital letters in the Latin alphabet, are clearly
separated from each other. Most of them have the same shape wherever
they come in a word: the five exceptions are the final kof, mem, nun, pe, and
tsade (respectively, the Hebrew equivalents of our K, M, N, P, and a special
letter for the sound TS):
kof
mem
nun
pe
tsade
Regular form
D
D
3
B
X
Final form 1
*1
D
1
1
r
Fig. 17.7.
Black-letter (“square”) Hebrew script is relatively simple and well-
balanced, but care has to be taken with those letters that have quite similar
graphical forms and which can mislead the unwary beginner:
Letter
Name
Sound
Value ■
Letter
Name
Sound
Value
K
aleph
(h) a
1
■?
lamed
1
30
a
bet
b
2
a
mem
m
40
3
gimmel ^
g
3
3
nun
n
50
1
daleth 1
d
4
D
samekh
s
60
n
he
h
5
V
ayin
guttural
70
vov 1
V
6
B
pe
P
80
t
zayin ,
z
7
X
tsade
ts
90
n
het
kh
8
P
kuf
k
100
B
tet
t 1
9
“1
resh
r
200
yod
y
10
shin
sh
300
a
kof
k
20
n
tav
t
400
Fig. 17.10. Hebrew numerals
Compound numbers are written in this system, from right to left, by
juxtaposing the letters corresponding to the orders of magnitude in
descending order (i.e., starting with the highest). Numbers thus fit quite
easily in Hebrew manuscripts and inscriptions. But when letters are used as
numbers, how do you distinguish numbers from “ordinary” letters?
333
I'M
□ B
1 T
b k p
d r k final
m t
V z
3 3
n n n
0 □
V X
g n
h kh t
s m final
(guttural) ts
Fig. 17.8.
Hebrew numerals use the twenty-two letters of the alphabet, in the same
order as those of the Phoenician alphabet from which they derive, to repre-
sent (from aleph to tet) the first nine units, then from jW to tsade, the nine
"tens”, and finally from kof to tav, the first four hundreds (see Fig. 17.10).
X aleph
1 vov
D kof
V ayin
GJ shin
3 bet
t zayin
*7 lamed
B pe
n tav
3 gimmel
fT het
Q mem
tsade 1
*1 daleth
B tet
^ nun
p kuf
n he
'-C
O
CL
O samekh
*7 resh 1
Fig. 17.9. The Modern Hebrew alphabet
THIS IS THE MONUMENT
OF ESTHER DAUGHTER
OF A DAIO, WHO DIED IN
THE MONTH OF SHIVRAT
OF YEAR 3 ( 31) OF THE
“shemita”. YEAR THREE
HUNDRED AND 46 (\*)
AFTER THE DESTRUCTION
OF THE TEMPLE (OF
JERUSALEM)*
PEACE! PEACE BE
WITH HER!
* Year 346 of the Shemita +70 = 416 CE
Fig. 17.11. Jewish gravestone (written in Aramaic), dated 416 CE.
From the southwest shore oj the Dead Sea. Amman Museum (Jordan).
See IR, inscription 174
LETTERS AND NUMBERS
HERE LIES AN
INTELLIGENT WOMAN
QUICK TO GRASP ALL
THE PRECEPTS OF FAITH
AND WHO FOUND THE
FACE OF GOD THE
MERCIFUL AT THE
TIME THAT COUNTS (?)
WHEN HANNA DEPARTED
SHE WAS 56 YEARS OLD
r-> = n
6 50
<-
56
Fig. 17 . 12 . Part of a bilingual (Hebrew-Latin) inscription carved on a soft limestone funeral stela
found at Ora (southern Italy), seventh or eighth century CE. See CIl, inscription 634 (vol. 1, p. 452).
Numbers that are represented by a single letter are usually distinguished
by a small slanted stroke over the upper left-hand corner of the character,
thus:
'a '3 '*? '3 X
300 80 30 3 1
Fig. 17 . 13 .
When the number is represented by two or more letters, the stroke is
usually doubled and placed between the last two letters to the left of
the expression (Fig. 17.14). But as these accent-strokes were also used
as abbreviation signs, scribes and stone-cutters sometimes used other
types of punctuation or “pointing” to distinguish numbers from letters
(Fig. 17.15):
1'3 to n'*?
2 + 50 + 300 5 + 30
<■ <r
Fig. 17 . 14 .
216
Fig . 17 . 15 . Numerical expressions found in mediaeval Hebrew manuscripts and inscriptions.
See Cantera and Millas
The highest Hebrew letter-numeral is only 400, so this is how higher
numbers were expressed:
pnn
nn
ton
“in
pn
100 400 400
400 400
300 400
200 400
100 400
900
800
700
600
500
Fig. 17 . 16 .
So for numbers from 500 to 900, the customary solution was to combine
the letter tav (= 400) with the letters expressing the complement in
hundreds. Compound numbers in this range were written as follows:
943
IHE, 108
date: 1183 CE
Fig. 17 . 17 . Expressions found on
Jewish gravestones in Spain
217
HEBREW NUMERALS
The numbers 500, 600, 700, 800, and 900 could also be represented by
the final forms of the letters kof, mem, nun,pe, and tsade (see Fig. 17.7 above).
However, this notation, which is found, for example, in the Oxford manu-
script 1822 quoted by Gershon Scholem, was adopted only in Cabbalistic
calculations. So in ordinary use, these final forms of the letters simply had
the numerical value of the corresponding non-final forms of the letters.
Fig. i 7 . i 8 . Page from a Hebrew codex, 1311 CE, giving Psalms 117 and 118. The numbers can be seen
in the right-hand margin, in Hebrew letter-numbers. (Vatican Library, Cod. Vat. ebr. 12,fol. 58)
To represent the thousands, the custom is to put two points over the
corresponding unit, ten, or hundred character. In other words, when a
character has two points over it, its numerical value is multiplied by 1,000.
n-
-*n
1 1,000
2
2,000
40 40,000
90 90,000
Fig. 17.19.
The Hebrew calendar in its present form was fixed in the fourth century
CE. Since then, the months of the Jewish year begin at a theoretical, calcu-
lated date and not, as previously, at the sighting of the new moon. The
foundation point for the calculation was the neomenia (new moon) of
Monday, 24 September 344 CE, fixed as 1 Tishri in the Hebrew calendar,
that is to say New Year’s Day. As it was accepted that 216 Metonic cycles, in
other words 4,400 years, sufficed at that point to contain the entire Jewish
past, the chronologists calculated that the first neomenia of creation took
place on Monday, 7 October 3761 BCE. As a result, the Jewish year 5739, for
example, corresponds to the period from 2, October 1978 to 21 September
1979, and it is expressed on Jewish calendars (of the kind you can find in
any kosher grocer’s or corner shop) as:
o'*? to n ii
9 30 300 400 5000
Fig. 17.20.
Jewish scribes and stone-carvers did not always follow this rule, but
exploited an opportunity to simplify numerical expressions that was
implicit in the system itself. Consider the following expression found on a
gravestone in Barcelona: it gives the year 5060 of the Jewish calendar
(1299-1300 CE) in this manner:
Q n (=5 x 1,000 + 60)
60 5
4
Fig. 17.21.
Here, the points simply signify that the letters are to be read as numbers,
not letters. But the expression appears to break one of the cardinal rules of
Hebrew numerals - that the highest number always comes first, counting
from right to left, which is the direction of writing in the Hebrew alphabet.
So in any regular numerical expression, the letter to the right has a higher
value than the one to its left. For that reason, the expression on the
Barcelona gravestone is entirely unambiguous. Since the letter he can only
have two values - 5 and 5,000, and the letter samekh counts for 60, the char-
acter to the right, despite not having its double point, must mean 5,000.
Fig. 17 . 22 . Fragment of a Jewish gravestone from Barcelona. The date is given as 804, for 4804
(4804 - 3760 = 1044 CE).
LETTERS AND NUMBERS
218
Here are some other examples:
• • •
5,109 CD p n
9 100 5
Toledo, 1349
CE; IHE,
no. 85
tit t
MS dated 1396;
5,156 1 D p n
BM Add. 2806,
6 50 100 5
fol. 11a
There is an even more interesting “irregularity” in the way some
mediaeval Jewish scholars wrote down the total number of verses in the
Torah [see G. H. F. Nesselmann (1842), p. 484], The figure, 5,845, was
written by using only the letters for the corresponding units, thus:
n'ta n n
He Mem Het He
5 40 8 5
Because of the rule that we laid out above, this expression is not ambigu-
ous. The letter het, for example, whose normal value is 8, cannot have a
lower value than mem, to its left, and whose value is 40; nor can it be 8,000,
since it is itself to the left of he, whose value must be larger. For that reason
the het can only mean 800.
It is not difficult to account for this particular variant of Hebrew numer-
als. In speech, the number 5,845 is expressed by:
KHAMISHAT ALAF 1 M SHMONEH MEOT ARBa‘iM VE KHAMISHA
“Five thousand eight hundred forty & five”
The names of the numbers thus make the arithmetical structure of the
number apparent:
5 x 1,000 + 8 x 100 + 40 + 5
This could be transposed into English as “five thousand eight hundred
forty (&) five”, or in Hebrew as:
no mso n D’s^K'n
5 40 hundred 8 thousand 5
“Mixed” formulations like these, combining words and numerals, are
found on Hispano-Judaic tombstones (IHE, no. 61) and in some mediaeval
manuscripts (for example, BM Add. 26 984, folio 143b). It is easy to see
how such expressions can safely be abbreviated by leaving out the words
for “hundred” and “thousand”.
Another particularity arises in Hebrew numerals with the numbers 15
and 16. The regular forms would be:
rr r
5 10 6 10
<r <r
Fig. 17.24.
However, the letter-values of these numbers spell out parts of the name
of Yahweh - and it is forbidden, in Jewish tradition, to write the name of the
Lord, even if its literal form of four letters (the “divine tetragrammaton”,
mrr “yahve”) is perfectly well-known. To avoid writing the tetragramma-
ton, various abbreviations were devised (‘liT’, HI ,V ,rP) but these two
were covered by the prohibition on writing the name of God. So the regular
forms of the numbers 15 and 16 could not be used, and were replaced by
the expressions 9 + 6 and 9 + 7 respectively:
10 TO
6 9 7 9
< <:
Fig. 17.25.
These are the main features of Hebrew numerals. It was by no means the
only one to use the letters of the alphabet for expressing numbers. Let us
now look at the Greek system of alphabetic numbering.
GREEK ALPHABETICAL NUMERALS
The Greek alphabet is absolutely fundamental for the history of writing and
for Western civilisation as a whole. As C. Higounet (1969) explains, the
Greek alphabet, quite apart from its having served to transmit one of the
richest languages and cultures of the ancient world, forms the “bridge”
between Semitic and Latin scripts. Historically, geographically, and also
graphically, it was an intermediary between East and West; even more
importantly, it was a structural intermediary too, in the sense that it first
introduced regular and complete representations of the vowel sounds.
There is no question but that the Greeks borrowed their alphabet from
Fig. 17.23.
219
GREEK ALPHABETICAL NUMERALS
the Phoenicians. Herodotus called the letters phoinikeia grammatika,
“Phoenician writing”; and the early forms of almost all the Greek letters as
well as their order in the alphabet and their names support this tradition.
According to the Greeks themselves, Cadmos, the legendary founder of
Thebes, brought in the first sixteen letters from Phoenicia; Palamedes was
supposed to have added four more during the Trojan War; and four more
were introduced later on by a poet, Simonides of Ceos.
ARCHAIC
PHOENICIAN
ALPHABET
GREEK ALPHABETS
CLASSICAL
GREEK
ALPHABET
ARCHAIC
THERA
EASTERN
MILETUS CORINTH
WESTERN
BOEOTIA
aleph fcl
<
A A
A
A A
A Pi 4
A a
alpha
bet 5
$
F
6
Tnr>
*
Bp
beta
gimmel ^
A
r
r
< C 1
h
Ty
gamma
daleth ^
<4
A
z
A
D
A 5
delta
he
F
£
p- e
f E
E e
epsilon
vov y
Y
f
F
F c
[ ^"digamma*
zayin J
I
Z£
zeta
het p
00
0
0 H
0
B H
Hn
eta
tet 0
©
®
®
® S €
e e
theta
yod
Z
1
* *
i
1 1
iota
kof \L*
■v
k
k
K
Y
K K
kappa
lamed £
i
F
A A
h A
V
A A
lambda
mem ^
>
x
r
r*
M p.
mu
nun L)
bf
y n
AS
y
N v
nu
samekh
¥
%
2
5
HS
ksi
ayin 0
o
o O
o
o
O
0 o
omicron
pe J
r
r
r
r r n
n tt
P'
tsade
h.
M
M
san*
kuf
f
9 9
9
f ?
$ ?
koppa*
resh ^
9
1"
p
p
p P
p P
rho
shin
W
i
i z
i
2 o-
sigma
tav ^
X
T
T
T
T
Tt
tau
Y
Y
V
Y v
Y v
upsilon
©
9
O
<t> <j>
phi
X
X +
-+-
x x
chi
9 r
t
T 9
psi
XL
n (0
omega
*Greek letters that were eventually dropped from the alphabet
Fig. 17 . 26 . Greek alphabets compared to the archaic Phoenician script.
See Fevrier (1959) and Jensen (1969)
The oldest extant pieces of writing in Greek date from the seventh
century BCE. Some scholars believe that the original borrowing from
Phoenician occurred as early as 1500 BCE, others think it did not happen
until the eighth century BCE; but it seems most reasonable to suppose
that it happened around the end of the second millennium or at the
start of the first. At any rate, the Greek alphabet did not arise in its
final form at all quickly. There was a whole series of regional variations
in the slow adaptation of Phoenician letters to the Greek language, and
these non-standard forms are generally categorised under the following
headings: archaic alphabets (as found at Thera and Melos), Eastern
alphabets (Asia Minor and its coastal archipelagos, the Cyclades, Attica,
Corinth, Argos, and the Ionian colonies in Sicily and southern Italy),
and Western alphabets (Eubeus, the Greek mainland, and non-Ionian
colonies). Unification and standardisation did not occur until the fourth
century BCE, following the decision of Athens to replace its local script
with the so-called Ionian writing of Miletus, itself an Eastern form of
the alphabet.
Early Greek writing was done right to left, or else in alternating lines
( boustrophedon ), but it settled down to left-to-right around 500 BCE. Since
letters are formed from the direction of writing, this change of orientation
has to be taken into account when we compare Greek characters to their
Semitic counterparts.
The names of the original Greek letters are:
alpha, beta, gamma, delta, epsilon, digamma,
zeta, eta, theta, iota, kappa, lambda, mu, nu,
ksi, omicron, pi, san, koppa, rho, sigma, tau.
Of these, the digamma was lost early on, and the san and koppa were
also subsequently abandoned. However, a different form of the Semitic vov
provided the upsilon, and three new signs, phi, chi, and psi, were added to
represent sounds that do not occur in Semitic languages. Finally, omega
was invented to distinguish the long o from the omicron. So the classical
Greek alphabet, from the fourth century BCE, ended up having twenty-four
letters, including vowels as well as consonants.
Semitic languages can be written down without representing the vowels
because the position of a word in a sentence determines its meaning and
also the vowel sounds in it, which change with different functions. In
Greek, however, the inflections (word-endings) alone determine the
function of a word in a sentence, and the vowel sounds cannot be guessed
unless the endings are fully represented. The Phoenician alphabet had
letters for guttural sounds that do not exist in Greek; Greek, for its part,
had aspirated consonants with no equivalents in Semitic languages. So the
LETTERS AND NUMBERS
220
Greeks converted the Semitic guttural letters, for which they had no use,
into vowels, which they needed. The “soft breathing sound” aleph became
the Greek alpha, the sound of a; the Semitic letter he was changed into
epsilon ( e ), and the vov first became digamma then upsilon (u); the Hebrew
yod was converted into iota (/); and the “hard breathing sound” ayin became
an omicron (o). For the aspirated consonants, the Greeks simply created new
letters, the phi, chi and psi. In brief, the Greeks adapted the Semitic system
to the particularities of their own language. But despite all that is clear
and obvious about this process, the actual origin of the idea of representing
the vowel sounds by letters remains obscure.
This survey of the development of the Greek alphabet allows us now to
look at the principles of Greek numbering, often called a “learned” system,
but which is in fact entirely parallel to Hebrew letter-numbers.
We can get a first insight into the system by looking at a papyrus (now in
the Cairo Museum, Inv. 65 445) from the third quarter of the third century
BCE (Fig. 17.31).
O. Gueraud and P. Jouguet (1938) explain that this papyrus is a “kind
of exercise book or primer, allowing a child to practise reading and
counting, and containing in addition various edifying ideas ... As he
learned to read, the child also became familiar with numbers. The place
that this primer gives to the sequence of the numbers is quite natural,
coming as it does after the table of syllables, because the Greek letters
also had numerical values. It was logical to give the child first the combi-
nation of letters into syllables, and then the combinations of letters into
numbers.”
The numeral system the papyrus gives uses the twenty-four letters of
the classical Greek alphabet, plus the three obsolete letters, digamma, koppa
and san (see Fig. 17.26 above). These twenty-seven signs are divided
into three classes. The first, giving the units 1 to 9, uses the first eight letters
of the classical alphabet, plus digamma (the old Semitic vov), inserted in
the sequence to represent the number 6. The second contains the eight
following letters, plus the obsolete koppa (the old quf), to give the sequence
of the tens, from 10 to 90. And the third class gives the hundreds from
100 to 900, using the last eight letters of the classical alphabet plus the
san (the Semitic tsade) (for the value of 900) (see Fig. 17.27).
Intermediate numbers are produced by additive combinations. For
11 to 19, for instance, you use iota, representing 10, with the appropriate
letter to its right representing the unit to be added. To distinguish the
letters used as numerals from “ordinary” letters, a small stroke is placed
over them. (The modern printing convention of placing an accent mark
to the top right of the letter is not used in most Greek manuscripts.)
UNITS
TENS
HUNDREDS
A
a
alpha
1
I
L
iota
10
P
p
rho
100
B
p
beta
2
K
K
kappa
20
2
tr
sigma
200
r
7
gamma
3
A
\
lambda
30
T
T
tau
300
A
8
delta
4
M
p-
mu
40
Y
V
upsilon
400
E
e
epsilon
5
N
V
nu
50
<t>
4>
phi
500
r
c
digamma*
6
H
e
ksi
60
X
X
chi
600
z
t
zeta
7
0
0
omicron
70
ip
>l<
psi
700
H
■n
eta
8
n
77
P>
80
ft
CO
omega
800
0
•»
theta
9
<r
?
koppa
90
rr\
*
san
900
(sampi)
*In manuscripts from Byzantium, 6 is written ctt (sigma+tau). In Modern Greek, where alphabetic numerals
are still used for specific purposes (rather like Roman numerals in our culture), this sign is called a stigma.
Fig. 17.27. Greek alphabetic numerals
The beginning of the primer scroll has the remnants of the number
sequence up to 25:
IT
8
K
20
■0
9
KA
21
I
10
KB
22
LA
11
KT
23
IB
12
KA
24
IT
13
Kt
25
Fig. 17.28.
Gueraud & Jouguet (1938) note that the list is an elementary one, and
does not even include all the symbols the pupil would need to understand
the table of squares given at the end of the primer (Fig. 17.31). However, the
table of squares itself, besides giving the young reader some basic ideas
of arithmetic, also served to show the sequence of numbers beyond those
given at the start of the scroll and to familiarise the learner with the
principles of Greek numbering from 1 to 640,000, and that may have been
its real purpose.
How could the scribe represent numbers from 1 to 640,000 when the
highest numeral in the alphabet was only 900? For numbers up to 9,000, he
just added a distinctive sign to the letters representing the units, thus*:
A 'B T 'A 'E Z 'H '0
1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000
Fig. 17.29.
* Printed Greek usually puts the distinctive sign (a kind of iota) as a subscript, to the lower left comer of
the character.
221
When he got to 10,000, otherwise called the myriad (Mupioi)*, the
second “base” of Greek numerals, he put an M (the first letter of the Greek
word for “ten thousand”) with a small alpha over the top. All following
multiples of the myriad could therefore be written in the following way:
a P y 8 e la i(3
M M M M M...M
10,000 20,000 30,000 40,000 50,000 110,000 120,000 6,690,000
Fig. 17-30-
As he gave these numbers in the form 1 myriad, 2 myriads, 3 myriads,
etc. the scribe could reach 640,000 without any difficulty. He could obvi-
ously have continued the sequence up to the 9,999th myriad, which he
would have written thus:
'e-S?9 9999
M (M = 99,990,000)
TRANSCRIPTION TRANSLATION
^ IG - 17-31- Fragment of a Greek papyrus, third quarter of the the third century BCE ( Cairo
Museum, inv. 65 445). See Gueraud & Jouguet (1938), plate X. The papyrus gives a table of squares,
from 1 to 10 and then in tens to 40 ( left-hand column), and from 50 to 800 (right-hand column). The
squares of 1,2 and 3 are missing from the start of the table.
*When the accent is on the first syllable, the word means “ten thousand”; when the accent is on the second
syllable, it has the meaning “a very large number”.
GREEK ALPHABETICAL NUMERALS
These kinds of notation for very large numbers were frequently used by
Greek mathematicians. For example, Aristarch of Samos (?310-?230 BCE)
wrote the number 71,755,875 in the following way, according to P. Dedron
&J. Itard (1959), p. 278:
'^poeM'ecooe
*
7,175 x 10,000 + 5,875
Fig. 17.32.
We find a different system in Diophantes of Alexandria (c. 250 CE): he
separates the myriads from the thousands by a single point. So for him the
following expression meant 4,372 myriads and 8,097 units, or 43,728,097
[from C. Daremberg & E. Saglio (1873), p. 426]:
8to(3 't] ? (
»
4,372 x 10,000 + 8,097
Fig. 17-33-
The mathematician and astronomer Apollonius of Perga (c. 262-c. 180
BCE) used a different method of representing very large numbers, and it
has reached us through the works of Pappus of Alexandria (third century
CE). This system was based on the powers of the myriad and used the
principle of dividing numbers into “classes”. The first class, called the
elementary class, contained all the numbers up to 9,999, that is to say
all numbers less than the myriad. The second class, called the class of
primary myriads, contained the multiples of the myriad by all numbers
up to 9,999 (that is to say the numbers 10,000, 20,000, 30,000, and so
on up to 9,999 x 10,000 = 99,990,000). To represent a number in^ this
class, the number of myriads in the number is written after the sign M. A
reconstructed example:
Fig. 17.34.
> means 664 x 10,000 = 6,640,000
664
Next comes the class of secondary myriads, which contains the multiples
of a myriad myriads by all the numbers between 1 and 9,999 (that is to
say, the numbers 100,000,000, 200,000,000, 300,000,000, and so on up to
LETTERS AND NUMBERS
222
9,999 x 100,000,000 = 999,900,000,000. A number in this range is
expressed by writing beta over M before the number (written in the
classical letter-number system) of one hundred millions that it contains.
A reconstructed example:
M 'ea)^7
»
5,863
Fig. 17 - 35 -
This notation thus means: 5,863 x 100,000,000 = 586,300,000,000, and
is “read” as 5,863 secondary myriads.
Next come the tertiary myriads, signalled by gamma over M, which begin
at 100,000,000 x 10,000 = 1,000,000,000,000; then the quaternary myriads
(signalled by delta over M), and so on.
The difference between the system used in the papyrus of Fig. 17.31 and
the system of Apollonius is that whereas for the papyrus the superscribed
letter over M is a multiple of 10,000, for Apollonius the superscript repre-
sents a power of 10,000.
In the Apollonian system, intermediate numbers can be expressed by
breaking them down into a sum of numbers of the consecutive classes.
Pappus of Alexandria [as quoted in P. Dedron & J. Itard (1959) p. 279] gave
the example of the number 5,462,360,064,000,000, expressed as 5,462
tertiary myriads, 3,600 secondary myriads, and 6,400 primary myriads (in
which the Greek word Kai can be taken to mean “plus”):
Fig. 17.36.
Archimedes (?287- 212 BCE) proposed an even more elaborate system
for expressing even higher magnitudes, and laid it out in an essay on the
number of grains of sand that would fill a sphere whose diameter was
equal to the distance from the earth to the fixed stars. Since he had to
work with numbers larger than a myriad myriads, he imagined a “doubled
class” of numbers containing eight digits instead of the four allowed for
by the classical letter-number system, that is to say octets. The first octet
would contain numbers between 1 and 99,999,999; the second octet,
numbers starting at 100,000,000; and so on. The numbers belonged to the
first, second, etc. class depending on whether they figure in the first,
second, etc. octet.
As C. E. Ruelle points out in DAGR (pp. 425-31), this example suffices to
show just how far Greek mathematicians developed the study and applica-
tions of arithmetic. Archimedes’s conclusion was that the number of grains
of sand it would take to fill the sphere of the world was smaller than the
eighth term of the eighth octet, that is to say the sixty-fourth power of 10
(1 followed by 64 zeros). However, Archimedes’s system, whose purpose
was in any case theoretical, never caught on amongst Greek mathemati-
cians, who it seems preferred Apollonius’s notation of large numbers.
From classical times to the late Middle Ages, Greek alphabetic numerals
played almost as great a role in the Middle East and the eastern part of the
Mediterranean basin as did Roman numerals in Western Europe.
Fig. 17. 37 a. Part of a portable sundial from the Byzantine era (Hermitage Museum,
St Petersburg). This disc gives the names of the regions where it can be used, with latitudes
indicated in Greek alphabetic numerals in ascending clockwise order.
TRANSCRIPTION
TRANSLATION
2 2 3
INAIA
H
India
8
MEPOH
Is<
Meroe
16 7 2
COHNH
Kr<
Syena
23 V 2
BEPONIKH
Kr<
Beronika
23 V 2
MEM3>IC
A
Memphis
30
AAEEANAPI
AA
Alexandria
31
riENTAITOAIC
AA
Pentapolis
31
BOCTPA
AA<
Bostra
31 V 2
NEAIIOAIC
AA To
Neapolis
31 2 / 3
KECARIA
AB
Caesaria
32
KAPXHAHN
AB To
Carthage
32 7s
AB<
32 72
Ar To
33 7s
rOPTYNA
AA<
Gortuna
34 72
ANTIOXIA
AE<
Antioch
35 72
POAOC
As
Rhodes
36
nAMO>YAIA
As
Pamphilia
36
AProc
As<
Argos
36 72
COPAKOYCA
AZ
Syracuse
37
A0HNAI
AZ
Athens
37
AEA3>OI
AZ To
Delphi
37 7s
TAPCOC
AH
Tarsus
38
AAPIANOYIIOAIC
A0
Adrianopolis
39
ACIA
M
Asia
40
HPAKAEIA
MA To
Heraklion
4173
PHMH
MA To
Rome
41 7.3
ArKYPA
MB
Ankara
42
©ECCAAONIKH
Mr
Thessalonika
43
AIIAMIA
A0
Apamea
39
EAECA
Mr
Edessa
43
KHNCTATINOYIII
Mr
Constantinople
43
TAAAIAI
MA
Gaul
44
APABENNA
MA
Aravenna
44
©PAKH
MA
Thrace
41 (?44)
AKYAHIA
ME
Aquileia
45
< = */ 2 To = 2 /z
Fig. 17.37B.
GREEK ALPHABETICAL NUMERALS
/ un 4 & pent mimm frpuMuh qutfchberibi
jk^Jbctt crdtric ~ fmucntxfbocmodo.
.1 . 11 . uimjvvij l*r-
JUt f A £* H * i fc X* M-K5
]^9c. c.cc.ccc. ucc ■T> ’bC.V<C3>CCC.
t n. t C. T V 4 » X f.iv-
iutnUtf Muttr
mimtro utu dumj^udtxlm
cu^f^uotdV^pe. 3. rt'pomr tmutiMn^.
JUia cjuwtr: cojn. cutuf fujvm tyc eft .C|.
^jMmrinwnM/ t(tiwn^m». 1 <rci^
nomtmra:. cut’&piia -pipe. ^ . f^onar
mTMunw.^ mttvjcmof.
(X.tu<rufco m^enaiimror d^paffupttfi
cateduficermr ntOfeirasritanotnuM*.
Utmf jwrtt*r iftten prefypre <«
u*iT' Vsru yc Wtnuft mmcJdtgpft*- ''
f^n** (ft Aw didtiunaji
drt* admuarr dv^Ur.'gqmmuU. uem
Amuf
Fig. 17.38. Fragment of a Spanish manuscript concerning the Venerable Bale’s finger-counting
system, copied in c. 1130 CE, probably at Santa Maria de Ripoll (Catalonia). Madrid, National
Library, Cod. A 19 folio 2 (top left). To explain the finger diagrams given on the following pages, the
scribe uses two different numerical notations - Roman numerals and the Greek alphabetic system,
with their correspondence.
LETTERS AND NUMBERS
224
Fig. 17.39. Coptic numerals. [From Mallon, (1956); Till, (1955)]. The script of Egyptian Christians
has 31 letters, of which 24 derive directly from Greek , and the others from demotic Egyptian writing.
However, Coptic numerals use the same signs as the Greek system (that is to say, the 24 signs of the
classical alphabet plus the three obsolete letters, digamma, koppa and san, with the same values as
in Greek). In Coptic, letters used as numbers have a single superscripted line up to 999, and a double
superscript for 1,000 and above.
ARMENIAN
LETTERS
NAMES
OF THE
LETTERS
SOUNDS
NUMERICAL
VALUES
UPPER-
CASE
LOWER-
CASE
WESTERN
ARMENIAN
EASTERN
ARMENIAN
a
lii
ayp/ayb
a
a
1
p
F
pen/ben
P
b
2
9*
?
kim/gim
k
g
3
7*
t
ta/da
t
d
4
b
h
yetch
e
ye/e
5
5
1
za
z
z
6
t
k
e
e
e
7
C
F
et
e
e
8
P
to
t
t/th
9
*
je
j
j
10
h
b
ini
i
i
20
L
L
lyoun
1
1
30
hf
b*
khe
kh
kh
40
IT
A
dza/tsa
dz
ts
50
If
k
genAen
g
k
60
4
<
ho
h
h
70
2
d
tsa/dza
tz
dz
80
1
1
ghad
g h
gh
90
Fig. 17.40. Armenian numerals. Armenian uses an alphabet of 32 consonants and 6 vowels,
designed specifically for this language in the fifth century CE by the priest Mesrop Machtots
(c.362-440CE). The alphabet was based on Greek and Hebrew.
225
GREEK ALPHABETICAL NUMERALS
ARMENIAN
LETTERS
NAMES
OF THE
LETTERS
SOUNDS
NUMERICAL
VALUES
UPPER-
CASE
LOWER-
CASE
WESTERN
ARMENIAN
EASTERN
ARMENIAN
d
6
dje/tche
dj
tch
100
IT
d
men
m
m
200
3
J
hi
y
y/h
300
l
h
nou
h
n
400
n
2
cha
ch
ch
500
n
n
VO
0
0
600
2
t
tcha
tch
tch
700
•H
be/pe
b
P
800
2
£
tche/dje
tch
dj
900
fh
n
ra
r
rr
1,000
u
u
se
s
s
2,000
'L
vev
V
V
3,000
s
in
dyoun/tyoun
d
t
4,000
p
P
re
r
r
5,000
3
3
tso
ts
ts
6,000
A
L
hyoun
u
iu
7,000
0
pyour
p
p
8,000
•fi
P
ke
k
k
9,000
0
O
0
6
0
9>
V
fe
f
f
Fig. 17. 40 (continued). Like Greek, Armenian uses the first 9 letters to represent the units, the
second 9 for the tens, the third 9 for the hundreds. However, as it has more letters than Greek, it can
use the fourth set of 9 letters to represent the thousands. Note than only 36 of the 38 letters are
used for numerical purposes.
GEORGIAN
LETTERS
VALUES
GEORGIAN
LETTERS
VALUES
UPPER-
CASE
LOWER-
CASE
PHONETIC
NUMERICAL
UPPER-
CASE
LOWER-
CASE
PHONETIC
NUMERICAL
K
X
a
1
ih
r
100
H.
b
2
b
U
s
200
"v
g
3
S
1!
t
300
5
y
d
4
O,
m
u
400
n
*n
e
5
<p
vi
500
T*
•n*
V
6
V
p
600
'b
*b
z
7
+
k’
700
F
K
h
ft
n
•n
V
800
a*
i*
m
t’
9
H
q
900
y
S
1,000
*1
i
10
t«
ts
2,000
k
20
ft
e
ts
3,000
hi
1
30
dz
4,000
<h
R
7
m
n
40
50
B
S
r
5
ts’
ts’
5.000
6.000
0
1
60
K
u
h
7,000
a
m
0
70
V
V
h
8,000
\j
11
p
80
>
7
dz
9,000
z
90
%
•m
h
10,000
Fig. 17-41- Georgian alphabetic numerals. An example of a script and numeral system influenced
by Greek in the Christian era. There are two distinct styles of writing the Georgian alphabet:
the “priestly" script, or khoutsouri, reproduced above, and the “military", or mkhedrouli. Both have
38 letters.
LETTERS AND NUMBERS
226
GOTHIC
LETTERS
VALUES
GOTHIC
LETTERS
VALUES
GOTHIC
LETTERS
VALUES
phonetic
NUMERICAL
PHONETIC
NUMERICAL
PHONETIC
NUMERICAL
A
a
1
I
i
10
K
r
100
is
b
2
K
k
20
s
s
200
r
8
3
X
1
30
T
t
300
a
d
4
M
m
40
V
w
400
6
e
5
N
n
50
*
f
500
u
q
6
9
y
60
X
ch
600
X
z
7
n
u
70
©
hw
700
h
h
8
n
P
80
R
0
800
0>
th
9
M
90
+
900
Fig. 17.43. Numeral alphabet used by some mediaeval and Renaissance mystics. This adaptation
of the Greek system to the Latin alphabet is described by A. Kircher in Oedipi Aegyptiaci, vol. Il/l,
p. 488 (1653).
Fig. 17.42. Gothic: Another alphabetical numeral system influenced by Greek in the Christian era.
The Goths - a Germanic people living on the northeastern confines of the Roman Empire, were
Christianised by Eastern (Greek-speaking) priests in the second and third centuries CE. Wulfila
(311-384 CE), a Christianised Goth who became a bishop, translated the Bible into his own tongue,
and invented the Gothic alphabet, based on Greek together with some additional characters, in order
to do this. The Goths eventually merged into other peoples, from Crimea to North Africa, and
disappeared, leaving only the term “ Gothic " with its various acquired meanings.
227
THE NUMERALS OF THE NORTHWESTERN SEMITES
CHAPTER 18
THE INVENTION OF
ALPHABETIC NUMERALS
Greek alphabetic numerals were, as we have seen, pretty much identical
to the system of Hebrew numerals, save for a few details. The similarity is
such as to prompt the question: which came first?
What follows is an attempt to answer the question on the basis of
what is currently known.
First of all, though, we have to clear away a myth that has been handed
down uncritically as the truth for more than a hundred years.
THE MYTH OF PHOENICIAN LETTER-NUMBERS
It has long been asserted that, long before the Jews and the Greeks, the
Phoenicians first assigned numerical values to their alphabetic signs and
thus created the first alphabetic numerals in history.
However, this assumption rests on no evidence at all. No trace has yet
been discovered of the use of such a system by the Phoenicians, nor by their
cultural heirs, the Aramaeans.
The idea is in fact but a conjecture, devoid of proof or even indirect
evidence, based solely on the fact that the Phoenicians managed to simplify
the business of writing down spoken language by inventing an alphabet.
As we shall see, Phoenician and Aramaic inscriptions that have come to
light so far, including the most recent, show only one type of numerical
notation - which is quite unrelated to alphabetic numerals.
In the present state of our knowledge, therefore, we can consider
only the Greeks and the Jews as contenders for the original invention of
letter-numerals.
THE NUMERALS OF THE NORTHWESTERN SEMITES
The numerical notations used during the first millennium BCE by
the various northwestern Semitic peoples (Phoenicians, Aramaeans,
Palmyreneans, Nabataeans, etc.) are very similar to each other, and
manifestly derive from a common source.
Leaving aside the cases of Hebrew and Ugaritic, the earliest instance of
“numerals” found amongst the northwestern Semites dates from no earlier
than the second half of the eighth century BCE. It is in an inscription on a
monumental statue of a king called Panamu, presumed to have come
from Mount Gercin, seven km northeast of Zencirli, Syria (not far from
the border with Turkey). Semites generally liked to “write out” numbers,
that is to say to spell out number-names, and this tradition, which contin-
ued for many centuries, no doubt explains why specific number-signs
made such a late appearance. But that does not mean to say that their
system of numerals is at all obscure.
The Aramaeans were traders who, from the end of the second millen-
nium BCE, spread all across the Middle East; their language and culture
were adopted in cities and ports from Palestine to the borders of India,
from Anatolia to the Nile basin, and of course in Mesopotamia and Persia,
over a stretch of time that goes from the Assyrian Empire to the rise of
Islam. Thanks to the economic and legal papyri that constitute the archives
of an Aramaic-speaking Jewish military colony established in the fifth
century BCE at Elephantine in Egypt, we can easily reconstruct the
Aramaeans’ numeral system.
Aramaic numerals were initially very simple, using a single vertical bar
to represent the unit, and going up to 9 by repetition of the strokes. To
make each numeral recognisable at a glance, the strokes were generally
written in groups of three (Fig. 18.1 A). A special sign was used for 10, and
also (oddly enough) for 20 (Fig. 18.1 B and 18.1 C), whereas all other
numbers from 1 to 99 were represented by the repetition of the basic signs.
Aramaic numerals to 99 were thus based on the principle that any number
of signs juxtaposed represented the sum of the values of those signs. As we
shall see (Fig. 18.2), Aramaic numerals up to this point were thus identical
to those of all other western Semitic dialects, namely:
Sources
S 18
1
i
S 61
it
2
S 8
w
3
S 19
i iff
4
S 61
V iff
5
S 19
iff tff
6
S 61
t iff iff
7
CIS. IF 147
if iff iff
8
S 62
fit iff iff
9
Fig. i 8 . i a . A ramaic figures for
the numbers 1 to 9. Copied from
Sachau (1911), abbreviated as
S, from fifth century BCE papyri
from Elephantine (same source
for Fig. 18.1 B-E)
THE INVENTION OF ALPHABETIC NUMERALS
228
SIGNS FOR THE NUMBER 10
4,
—
S 61
KR 5
KR 5
S 8
o
S 61
S 7
KR 5
S 61
Fig. i 8 .ib.
SIGNS FOR THE NUMBER 20
S 18
S 18
S 25
S 18
9
S 19
S 61
S 15
S 7
Fig. 18.1c.
REPRESENTATIONS OF THE TENS
Sources
S 7
30
S 19
3*
40
KR 5
50
S 18
AW
60
S 61
70
S 18
80
S 18
90
Fig. i 8 .id.
NUMBERS BELOW 100
KR2
18
KR 5
ji tutu
38
KR 9
\f III W
98
Fig. i 8 .ie.
KHATRA NABATAEA PALMYRA PHOENICIA
UNITS
UNITS
UNITS
UNITS
a
>
5
HU
4
1
1
b
b
5
a
XoAV»
4
/
1
a
y
5
////
4
1
1
im
5
wt
4
/
1
mi>
9
- m/w
9
my
9
mm m
9
TENS
TENS
TENS
TENS
d
1
b
—1
f
e
”1
d
c
c
b
C
b
a
e
d
f
e
d
''J
TWENTY
TWENTY
TWENTY
TWENTY
h
*
8
f
e
<
i
a
h
g
>
h
3
g
3
f
3
i
*
h
g
%
i
k
3
j
3
k
3
j
%
i
%
i
N
k
j
Fig. 18.2.
KHATRA
NAB AT A EA
PALMYRA
PHOENICIA
References;
B. Aggoula (1972);
Milik (1972);
Naveh (1972)
References:
G. Cantineau (1930)
References:
M. Lidzbarski (1962)
References:
M. I.idzbarski (1962)
a Khatra no. 65
a CIS 11'. 161
a CIS II 3 . 3 913
a CIS I 1 , 165
b Khatra no. 65
b CIS II 1 , 212
b CIS IP, 3 952
b CIS I', 165
c Khatra no. 62
c CIS II 1 , 158
c CIS IP. 4 036
c CIS I 1 , 93
d Abrat As-Saghira
d CIS II 1 , 147 B
d CIS II 3 . 3 937
d CIS P, 88
e Abrat As-Saghira
e CIS II 1 . 349
e CIS IP, 3 915
e CIS I 1 , 165
f Khatra no. 62
f CIS II 1 . 163 D
f CIS IP, 3 937
f CIS I 1 , 3 A
g Abrat As-Saghira
g CIS II 1 , 354
g CIS IP, 4 032
g CIS I 1 , 87
h Khatra nos. 34, 65, 80
h CIS II 1 . 211
h CIS IP. 3 915
h CIS I', 93
i Doura-Europos
i CIS II 1 . 161
i CIS IP, 3 969
i CIS 1', 7
j Ashoka
j CIS II 1 , 213
j CIS II*. 3 969
j CIS I 1 , 86 B
k Ostraca nos. 74 & 113 from Nisa
k CIS II 1 , 204
k CIS IP, 3 935
k CIS I 1 , 13
1 Khatra nos. 62 & 65
1 CIS II 1 , 204
1 CIS IP, 3 915
1 CIS I 1 , 165
mN, II, 12
m CIS IP, 3 917
m CIS I 1 , 143
n CIS II 1 , 163D
n CIS I 1 , 65
o CIS II 1 , 161
o IS I', 7
p CIS I 1 , 217
• Phoenician, the language of a people of traders and sailors
who settled, from the third millennium BCE, in Canaan (on the
Mediterranean shore of Syria and Palestine); but Phoenician
numerals are not found earlier than the sixth century BCE;
• Nabataean, spoken by people who, from the fourth century
BCE, were settled at Petra, a city (now in Jordan) at the crossroads
of trails leading from Egypt and Arabia to Syria and Palestine, and
whose numeral system is attested from the second century BCE;
• Palmyrenean, spoken at Palmyra (east of Homs, in the Syrian
desert), from around the beginning of the Common Era;
• Syriac, in use from the beginning of the Common Era;
• the dialect of Khatra, spoken in the early centuries of the
Common Era by the inhabitants of the city of Khatra, in upper
Mesopotamia, southwest of Mosul;
• Indo-Aramaic, a numeral system found in Kharoshthi inscrip-
tions in the former province of Gandhara (on the borders of
present-day Afghanistan and the Punjab), from the fourth century
BCE to the third century CE;
• Pre-Islamic Arabic, in the fifth and sixth centuries CE.
However, despite affirmations to the contrary, the existence in these
systems of a special sign for 20 is not a trace of an underlying vigesimal
system borrowed by the Semites from a prior civilisation. The Semitic
229
sign for 10 was originally a horizontal stroke or bar, and the tens were
represented by repetitions of these bars, two by two:
Fig. 18 . 3 . Figures for the tens on the Aramaic inscription at Zencirli (eighth century BCE).
Donner & Rollig, Inscr. 215
By a natural process of graphical development, which is found in all cursive
scripts written with a reed brush on papyrus or parchment, the stroke
became a line rounded off to the right. The double stroke for the number
20 developed into a ligature in rapid notation, and that “joined-up” form
then gave rise to a whole variety of shapes, all deriving simply from writing
two strokes without raising the reed brush.
Fig. 18 . 4 . Origin and development of the figure for 20
Aramaic numerals are thus strictly decimal, and do not have any trace of
a vigesimal base. It was identical in principle to the Cretan Linear system
for numbers below 100 - but that does not mean that it was a “primitive”
form of number-writing nor that it lacked ways of coping with numbers
above the square of its base. In fact, the system had a very interesting
device for representing higher numbers which makes it significantly more
sophisticated than many numeral systems of the Ancient World.
The Elephantine papyrus shows that Semitic numbering possessed
distinctive signs for 100, 1,000 and 10,000 (though this last is not found
on Phoenician or Palmyrenean inscriptions). What is more, the system
did not require these higher signs to be repeated on the additive principle,
but put unit expressions to the right of the higher numeral, that is to say
used the multiplicative principle for the expression of large numbers (see
Fig. 18.7 and 18.8).
THE NUMERALS OF THE NORTHWESTERN SEMITES
SOURCES
a CIS II 147
h Khatra
o CIS II 4 021
u CIS II 147
b S 19
i S 15
p CIS II 3 935
v CIS I 7
c S61
j KR 4
q Sumatar Harabesi
w Assur
d Sari inscription
e Nisa ostracon 113
k S 61
1 CIS 1165
r CIS II 161
x En-Namara
(Cantineau, p. 49)
f Qabr Abu Nayf
g Khatra
m CIS 1143
n CIS II 3 999
s CIS II 163 D
t CIS II 3 915
y Biihler, p. 77
Fig. 18 . 5 . Variant forms of the Semitic numeral 100
THF. INVENTION O F ALPHABETIC NUMERALS
230
10 ** '
10
A
P 7 ;: <_
K
A
A
v M
vi
Fig. i 8 . 6 . Origin and development of the figure 100. All these signs derive from placing two
variants of the sign for 10 one above the other. This multiplicative combination has a kind of
additional superscript to avoid confusing it with the sign for 20, and produced widely different
graphical representations of the number 100.
ARAMAIC (ELEPHANTINE PAPYRI)
S61
4#"
100 x 5
500
S 19
-0'
100 X 1
100
CIS II'
v/\tf
100 X 8
800
s
fragm. 3
100 x 2
200
S 61
’*>111111111
100 x 9
900
S 19
100 x 4
400
KHATRA NABATAEA PALMYRA PHOENICIA
k
100x1
j
100x1
Ai
100x1
100x1
100x1
0
L*
100 x 1
n
Si
100x1
m
'V
100 X 1
1 UL
100x2
m
*1//
100x2
100x2
100 x 2 100 x 2
>HII
100x3
100x3
^O///
100x3
100 x 3 100 x 3
P>HII
100x4
100x4
m
100x4
/.fl 1
100x4
1
100x4
THOUSANDS AND TENS OF THOUSANDS
THOUSAND FIGURES
/
/
S 61
S 61
S fragm. 3
CIS II 1 147
This sign is visibly made up from the Aramaic
letters
^ and j
L F
and thus constitutes an abbreviation of the
word alf ( jC ^ )
F L ‘A
<■
the Western Semitic word for “thousand”
1,000
j> OB
2,000
CIS II* 14
col 1, 1.3
J
i>\u
3,000
4,000
S6i
1.3
5,000
S6i
1.14
7^ if If
8,000
TEN THOUSAND FIGURES
%
&
a
CIS II 1 147
S 62
S 61
this figure d
signs for 10
the multiplic
100
10 ^
>rivcs from the Aramaic
and 1,000 combined by
ative principle as follows
* i
100.10.10 10,000
S6i
1.14
a-
10,000
20,000
S 62
1.14
&
30,000
40,000
wm
50,000
turn
80,000
Fig. i 8 . 8 . Aramaic representations of the numbers 1,000 and over. Figures for these numbers have
not been found in other northwestern Semitic numeral systems.
Fig . 18 . 7 . Semitic representations of the number 100. Attested examples are given in solid lines;
reconstructed examples in outline. For sources, see list oj references in Fig. 18.2 and 18.5.
231
THE NUMERALS OF THE NORTHWESTERN SEMITES
In other words, the Semites used the additive principle for numbers from
1 to 99, but for multiples of 100, 1,000 and 10,000, they adopted the
multiplicative principle by writing the numbers in the form 1 x 100, 2 x 100,
3 X 100, etc.; 1 x 1,000, 2 x 1,000, 3 x 1,000, etc. So for intermediate
numbers above 100 they used a combination of the additive and multi-
plicative principles.
This corresponds with the general traditions of numbering amongst
Semitic peoples. It is found amongst all the northwestern Semites
(Phoenicians, Palmyreneans, Nabataeans, etc.) who used, as we have
seen, numerical notations of the same kind as the Aramaic system of
Elephantine. But it is also found amongst the so-called eastern Semites.
The Assyrians and the Babylonians certainly inherited the additive
sexagesimal system of the Sumerians, but they modified it completely
even whilst adopting the cuneiform script for writing it down. Precisely
because of their tradition of counting in hundreds and thousands, and
finding no numeral for 100 or 1,000 in the Sumerian system, their
scribes wrote those two numbers in phonetic script and represented their
multiples not by addition of a sequence of signs, but by multiplication
(Fig. 18.9).
So we can say that with the obvious exception of late Hebrew, none of the
Semitic numeral systems had anything to do with the use of letters as
numbers.
ASSYRO-BABYLONIAN EXPRESSIONS OF
NUMBERS UP TO 100
T
TT
TR
V
YT
¥
¥
¥
1
2
3
4
5
6
7
8
9
<
«
Iff
O
t—
T<
60+10
T«
60+20
Y«<
60+30
10
20
30
40
50
60
70
80
90
Fig. 18.9. Assyro-Babylonian “ordinary" numerals - an adaptation of Sumerian numerals to
Semitic numbering traditions
AKKADIAN SIGN FOR 100
AKKADIAN SIGN FOR 1,000
V
This is the syllable “ME” , the initial
This is the syllable “LIM”, the phonetic
letter of
spelling of the Assyro-Babylonian word
for “thousand”. It is visibly composed
"ME-AT",
of the signs:
and
the name of the number 100 in
Assyro-Babylonian
10 100
100
T
l
F
»
100
200
T
2
r f
»
100
300
T
3
TF
>
100
400
Tf-
»
4 100
500
ffF
»
5 100
1,000
T < 1 ^-
*
1 1,000
2,000
t r<F
>
2 1,000
3,000
TIFF
»
3 1,000
4,000
y<F
>
4 1,000
5,000
5 1,000
Fig. 18.9 (Continued).
7 + 20 + 20 + 20 + 20 + 100 x 8 + 1,000 x 3 + 10,000 + x?
x x 10,000 + 3,887
(Ref. CIS, H\ no. 147, col. 1, 1. 3)
Fig. 18 . 10 . Facsimile and interpretation of numerical expressions in the Elephantine papyrus
0000
6 + 10 + 20 + 20 + 20 + 100 x 4 1 + 5 + 10 + 20 + 20 + 20 + 100 x 4
<-
476 476
Fig. i8.ii. Tracing and interpretation of two examples from Syriac inscriptions at Sumatar
Harabesi, dated 476 of the Seleucid era (165-166 CE). Source: Naveh
Fig. 18.12. Phoenician inscription, fifth century BCE. Source: CIS I', 7
Fig. 18.13. The number 547 on a Syriac inscription at Sari. Source: Naveh
THE OLDEST ARCHAEOLOGICAL EVIDENCE OF
GREEK ALPHABETIC NUMERALS
Amongst the oldest known uses of Greek alphabetical numerals are those to
be found on coins minted in the reign of Ptolemy II (286-246 BCE), the
second of the Macedonian kings who ruled over Egypt after the death of
Alexander the Great (Fig. 18.14).
233
HtBREW ALPHABETIC NUMERALS
Coin
inventory
numbers
Date
symbols
Transcription
and
translation
CGC 61
A
A
30
CGC 63
AA
AA
31
CGC 68
A®
AB
32
CGC 70
AT
Ar
33
CGC 73
AA
AA
34
CGC 99
Af
AE
35
CGC 100
AC
A
36
CGC 101
AS
AZ
37
CGC 77
A W
AH
38
Coin
inventory
numbers
Date
symbols
Transcription
and
translation
CGC 44
K
K
20
CGC 45
*
KA
21
CGC 46
V
KA
21
K
CGC 48
0
KB
22
CGC 49
r
Kr
23
CGC 50
K
KA
24
CGC 53
6*
KE
25
CGC 57
1
KZ
27
CGC 50
M
KH
28
Fig. 18.14. Coins from the British Museum, catalogued by R. S. Poole
Even earlier, in a Greek papyrus from Elephantine, we find a marriage
contract that states that it was drawn up in the seventh year of the reign of
Alexander IV (323-311 BCE), that is to say in 317-316 BCE, in which the
dowry is expressed as alpha drachma, thus:
(transcription: 1- A
translation: drachma A)
Fig. 18.15.
The alphabetic numeral alpha probably means 1,000 in this case, unless
the father of the bride was a real miser, since alpha could either mean
1,000 -or 1!
It therefore seems that the use of Greek alphabetic numerals was
common by the end of the fourth century BCE.
Moreover, relatively recent excavations of the agora and north slope of
the Acropolis in Athens prove that the system arose even earlier, in the
fifth century BCE, since it is found on an inscription on the Acropolis that
is assumed to date from the time of Pericles (see N. M. Tod, in ABSA,
45/1950).
THE OLDEST ARCHAEOLOGICAL EVIDENCE OF
HEBREW ALPHABETIC NUMERALS
Amongst the earliest instances of Hebrew alphabetic numerals are those
found on coins struck in the second century CE by Simon Bar Kokhba, who
seized Jerusalem in the Second Jewish Revolt (132-134 CE). The shekel
coin shown in Fig. 18.16 bears an inscription in what were already the
obsolete forms of the palaeo-Hebraic alphabet* that gives the date as bet,
that is to say “Year 2”, in alphabetic numerals, which corresponds (as Year
2 of the Liberation of Israel) to 133 CE.
Fig. 18.16. Coin from the Second Jewish Revolt (132-134 CE). Kadman Numismatic
Museum, Israel
Other earlier instances are found on coins from the First Jewish Revolt
in 66-73 CE (Fig. 18.17), and Hasmonaean coins dating from the end of
the first century CE. These inscriptions, such as the one reproduced as
Fig. 18.18 (from a coin minted in 78 CE), are in the Aramaic language
but written in palaeo-Hebraic script.
B
5*
9m®* 9p®
a®
LT1WK’ lYw
1 W
5m 0' 5»®
i 0
9m®' 5 p®
n®
“SHEKEL [OF]
ISRAEL
YEAR 2 ”
“SHEKEL |OF]
ISRAEL
YEAR 3 ”
“SHEKEL [OF]
ISRAEL
YEAR 5 ”
Fig. 18.17. Coins struck during the First Jewish Revolt (66-73 CE): shekels dated Year 2
(A: 67 CE), Year 3 (B: 68 CE), and Year 5 (C: 70 CE) with alphabetic numerals in palaeo-Hebraic
script. Kadman Numismatic Museum, Israel. See Kadman (I960), plates I-III.
* Palaeo-Hebraic letters are close to Phoenician script. They were replaced by Aramaic script (which gave
rise to modern square-letter Hebrew around the beginning of the CE) in the fifth century BCE (see Fig. 17.2
above). However, the archaic forms of the letters continued to be used sporadically up to the second century
CE, most particularly by the leaders of the two Jewish revolts, to signify a return to the “true traditions of
Israel". The alphabet of the present-day Samaritans is derived directly from palaeo-Hebraic script.
THE INVENTION OF ALPHABETIC NUMERALS
ABC
... 1 t*J flj) RJfe ©11^
(h]j i o n a nJP o
25
“king ALEXANDER YEAR 25”
Fig. 1 8 . 1 8 . Coins struck in 78 BCE under Alexander Janneus. Kadman Numismatic Museum,
Israel. See Naveh (1968), plate 2 (nos. 10 & 12) and plate 3 (no. 14).
We must also mention a clay seal in the Jerusalem Archaeological
Museum which must have originally served to fix a string around a papyrus
scroll (Fig. 18.19). The seal bears an inscription in palaeo-Hebraic charac-
ters which can be translated as: “Jonathan, High Priest, Jerusalem, M". The
letter mem at the end is still a puzzle, but it could be a numeral, with a value
(= 40) referring to the reign of Simon Maccabeus, recognised by Demetrius
II in 142 BCE as the “High Priest, leader and ruler of the Jews”. If this were
so, then the seal would date from 103 BCE (the “fortieth year” of Simon
Maccabeus) and thus constitute the oldest known document showing the
use of Hebrew alphabetic numerals.
Fig. 18 . 19 . Bulla of mean period (second
century BCE). Israel Museum, Jerusalem, item /o.JS. See Avigad (1V/5J, tig. 1 and Plate LA.
Finally, there is this fragment of a parchment scroll from Qumran (one
of the “Dead Sea Scrolls”):
Fig. 18 . 20 . Fragment of
a parchment scroll, recently
found at Khirbet Qumran.
Scroll 4QSd, no. 4Q259.
See Milik (1977).
234
The scroll contains a copy of the Rule of the Essene community, written in
square-letter Hebrew of a style that dates from the first century BCE at the
earliest. The fragment comes from the first column of the third sheet of
the scroll as it was found in the caves at Qumran. In the top right-hand
corner there is a letter, gimmel : since this is the third letter of the Hebrew
alphabet, people have assumed that the letter gives the sheet number, 3.
However, the gimmel was not written by the same hand as the rest of the
scroll; J. T. Milik has explained that the page-numbering was probably
the work of an apprentice, using what was then a novel procedure for
numbering manuscripts by the letters of the alphabet, whereas the main
scribe used an older form of writing.
JEWISH NUMERALS FROM THE PERSIAN TO
THE HELLENISTIC PERIOD
The preceding section shows that in Palestine Hebrew letters were only just
beginning to be used as numerals at the start of the Common Era.
This is confirmed by the discovery, in the same caves at Qumran, of
several economic documents belonging to the Essene sect and dating from
the first century BCE. One of them, a brass cylinder-scroll (Fig. 18.21), uses
number-signs that are quite different from Hebrew alphabetic numerals.
1
5
10
15
Fig. i 8 . 2 i a . Fragment of a brass cylinder- scroll, first century BCE, from the third of the Qumran
caves. See DJD III, 3Q plate LXIl, column VIII.
iifrp*
3>yRvnfphKi?' > '
y.""^ .inn .
yi
tvttt'Oort' ijii i i*i i
235
Lines
NUMERALS FOUND
ON THE DOCUMENT
SHOWN IN FIG. 21A
VALUES
HAD THE SCRIBE USED
LETTER-NUMERALS,
HE WOULD HAVE
WRITTEN:
7
2 + 5 + 10
17
L «•>
13
v'»Wp\
2 + 4 + 20 + 20 + 20
<■
66
^ 0o)
6 + 60
<-
Fig. 18 . 21 B.
Further confirmation is provided by the many papyri from the fifth
century BCE left by the Jewish military colony at Elephantine (near Aswan
and the first cataract of the Nile). These consist of deeds of sale, marriage
contracts, wills, and loan agreements, and they use numerals that are
identical to those of the Essene scroll. For example, one such papyrus [E.
Sachau (1911), papyrus no. 18] uses the following representations of 80
and 90, which are obviously unrelated to the Semitic letter-numbers pe (for
80) and tsade (for 90).
20 + 20 + 20 + 20 10 + 20 + 20 + 20 + 20
+ <■
80 90
Fig. 18 . 22 .
An even more definitive piece of evidence comes from the archaeological
site of Khirbet el Kom, not far from Hebron, on the West Bank (Israel). It is
a flat piece of stone that was used, at some point in the third century BCE,
for writing a receipt for the sum of 32 drachmas loaned by a Semite called
Qos Yada to a Greek by the name of Nikeratos - and is thus written in both
Aramaic and Greek.
JEWISH NUMERALS
H kj*4ir**t r«A\
TRANSCRIPTION
Greek Text
Aramaic Text
* *
l IBmhnox ii a
NHMOY EXEI NI
KHPATOS SOBBA
0O EIAPA KOXIAH KA
nHAOY i-AB
hi m mar non? H-t 3
D 5 'Bp Kin *13 jh'Dlp
|nt Djrip-j [?) jn, , n
2 10 20
TRANSLATION
6th year, the 12th of the month The 12th [of the month] of
of Panemos, Nikeratos, son of Tammuz [of] the 6th year Qos
Sobbathos, received from Yada son of Khanna the trader
Koside the moneylender [the gave Nikeratos in “Zuz”: 32.
sum of] 32 drachma
Fig. 18 . 23 . Bilingual ostracon from Khirbet el Kom (Israel), probably dating from 277 BCE
(Year 6 of Ptolemy II). See Geraty (1975), Skaist (1978).
THF. INVENTION OF ALPHABETIC NUMERALS
236
Close scrutiny of the inscription shows first of all that the two languages
are written by different hands: probably the moneylender wrote the
Aramaic and the borrower wrote the Greek. Moreover, we can see that
Nikeratos the Greek wrote the sum he had borrowed and the date of
the loan (“6th year, on the 12th of the month of Panemos") using Greek
alphabetic numerals: C, digamma ( = 6), i(3 iota-beta ( = 12), and \(3
lambda-beta (=32). On the other hand, Qos Yada the Semite wrote the sum
of the loan (32 zuz) using the numeral system we have seen on the Essene
scroll above, broken down as:
20+10 + 1 + 1
It seems indisputable that if Hebrew alphabetic numerals had been in use
in Palestine at this time, then Qos Yada would have used them, and written
the number 32 much more simply as
j\ ” 3 *7
2 + 30
Fig. 18.24. *
We can therefore conclude that in all probability the inhabitants of Judaea
did not use alphabetic numerals in ordinary transactions until the dawn of
the Common Era.
The numeral system we have found in use amongst Jews from the Persian
to the Hellenistic period (fifth to second centuries BCE) is in fact nothing
other than the old western Semitic system, borrowed by the Hebrews
from the Aramaeans together with their language (Aramaic) and script.
Because the Aramaeans were very active in trade and commerce - their
role across the land-mass of the Middle East was similar to that of the
Phoenicians around the shores of the Mediterranean Sea - Aramaic script
spread more or less everywhere. It finally killed off the cuneiform writing
of the Assyro-Babylonians, and became the normal means of international
correspondence.
ACCOUNTING IN THE TIME OF THE
KINGS OF ISRAEL
How did the Jews do their accounting in the age of the Kings, roughly from
the tenth to the fifth centuries BCE? In the absence of archaeological
evidence, it was long thought that numbers were simply written out as
words, for the numeral system explained below remained undiscovered
until less than a hundred years ago.
That was when excavations in Samaria uncovered a hoard of ostraca in
palaeo-Hebraic script in the storerooms of the palace of King Omri. An
ostracon is a flat piece of rock, stone or earthenware used as a writing
surface. (The use of ostraca as “scribble-pads” for current accounts, lists
of workers, messages and notes of every kind was very common amongst
the Ancient Egyptians, the Phoenicians, the Aramaeans, and the Hebrews.)
The Samarian ostraca consist of bills and receipts for payments in kind
to the stewards of the King of Israel, and reveal that the Jews wrote out their
numbers as words and also used a real system of numerals.
Subsequent discoveries confirmed the existence of these ancient Hebrew
numerals. They have been found on a hoard of about a hundred ostraca
unearthed at a site at Arad (in the Negev Desert, on the trail from Judaea
to Edom); on another score of ostraca found at Lakhish in 1935, which
contain messages from a Jewish military commander to his subordinates,
written in the months prior to the fall of Lakhish to Nebuchadnezzar II
in 587 BCE; numerous Jewish weights and measures; and on various
similar discoveries made at the Ophel in Jerusalem, at Murabba’at and
at Tell Qudeirat.
Although it took a long time to decipher these inscriptions, there is
no longer any doubt (Fig. 18.26) but that these number-signs are Egyptian
hieratic numerals in their fully developed form from the New Empire
(shown in Fig. 14.39 and 14.46 above). This incidentally provides addi-
tional confirmation of the significant cultural relations between Egypt and
Palestine which historians have revealed in other ways. In other words, in
the period of the Kings of Israel, the Jews were influenced by the civilisation
of the Pharaohs to the extent of adopting from it Egyptian cursive hieratic
numerals (Fig. 18.25 and 18.27).
Fig. 18.25. Hebrew ostracon from Arad, sixth century BCE (ostracon no 17). Written in
palaeo-Hebraic script, side 2 has the number 24 written as: See Aharoni (1966).
4 20
<r
237
DATES BCE SOURCES
i
2
3
4
5
6
7
8
9
9TH C
ARAD Ostracon no. 72
i
ii
IN
8TH C
SAMARIA
Ostraca published
in 1910
i
n
T
Ostracon C 1101
1
8TH-
7TH C
Inscribed Jewish weights
i
c
B
LATE
8TH C
Jerusalem
Ophel
Ostr. no. 2
1
Ostr. no. 3
1
St
Ostr. no. 4
MURABBA’AT Papyrus no. 18
D
<
ARAD Ostracon no. 34
E
B
6TH C
LAKHISH
i
u
1
i
1
ARAD
/
D
D
1
Ostr. no. 16-18
i
MW
O
Ostr. no. 24-29
EGYPTIAN HIERATIC NUMERALS
(NEW KINGDOM, CURSIVE). FROM
MOLLER (1911).
1
2 '
3
4
5
6
7
8
9
1
M
iff
i m
*1
&
O’
\
Fig. 18.26. Table showing the identity of numerals used in Palestine under the Jewish Kings with
Egyptian hieratic numerals
IU *
ACCOUNTING IN THE TIME OF THE KINGS OF ISRAEL
10
20
30
40
50
60
70
80
90
100
200
300
K
*0*
D
D
B
1
■
1.
■
L-f
B
■
B
A
10
20
1
30
40
50
60
70
80
90
100
200
300
1
i
A
*
i
X
A
1
1
4
i
*
A
JJ
THE INVENTION OE ALPHABETIC NUMERALS
238
Fig. 18 . 27 . Ostracon no. 6 from Tell Qudeirat, late seventh century BCE, the largest known
palaeo-Hebraic ostracon, found by R. Cohen in 1979. This text confirms the results of Fig. 18.2S,
since it gives almost the whole series of the hundreds and thousands in Egyptian hieratic script.
EGYPT Coptic inscription concerning Luke and two of his
works. See ASAE, X, 1909, p.51
KN
KA
KZ
28
- - — ->
24
27
Jewish funerary stelae from Tell el Yahudieh (10 km
north of Cairo), dating from the first century CE.
See CII 1454, 1458 and 1460
IB
1 r
Kr
AC
N
PB
» ->
—
■>
»
>
12
13
23
35
50
102
PHRYGIA Jewish inscription dated 253-254 CE. See CII 773
TAH
*
338
ETHIOPIA
Aksum inscription, third century CE. See DAE 3 and 4
KA TPtB ?CKA
* »
24 3112 6224
LATIUM
Jewish catacombs on the Via Nomentana. Via Labicana
and Via Appia Pignatelli. See CII 44, 78, 79
Ar KA 26
— > — > — >
33 21 69
NORTHERN SYRIA
Synagogue mosaic. Jewish inscription dated 392 CE.
See CII 805
y r
— ^
703
SOUTH OF THE
DEAD SEA
Jewish grave marking dated 389-390 CE. See CII 1209
tt' cwr
— » — »
86 283
Fig. 18.28A.
Fig. 18.28B.
239
SUMMARY
JEWISH LAPIDARY NUMERALS AT THE DAWN
OF THE COMMON ERA
There is a final curiosity to add to this story. From the first century BCE to
the seventh century CE, the use of Hebrew alphabetic numerals grew ever
more common amongst Jews all over the Mediterranean basin, from Italy
to Palestine and northern Syria, from Phrygia to Egypt and even Ethiopia.
However, during this period, Jewish stone-carvers, who could write just as
well in Hebrew as in Greek or Latin, most often put dates and numbers not
in Hebrew, but in Greek alphabetic numerals, as the examples reproduced
in Fig. 18.28 show.
THE jews: national identity and
CULTURAL COMPLEXITY
The people of Israel certainly played a major role in the history of the
world’s religions; but at the same time, Jewish culture has, throughout its
history, accepted and adopted influences of the most diverse kinds.
The most notable of these “foreign influences” include:
• the adoption of the Phoenician alphabet in the period of
the Kings;
• the adoption of the Assyro-Babylonian sexagesimal system for
weights and measures (see Ezekiel XLV:12, where the talent is
set at 60 maneh, and the maneh at 60 shekels)',
• the presumed adoption of the Canaanites' calendar, in which
each month starts with the appearance of the new moon;
• the borrowing of the names of the months from the ancient
calendar of Nippur, used throughout Mesopotamia from the time
of Hammurabi ( Nisan , Ayar, Siwan, Tammuz, Ab, Elul, Teshret,
Amshamna, Kisilimmu, Tebet, Shebat, and Adar). In Modern
Hebrew, the names are still almost identical;
• the adoption of Aramaic and its script (the only ones in general
use in Judaea at the time of Jesus).
What is remarkable about Jewish culture is that despite these numerous
borrowings it retained a separate identity. Since the expulsion of the
Jews from Palestine in the first century CE, and for the following 1,800
years, it has not ceased to adapt itself to the most diverse situations and to
incorporate new elements, whilst also exercising a determining influence
over developments in Western and Islamic culture. As Jacques Soustelle
sees it, this long history of a cultural identity within a complex of cultural
influences is what accounts for the successful re-founding of a Jewish
nation-state in the twentieth century: Israel today is made of more than a
score of distinct ethnic groups with many different mother-tongues, but
sharing a common cultural identity.
SUMMARY
From the tenth to the sixth century BCE (the era of the Kingdom of Israel),
the Hebrews used Egyptian hieratic numerals; from the fifth to the second
century BCE, they used Aramaic numerals; and from around the start of
the common era, many Jews used Greek alphabetical numerals.
In the present state of knowledge, it seems that Greek alphabetic
numerals go back at least as far as the fifth century BCE; whereas Hebrew
alphabetic numerals are not found before the second century BCE.
Does that mean to say that the Greeks invented the idea of representing
numbers by the letters of their alphabet, and that the Jews copied it during
the Hellenistic period? It seems very likely, and all the more plausible in
the light of the Jews’ adoption of numerous other “outside” influences.
However, this is not the only possible conclusion. Many passages in the
Torah (the Old Testament) suggest very strongly that the scribes or authors
of these ancient texts were familiar with the art of coding words according
to the numerical value of the letters used (see further explanations in
Chapter 20 below). It is currently reckoned that the oldest biblical texts
were composed in the reign of Jeroboam II (eighth century BCE) and that
the definitive redaction of the main books of the Torah took place in the
sixth century BCE, around the time of the Babylonian exile.
Do Hebrew alphabetic numerals go so far back in time? Or are the
passages showing letter-number coding later additions?
If the system is as old as it seems, and which would imply that Hebrew
letter-numbers were invented independently of the Greek model, we would
still have to explain why they had no use in everyday life until the Common
Era. One plausible answer to that question would be that since the letters of
the Hebrew alphabet acquired a sacred character very early on, the Jews
avoided using sacred devices for profane purposes.
In conclusion, let us say that the “Greek hypothesis” seems to have most
of the actual evidence on its side; but that the possibility of an independent
origin for Hebrew alphabetic numerals and of their restriction over several
centuries to religious texts alone is not to be rejected out of hand.
OTHER alphabetic: number- systems
CHAPTER 19
OTHER ALPHABETIC
NUMBER-SYSTEMS
SYRIAC LETTER-NUMERALS
The Arabic-speaking Christians of the Maronite sect have maintained,
mainly for liturgical use, a relatively ancient writing system which is known
as serto or Jacobite script.
Christians of the Nestorian sect, who are found mainly in the region of
Lake Urmia (near the common frontier between Iraq, Turkey, Iran and the
former Soviet Union), still speak a dialect of Aramaic which they write in a
graphical system called Nestorian writing.
Each of these two writing systems has an alphabet of twenty-two letters,
and is derived from a much older script called estranghelo, formerly used to
write Syriac, a ancient Semitic language related to Aramaic.
Graphically, the Nestorian form, which is more rounded than the
estranghelo, is intermediate between this and serto which in turn has a
more developed and cursive form (Fig. 19.1). The letters themselves are
written from right to left, are joined up, and, as in the writing of Arabic,
undergo various modifications according to their position within a
word, i.e. according to whether they stand alone or are in the initial,
medial, or final position (Fig. 19.1 only shows the independent forms of
Syriac letters).
The oldest known Syriac inscriptions seem to date from the first century
BCE. Estranghelo writing seems to have been used only up to the sixth or
seventh century. As used by the Nestorian Christians, fairly numerous in
Persia in the period of the Sassanid Dynasty (226-651 CE), it gradually
evolved until, around the ninth century, it attained its canonical Nestorian
form. With the Jacobites, who mainly lived in the Byzantine Empire, it
seems to have evolved more rapidly towards the serto form, since it was
gradually replaced by this after the seventh or eighth century.
Finally, estranghelo (which is simply a variant of Aramaic script and
therefore ultimately derives from the Phoenician alphabet) has preserved in
its entirety the order of the original twenty-two Phoenician letters (the
same order which is to be found with all the western Semites).
In serto, however, as in Nestorian, letters have been used (and still are
used) as number-signs. This is confirmed by the fact that in all Syriac manu-
scripts (at least those later than the ninth century), codices are made up of
240
serially numbered quires, ensuring the correct order of composition of the
bound book. (The manuscript folios, however, were only numbered later,
often using Arabic numerals.)
The numerical values of the Syriac letters are assigned as follows. The
first nine letters are assigned to the units, the next nine letters to the tens,
and the remaining four are assigned to the first four hundreds. Also, as in
Hebrew, the numbers from 500 to 900 are written as additive combinations
of the sign for 400 with the signs for the other hundreds, according to the
schema:
500 = 400 + 100
600 = 400 + 200
700 = 400 + 300
800 = 400 + 400
900 = 400 + 400 + 100
The thousands are represented by a kind of accent mark placed beneath
the letters representing the units, and the tens of thousands by a short
horizontal mark beneath these same letters:
f 10,000
m
J 1,000
I 1
f 20,000
2,000
O 2
^ 30,000
^ 3,000
^ 3
i 40,000
! 4,000
S
t 4
Similar conventions allowed the Maronites to represent numbers greater
than the tens of thousands. With a few exceptions, this number-system
is quite analogous to that of the Hebrew letter-numerals. It is however a
relatively late development in Syriac writing, since the oldest documents
show that it does not go back earlier than the sixth or seventh centuries.
Older Syriac inscriptions only reveal a single kind of numerical notation
related to the “classical” Aramaic system.
241
SYRIAC I.ETTER-NUMERALS
HEBRAIC
LETTERS
ARCHAIC
PHOENICIAN
PALMYRENEAN
ESTRANGHELO
NESTORI AN
SERTO
NAMES
TRANSCRIPTIONS
AND NUMERICAL
VALUES OF SYRIAC
LETTERS
Aleph
K
K
*
re
2
* 1
Olap
■
L
Bet
2
9
r=>
*3
Bet
b bh
2
Gimmel
:
A
A
-X
A
Gomal
ggh
3
Dalet
T
a
X
a
♦
Dolat
ddh
4
He
n
%
A
<n
m
a*
He
h
5
Vov
x
Y
?
a
A
a
Waw
w
6
Zayin
r
1
I
i
i
1
Zayn
z
7
Het
n
H
X K
ji
V*
Het
h
8
Tet
D
9
6
Ar
A,
Tet
t
9
Yod
*
7
f
V*
Yud
y
10
Kof
3
*
3
r
Kop
kkh
20
Lamed
5
h
Lomad
i
30
Mem
0
%
n>
Jo
"P
ip
Mim
m
40
Nun
i
X
4
„
\
V
Nun
n
50
Samekh
D
w
CO
vft>
Semkat
s
60
Ayin
?
©
y
NX,
**
'E
70
Pe
to
j
A
A
<0
Pe
pph
80
Tsade
X
h.
j<
-S
J
Sode
s
90
Quf
p
9
si
•JO
Quf
q
100
Resh
9
X
a
>
Rish
r
200
Shin
V
w
X/
**•
o*
Shin
sh
300
Tav
n
Jr x
S'
b
V
L
Taw
t
400
Fig. 19.1. Syriac alphabets compared with Phoenician, Aramaean (from Palmyra) and Hebraic
alphabets. The use of Syriac letters as number-signs is attested notably in a manuscript in the
British Museum (Add. 14 620) which features the above order. (See M. Cohn, Costaz, Duval.
Fevrier, Hatch, Pihand, W. Wright)
When did letter-numerals in Syriac writing first arise? In the absence of
documents, it is hard to say. But there are several good reasons to suppose
that the introduction of this system owed much to Jewish influence on the
Christian and Gnostic communities of Syria and Palestine.
One final question: a Syriac manuscript, now in the British Museum
(reference Add. 14 603), which probably dates from the seventh or eighth
century [W. Wright (1870), p. 587a], reveals some interesting information.
Its quires are numbered in the usual way, with Syriac letters according to
their numerical values; but these have alongside them the corresponding
older number-signs. Should we conclude that, at the date of this manu-
script, the system of letter-numerals had not been universally adopted? Or,
taking the question in the other sense, should we conclude that at that time
the use of the old system was already a traditional but archaic usage, and
the letter-numerals were by then not only widespread but considered by the
majority of Syrians to be the only normal and official system of notation?
The documentation which we have to hand does not give us an answer.
ARABIC LETTER-NUMERALS
Arabic has a number-system modelled not only on the Hebrew system, but
also on the Greek system of letter-numerals. But first we need to look at a
curious problem.
The order of the twenty-eight letters of the Arabic alphabet, in its
Eastern usage, is quite different from the order of the letters in the
Phoenician, Aramaic or Hebrew alphabets.
A glance at the names of the first eight Arabic letters compared with
the first eight Hebrew letters shows this straight away:
ARABIC
HEBREW
'alif
'aleph
ba
bet
ta
gimmel
tha
dalet
jim
he
ha
vov
kha
zayin
dal
het
We would expect to find the twenty-two western Semitic letters in the
Arabic alphabet, and in the same order, since Arabic script derives from
archaic Aramaic script. So how did the traditional order of the Semitic
letters get changed in Arabic? The answer lies in the history of their system
for writing numbers.
OTHER ALPHABETIC NUMBER-SYSTEMS
242
The Arabs have frequently used a system of numerical notation in which
each letter of their own alphabet has a specific numerical value (Fig. 19.3);
according to F. Woepke, they "seem to have considered [this system] as
uniquely and by preference their own”.
They call this huruf al jumal, which means something like “totals by
means of letters”.
But, if we look closely at the numerical value which this system assigns
to each letter, we are bound to note that the method used by the Arabs
of the East is not quite the same as the one adopted, later, by western
(North African) Arabs, since the values for six of the letters differ in the
two systems.
ITS VALUE
LETTER
IN THE MAGHREB
IN THE EAST
J* sin
300
60
sad
60
90
^ shin
1.000
300
dad
90
800
J? dha
800
900
£ ghayin
900
1,000
Fig. 19.2.
Now, let us first note that the numerical values of the Arabic letters can
be arranged into a regular series, as follows:
1; 2; 3; 4; . . . 10; 20; 30; 40; . . . 100; 200; 300; 400; . . . ; 1,000,
and if we set out, according to this sequence, the letter-numerals of the
eastern Arabic system (the more ancient of the two) we obtain the order of
the western Semitic letters of which we have just written (Fig. 17.2 and 17.4
above). Furthermore, if we tabulate the letter-numerals of the Arabic
system (as in Fig. 19.4) and compare this with the Hebrew letter-numerals
(Fig. 17.10) and also with the Syriac system of alphabetic numbering
(Fig. 19.1), then it is easy to see that for the numbers below 400 all three
systems agree perfectly. This shows that “in the initial system of numera-
tion, the order of the northern Semitic alphabet was preserved, and
additional letters from the Arabic alphabet were added later in order to go
up to 1,000” [M. Cohen (1958)].
LETTERS
NUMERICAL
VALUES
LETTERS
PHONETIC
LETTERS
LETTERS
LETTERS
ON
LETTER-
VALUES
IN
IN
IN
IN THE
IN THE
THEIR
OWN
NAMES
OE
LETTERS
INITIAL
POSITION
MEDIAN
POSITION
END
POSITION
EAST
MAGHREB
I
'Alif
i
l
l
1
1
o-J
Ba
b
;
7
-r
2
2
01
Ta
t
7
=
400
400
S±J
Tha
th
*
t
-L-
500
500
z
Jim
j
9?
E
3
3
z
Ha
h
>■
7K
C
8
8
t
Kha
h
9-
9*
C
600
600
s
Dal
d
J
a
JL
4
4
s
Dhal
dh
3
i
i
700
700
J
Ra
r
J
j
J
200
200
j
Zay
z
j
j
j
7
7
lT
Sin
s
-
-
60
300
J*
Shin
sh
A
A
*
300
1,000
Sad
s
-£>
90
60
J*
Dad
d
la
la
. r*
800
90
J*
fa
t
da
la
da
9
9
Dha
dh
Ji
da
da
900
800
t
'Ayin
c-
A
70
70
t
Ghayin
gh
i
A
1,000
900
Fa
f
j
A
80
80
J
Qaf
9
5
A
100
100
Kaf
k
7
5C
dd
20
20
J
Lam
1
J
1
J
30
30
r
Mim
m
M
A.
r
40
40
j
Nun
n
J
-
O
50
50
6
Ha
h
*
♦
4
5
5
3
Wa
w
J
>
6
6
t £
Ya
y
t
z
10
10
Fig. 19 . 3 . The Arabic alphabet, in its modern representation
243
ARABIC LETTER-NUMERALS
We may therefore conclude that the use of alphabetic numerals by the
Arabs was introduced in imitation of the Jews and the Christians of Syria
for the first twenty-two letters (numbers below 400), and according to the
example of the Greeks for the remaining six (values from 400 to 1,000).
I
’ Alif
•
i
t - r
Sin
s
60
u
Ba
b
2
t
*Ayin
*
70
i
Jim
j
3
Fa
f
80
$
Dal
d
4
Sad
s
90
6
Ha
h
5
J
Qaf
q
100
3
Wa
w
6
J
Ra
r
200
j
Zay
z
7
A
Shin
sh
300
c
Ha
h
8
cZJ
Ta
t
400
J*
Ta
t
9
s±J
Tha*
th
500
Ya
y
10
t
Kha*
kh
600
i)
Kaf
k
20
b
Dhal*
dh
700
J
Lam
i
30
J*
Dad*
d
800
f
Mim
m
40
Dha*
dh
900
j
Nun
n
50
Ghayin*
* subsequently added
g h
1,000
Fig . 19 . 4 . The order of Arabic letters as ordained according to the regular development of the values
of the alphabetic number-system of eastern Arabs
In fact, “following the conquest of Egypt, Syria and Mesopotamia,
numbers were habitually written, in Arabic texts, either spelled out in
full or by means of characters borrowed from the Greek alphabet” [A. P.
Youschkevitch (1976)].
Thus we find in an Arabic translation of the Gospels, the manuscript
verses have been numbered with Greek letters:
Similarly, in a financial papyrus written in Arabic and dating from the
year 248 of the Hegira (862-863 CE), the sums were written exclusively
according to the Greek system. [This document is, along with others of
similar kind, in the Egyptian Library, inventory number 283; cf. A.
Grohmann (1962)].
This usage persisted in Arabic documents for several centuries, but
disappeared completely in the twelfth century. For all that, we should not
conclude that Arabic letter-numerals were introduced only at that time.
The system certainly first arose before the ninth century. We have, in fact,
a mathematical manuscript copied at Shiraz between the years 358 and 361
of the Hegira (969-971 CE) in which all of the Arabic alphabetic numerals
are used according to the Eastern system.*
Likewise, there is an astrolabe 1 dating from year 315 of the Hegira
(927-928 CE) where this date is expressed in a palaeographic style known
as Kufic script (Fig. 19.10). Other older documents indicate that the intro-
duction of this system to the Arabs occurred as early as the eighth century,
or, at the earliest, the end of the seventh.
From then on, all becomes clear. After adding six letters to the western
Semitic alphabet which they inherited, and having established their system
of alphabetic numerals preserving the traditional order of the letters, the
Arab grammarians of the seventh or eighth century, apparently for
pedagogical reasons, completely changed the original order of the letters
by bringing together letters which had much the same graphical forms. At
that time these grammarians “worked mainly in Mesopotamia where
Jewish and Christian studies flourished, with Greek influences” (M. Cohen).
Thus it was that, from that time on, letters such as ba, ta, and tha, or jim,
ha, and kha were placed in sequence in the Arabic alphabet (Fig. 19.3).
♦
c
c
e
ki-
k&-
♦
kha
ha
jim
th a
ta
ba
600
8
3
500
400
2
Fig. 19.5. Excerpt from a Christian ninth-century manuscript. In this manuscript, which gives a
translation from the Gospels, the corresponding verses have been numbered by reference to Greek
letter-numbers, (first line, right: OH = v. 78: second line, right: OS = v. 79. Vatican Library, Codex
Borghesiano arabo 95, folio 173. (See E. Tisserant, pi. 55)
Fig. 19.6.
* “Treatise by Ibrahim ibn Sinan on the Methods of Analysis and Synthesis in Problems of Geometry", a
tenth-century copy of fifty-one works on mathematics (BN Ms.arab. 2457; see for example flf. 53v and 88).
t A scientific instrument for observing the position of stars and their height above the horizon. It was used
in particular by Arabian astrologers, but some examples have been found from the Graeco-Byzantine era.
OTHER ALPHABETIC NUMBER-SYSTEMS
244
The better to establish the order of the alphabetic numerals, the eastern
Arabs invented eight mnemonics which every user had to learn by heart
in order to be able to recall the number-letters according to their regular
arithmetic sequence (Fig. 19.7).
This clearly shows that the “ABC” order, pronounced Abajad (or
Abjad, Abujad, Aboujed, etc. depending on accent), which sometimes
governs the order of letters in the Arabic alphabet, does not correspond to
their phonetic value nor to their graphical form, but to their respective
numerical values according to the eastern Arabian system (Fig. 19.4).
In the usage of the Maghreb, it should be noted, the numerical values
given to six of the twenty-eight Arabic letters are different from those in
the Eastern system; also, the grouping of the number-letters is different,
being done according to nine mnemonics which yield the following
groups of values: (1; 10; 100; 1,000); (2; 20; 200); (3; 30; 300); etc. (Fig.
19.11).
MNEMONIC WORDS
DECOMPOSED AS
... • •
Abajad <
d j b 'a
4. 3. 2. 1
<■
JJ*
Hawazin ^
j 3 6
i w h
<r
7. 6. 5.
<r
Hutiya
y l h
10. 9. 8
vJLS
Kalamuna
n m 1 k
<r
50. 40. 30. 20
<
♦
Sa'fas ^ _
S f * f
90. 80. 70. 60
▲ **
d
°i J 1 J J
t sh r q
400. 300. 200. 100.
Thakhudh *
3 1 ^
dh Rh th
700. 600. 500.
gh dh d
1,000. 900. 800
Fig. 19.7. Mnemonic words enabling eastern practitioners to find the order of numerical values
associated with Arabic letters
604 j, £
4 600
<r
12 v S
- ♦ •*
2 10
«
472 l ^ ^
2 70 400
<r
58 Z. j
8 so
<r
3 B0 200 1,000 ^
96 Jj*
6 90
<r
1,631 | l i . •
t
1 30 600 1,000
<r
169 L...S
9 60 100
<•
<r
315 - » a
4^ a . *■
S 10 300 ••
<■
Fig. 19.8. The writing of numbers by reference to the number-letters of the eastern Arabic system
(transcribed into current characters) is always from right to left in descending order of values, start-
ing with the highest order. Moreover, these number-signs (as with ordinary Arabic letters) have an
inter-relationship generally by undergoing slight graphic modifications according to the position they
occupy within the body of the number- or word-combinations. Examples reconstituted from an Arabic
manuscript copied at Shiraz c. 970. Paris: Bibliotheque nationale, Ms. ar. 2457
On the other hand, this same order occurs not only with the Jews, but
also with all the northwestern Semites, as well as the Greeks, the Etruscans
and the Armenians, to cite but a few. It is a very ancient ordering since,
more than twenty centuries earlier than the Arabs, the inhabitants of Ugarit
were familiar with it.
Nonetheless the Arabs, lacking knowledge of the other Semitic
languages . . . sought other explanations for the mnemonics abjad,
etc. which had come down to them by tradition but which they
found incomprehensible. The best that they could propose on this
subject, interesting though it is, is pure fable. According to some, six
kings of Madyan arranged the Arabic letters according to their own
names. According to a different tradition, the first six mnemonics
were the names of six demons. According to a third, it was the names
of the days of the week. . . . We may none the less discern an
interesting detail amongst these fables. One of the six kings of
Madyan had supremacy over the others ( ra'isuhum ): this was Kalaman,
whose name bears perhaps some relation with the Latin elementa*.
In North Africa, the adjective bujadi is still used to mean beginner,
novice literally, someone who is still on his ABC. [G. S. Colin]
* According to M. Cohen (p. 137 ), the Latin word elementum goes back to an earlier alphabet that began
in the middle, with the letters I., M, N. So giving the LMN ( elemen-tum ) of a matter was the same as “saying
the ABC of it all”.
245
ARABIC LETTER-NUMERALS
Fig. 19.9. Seventeenth -century Persian astrolabe inscribed by Mohannad Muqim (Delhi, Red
Fort, Isa 8). Note that the rim is marked in fives to 360 degrees by means of Arabic letter-numbers.
(See B. von Dorn)
MNEMONIC WORDS
RETAINED BY THIS
USAGE
1 \ ’alif
10 ya
100 qaf
1,000 shin
Ayqash ^
4 -
2 V* ba
20 ^ kaf
200 J ra
Bakar
<r
3 ^ jim
30 J lam
300 ^ sin
Jalas
4
4 ^ dal
40 (* mim
400 O ta
Damat C-* J
4
5 * Ha
50 O Nun
500 Tha
Hanath
4 -
6 J Wa
60 Sad
600 £ Kha
Wasakh
•4
7 j Zay
70 'Ayin
700 j Dhal
Za'adh JC-J
4
8 C Ha
80 i Fa
800 Dha
Hafadh
4 -
9 L Ta
90 J* Dad
900 £Ghayin
Tad ugh
4
“WORK OF BASTULUS
YEAR 315”
Fig. 19 . 11 . Numeral alphabet used by African Arabs. (For mnemonic words see Fig. 19. 7 and foot-
note [same page})
Fig. 19 . 10 . Detail from an early oriental astrolabe, ostensibly once the property of King Farouk of
Egypt, inscribed by Bastulus and dating from 315 of the Hegira (927-928 CE). The date is
expressed by means of letter-numbers from the eastern number-system (“Kufic" characters with
diacritics). (Personal communication from Alain Brieux)
OTHER ALPHABETIC NUMBER-SYSTEMS
246
The eastern Arabs represented thousands, tens of thousands and
hundreds of thousands by the multiplicative method. For this purpose,
they adopted the convention of putting the letter associated with the
corresponding numbers of units, of tens or of hundreds to the right of
the Arab letter ghayin, whose value was 1,000 (Fig. 19.12).
Arabic letter-number attributed to 1,000*
i 6
isolated form final form
♦
1,000 X 8 £■>* 8,000
•
1,000 x 2 2,000
ghH
gh B
1,000 x 9 9,000
•
1,000 x 3 3,000
ghT
gh J
1,000 x 10 iu 10,000
•
1,000 x 4 4,000
gh Y
gh D
1,000 x 20 20,000
1,000 x 5 ^ 5,000
gh K
ghH
1,000 x 30 30,000
gh L
•
1,000 x 6 gj 6,000
gh W
♦
1,000 x 40 40,000
« •
1,000 x 7 7,000
gh M
ghZ
* i.e. the letter ghayin, twenty-eighth in the Abjad system (Fig. 19.4)
Fig. 19.12. Eastern Arabic notation for numbers above 1,000
THE ETHIOPIAN NUMBER-SYSTEM
The Ethiopians borrowed the Greek alphabetical numbering system during
the fourth century CE, no doubt under the influence of Christian mission-
aries who came from Egypt, Syria and Palestine.*
Starting, however, with 100, they radically altered the Greek system.
Having adopted the first nineteen Greek alphabetic numerals to represent
the first hundred whole numbers, they decided to indicate the hundreds
and thousands by putting the letters for the units and tens to the left
of the sign P (Greek rho) whose value was 100. That is to say that instead of
representing the numbers 200, 300, . . . 9,000 after the Greek fashion:
S T Y ... A 'A 'B ... '0
200 300 400 900 1,000 2,000 9,000
they expressed them as follows (see Fig. 19. 13 A):
BP ... HP ... KP ... np
2 x 100 8 X 100 20 X 100 80 x 100
> » » »
200 800 2,000 8,000
They denoted 10,000 by marking a ligature between two identical P signs
(making a composite sign equivalent to multiplying 100 by itself, which we
shall transcribe as P-P. Then multiples of 10,000 were expressed by placing
the symbol for the multiplier to the left of this symbol P-P for 10,000.
BPP ... HPP ... KPP ... npp
2 x 10,000 8 x 10,000 20 x 10,000 80 x 10,000
» * »
20,000 80,000 200,000 800,000
* The numerals that Ethiopians still sometimes use today are actually much more rounded stylisations of
the numerical signs found on the Aksum inscriptions (Aksum, near the modern port of Adowa, was the
capital of the Kingdom of Abyssinia from the fourth century CE). The modem signs follow the same princi-
ples as the ancient ones, which are themselves derived from the first nineteen letter-numerals of the Greek
alphabet. Since the fifteenth century Ethiopian numerals have always been written inside two parallel bars
with a curlicue at either end, signifying that they are to be taken as numbers, not as letters.
THE ETHIOPIAN NUMBER-SYSTEM
VALUES AND
ETHIOPIAN
INSCRIPTIONS
MODERN ETHIOPIAN
NUMBERS AND
ARITHMETICAL
TRANSLATIONS
MAGIC, MYSTERY, DIVINATION
248
CHAPTER 20
MAGIC, MYSTERY,
DIVINATION,
AND OTHER SECRETS
SECRET WRITING AND
SECRET NUMBERS IN THE
OTTOMAN EMPIRE
We shall close our account of alphabetic numerals with an examination
of the secret writing and secret numerals used until recently in the
Middle East and, especially, in the official services of the Ottoman
Empire.*
The Turks used cryptography with abandon. Documents on
Mathematics, Medicine and the occult, written or translated by
the Turks, teem with secret alphabets and numerals, and they made
use of every alphabet they knew. Usually they adopted such alphabets
in the form in which they came across them, but sometimes they
changed them; either deliberately, or as a result of the mutations
which attend repeated copying. [M. J. A. Decourdemanche (1899)]
Fig. 20.1 shows secret numerals which were used for a long time in Egypt,
Syria, North Africa, and Turkey. At first sight these would seem to have
been made up throughout. However, if they are put alongside the Arabic
letters which have the same numerical values, and then we put alongside
these the corresponding Hebrew and Palmyrenean characters, we can at
once see that the figures of these secret numerals are simply survivals of
the ancient Aramaic characters in their traditional Abjad order (Fig. 20.2;
see also Fig. 17.2, 17.4, 17.10 and 19.4).
* Such esoteric writing was used in a great variety of contexts: occultism, divination, science, diplomacy,
military reports, business letters, administrative circulars, etc. Until the beginning of this century, the
Turkish and Persian offices of the Ministry of Finance used a system of numerals known as Siyaq, whose
figures were used in balance sheets and business correspondence. These figures were abbreviations of the
Arabic names of the numbers, and their purpose was both to keep the sums of money secret from the public
and also to prevent fraudulent alteration (see Chapter 25).
Yi
T
1
ft
1
*
X
9
8
7
6
5
4
3
2
1
?) or &
x>
6
T**
h°' S
V
v»
90
80
70
60
50
40
30
20
10
is
5
w
V
H
1>
)
1,000
900
800
700
600
500
400
300
200
100
Fig. 20.i.
Among these secret numerals there were alternative forms for the values
20, 40, 50, 80 and 90. These are in fact the final forms of the Hebrew and
Palmyrenean letters kof, mem , nun, pe and tsade. The correspondences
noted here are confirmed by treatises on arithmetic. The Egyptian treatises
refer to this system as al Shamisi (“sunlit”), which was used in those parts
to designate things related to Syria. The Syrian documents themselves
called it al Tadmuri (“from Tadmor”), which was the former Semitic name
of Palmyra, an ancient city on the road linking Mesopotamia to the
Mediterranean via Damascus to the south and via Homs to the north.
The people who had devised these secret writings had therefore taken
the twenty-two Aramaic letters as they found them and (as has been explic-
itly mentioned by Turkish writers) they added six further conventional
signs in order to complete a correspondence with the Arabic alphabet and
to achieve a system of numerals which was complete from 1 to 1,000. This
system was used until recent times, not only for writing numbers, but also
as secret writing:
In 1869, in order to draw up for French military officers a comparison
between the abortive expedition of Charles III of Spain against
Algiers, and the French expedition of 1830, the Ministry of War
brought to Paris the original military report on the expedition of
Charles III, which had been written in Turkish by the Algerian Regency
at the Porte. This document was given to a military interpreter to
be summarised. The manuscript, which I have seen, carried the
stamp of a library in Algiers. After a whole wad of financial accounts
came the report from the Regency. Following this came a series of
annexes amongst which is an espionage report written as a long
letter in the Hebraic script called Khat al barawat.
249
SECRET WRITING AND SECRET NUMBERS
PALMYRENEAN
AND HEBRAIC
LETTERS
ARABIC
LETTERS
TADMURI
ALPHABET
PALMYRENEAN
AND HEBRAIC
LETTERS
ARABIC
LETTERS
TADMURI
ALPHABET
a H
*
a 1
1
■ V
b
■ J
S
30
b 3
b v
'L
2
D
n
m r
b S
40
g 3
>
1 z
X
3
n) 3
1
n ^
1 r
50
d 1
d J
'i
4
, D
V
s
60
h "
A
h *
A
5
•e 7
■« t
70
w 1
?
w ^
6
*1 *
P 1
J
f yJ
80
■ *
1
2 j
¥
7
,T *
90
h n
K
h z
8
, P
a
, 3
100
t
4
I J-
9
n
r
X
J
r
>
200
*
y
9
J
y "
*->
10
m
sh
X/
lT*
sh
V
300
k v
d
k ^
1 ^
20
n
t
r>
O
t
400
Fig. 20.2. Secret alphabet (still used in Turkey, Egypt, and Syria in the nineteenth century)
compared with the Arabic, Palmyrenean, and Hebraic alphabets
The signature was written in Tadmuri characters, not Latin:
Felipe, rabbi na Yusuf ben Ezer, nacido en Granada.
rz'e NB FWSWY ’ANB’AR pylf
e
ADANARGh N’E WDYSAN
4
Fig. 20.3.
Then, on exactly the same kind of paper as the letter, is a
detailed analysis of the Spanish land and sea forces, again written
in Tadmuri characters. Since this analysis is also reproduced line
for line in normal Turkish characters in the Regency report, it was
easy for me to discern the value of each of the Tadmuri signs.
As an example, here in reproduction is the first line of the
analysis, possibly for the army, possibly for the navy:
A3
5 80 100 %. UJbrtSlSjt
70
Fig. 20.4.
in which the following Spanish expressions are written in Tadmuri
script:
Regimento (del) Rey,
“King’s Army”
El Velasco,
“Navy”
185 (hombres)
185 men
70 (canones)
70 guns
[M. J. A. Decourdemanche (1899)]
We have no intention of presenting a general survey of the very many
clandestine systems of the East; nonetheless we shall discuss two other
systems of secret numerals which were used until recent times in the
Ottoman army.
We begin with the simplest case. This is a system of numerals used in
Turkish military inventories of provisions, supplies, equipment, and so on.
Fig. 20.5.
MAGIC, MYSTERY, DIVINATION
250
Here the numbers 1, 10, 100, 1,000 and 10,000 are represented by a
vertical stroke with, on the right, one, two, three, four, or five upward
oblique strokes. Adding one upward oblique stroke on the left of each of
these gives the figures for 2, 20, 200, 2,000, and 20,000; two strokes on
the left gives the figures for 3, 30, 300, 3,000, and 30,000; and so on, until
with eight oblique strokes on the left we have the figures for 9, 90, 900,
9,000, and 90,000.
The above system is very straightforward, which is not the case for the
next one. This was used in the Turkish army for recording the strengths of
their units.
) 8 7 6 5 4 3 2 1
*
9
p ^
0 8
0 70 60 50 40 30 20 10
1
9
00 8
00 700 600 5
if f f f Y
00 400 300 200 100
Fig. 20.6.
To the uninitiated, this system follows no obvious pattern. However, it
was used both for writing numbers and also as a means of secret writing,
which leads us to suppose that each of these signs corresponded to the
Arabic letter corresponding to the numeral in question.
Proceeding as we did before, placing each of these numerals beside the
Arabic letter corresponding to the same numerical value (see Fig. 19.4
above), we now consider the eight mnemonics for the letters of the Arabic
numerals (Fig. 19.7), and it becomes clear how the figures of this system
were formed.
For the numbers 1, 2, 3, 4 (corresponding to the first mnemonic,
ABJaD), we take a vertical stroke with one oblique upward stroke on its
right, and adjoin successively one, two, three, or four oblique upward
strokes on its left.
Then, for the second mnemonic, HaWaZin, we take a vertical stroke with
two upward oblique strokes on the right, and add successively one, two, or
three upward oblique strokes on the left, and so on (Fig. 20.7).
Fig. 20.7. Secret numerical notation based on the succession of eight mnemonic words in the
eastern Arab alphabetical numbering
THE ART OF CHRONOGRAMS
Jewish and Muslim writings since the Middle Ages abound in what are
called “chronograms”: these correspond to a method of writing dates,
but - like calligraphy or poetry - are an art form in themselves.
This is the Ramz of the Arab poets, historians and stone-carvers in North
Africa and in Spain, the Tarikh of the Turkish and Persian writers, which
“consists of grouping, into one meaningful and characteristic word or short
phrase, letters whose numerical values, when totalled, give the year of a
past or future event.” [G. S. Colin]
251
The following example occurs on a Jewish tombstone in Toledo [IHE,
inscr. 43]:
dsj^k n®an bv *?tp na®
THOUSAND FIVE ON DEW DROP YEAR
YEAR: ONE DROP OF DEW ON FIVE THOUSAND
Fig. 20.8.
If we take it literally, the phrase is meaningless. But if we add up the
numerical values of the letters in the phrase translated as “drop of dew”, we
discover that this phrase represents, according to the Hebrew calendar, the
date of death of the person buried here:
ONE DROP OF DEW
b cd •» b a x
30 9 10 30 3 1
Fig. 20.9.
This person died, in fact, in the year “eighty-three [= drop of dew] on five
thousand,” or, in plain language, in the year 5083 of the Hebrew era, i.e.
1322-1323 CE.
In the following two further examples from the Jewish cemetery in
Toledo we find the years 5144 (Fig. 20.10) and 5109 (Fig. 20.11, in two
different forms) shown in the chronograms: but note that the “5000” is not
indicated, since it would have been implicitly understood, much as we
understand “1974” when someone says “I was born in seventy-four”. Also,
note that in these examples the words whose letters represent numerals
have been marked with three dots.
nx rx na^n na®
2 1 50 1 5 10 5 YEAR
10 50 10
Fig. 2o.io.
year: we HAVE BEEN MADE FATHERLESS
D^n'h i b nrri a a
40 10 10 8 5 6 30 5 8 6 50 40
Fig. 20.11.
THE ART OF CHRONOGRAMS
The same procedure is found in Islamic countries, especially Turkey,
Iraq, Persia, and Bihar (in northwest India); but, like the oriental art of
calligraphy, it seems to go no further back than the eleventh century.* The
dreadful death of King Sher of Bihar in an explosion occurred in the year
952 of the Hegira (1545 CE), which is recorded in the following chronogram
[CAPIB, vol. X, p. 368]:
* - 1 •
•> J f wT J
4 200 40 300 400 1 7
<.
952 (of the Hegira)
Fig. 20.12.
Another interesting chronogram was made by the historian, mathemati-
cian and astronomer A1 Biruni (born 973 CE at Khiva, died 1048 at Ghazni)
in his celebrated Tarikh ul Hind. This learned man accused the Jews of delib-
erately changing their calendar so as to diminish the number of years
elapsed since the Creation, in order that the date of birth of Christ should
no longer agree with the prophecies of the coming of the Messiah; he boldly
asserted that the Jews awaited the Messiah for the year 1335 of the Seleucid
era (1024 CE), and he wrote this date in the following form:
JU9B4J J* cJjUell Sl^eJ
“MOHAMMED SAVES THE WORLD FROM UNBELIEF"
4 40 8 40 2 200 80 20 30 1 50 40 100 30 600 30 1 5 1 3 50
<r
1335
Fig. 20.13.
* In Persian and in Turkish certain letters have exactly the same numerical values as the equivalent Arabic
letters according to the Eastern usage. For instance:
the
letter
O
, or P, has the same value as
, or B;
the
letter
C
, or Ch, has the same value as
C
. orj;
the
letter
s
, or G, has the same value as
s
, or K.
“died of burns”
MAGIC, MYSTERY, DIVINATION
252
Chronograms were also common in Morocco, but only from the
seventeenth century CE (possibly the sixteenth, or earlier, according to
recent documentation). They were often used in verse inscriptions
commemorating events or foundations, and by writers, poets, historians
and biographers, including the secretary and court poet Muhammad Ben
Ahmad al Maklati (died 1630), and also the poets Muhammed al Mudara
(died 1734) and ‘Abd al Wahab Adaraq (died 1746) who both composed
instructional historical synopses on the basis of chronograms, which in
one case referred to the notabilities of Fez, and in the other to the saints
of Maknez.*
The following example comes from an Arabic inscription discovered by
Colin in the Kasbah of Tangier over fifty years ago, in the south chamber of
the building known as Qubbat al Bukhari, in the old Sultan’s Palace. We
make a brief detour in time so as to stand in the period when this building
was constructed.
The inscription was written to the glory of Ahmad ibn ‘Ali ibn ‘Abdallah.
This notable person was:
the son of the famous ‘Ali ibn ‘Abdallah, governor ( qa’id) of Tetuan
and chief of the Rif contingents destined for holy war ( mujahidin ) who,
after a long siege, entered Tangier in 1095 of the Hegira (1684 CE) after
its English occupiers had abandoned it . . .
When Qa’id ‘Ali ibn ‘Abdallah, commandant (amir) of all the people
of the Rif, died in year 1103 of the Hegira (1691-1692), Sultan Isma’il
gave to them as chief the dead man’s son, basa Ahmad ibn ‘Ali;
henceforth, almost all the history of northwest Morocco can be found
in this man’s biography . . . After 1139 of the Hegira (1726-1727),
following the death of Sultan Isma’il, he took the opportunity pro-
vided by the weakness of his successor, Ahmad ad Dahabi, to try to
seize Tetuan which was administered by another, almost indepen-
dent, governor (amir), Muhammed al Waqqas, but he was repulsed
with loss.
In 1140 of the Hegira (1727-1728), when Sultan Ahmad ad Dahabi
(who had been overturned by his brother ‘Abd al Malik) was restored
to the throne, Ahmad ibn ‘Ali refused to recognise him and declined
to send him a deputation (a snub which was imitated by the town
of Fez). The enmity between the Rif chieftain and the ‘Alawite
kings waxed from then on, and an impolitic gesture by Sultan
‘Abdallah, successor of Ahmad ad Dahabi, transformed this into overt
hostility . . .
* In epigraphic texts, chronograms were often written in a contrasting colour, and sometimes also in
manuscripts where, however, we also find them written with thicker strokes. Arab chronograms, like those
in Hebraic inscriptions, were always preceded by the preposition fi or by Sanat 'ama in the year, etc.
In 1145 of the Hegira, when a delegation of 350 holy warriors
from the Rif came from Tangier to Sultan ‘Abdallah to try to resolve
the differences between him and basa Ahmad ibn ‘Ali, he had them
killed. The Rif chieftain distanced himself from the King and came
closer to his brother and rival Al Mustadi. Thenceforth, until his
unfortunate death in 1156 of the Hegira (1743), he did not cease from
fighting with ‘Abdallah, son of Sultan Isma’il, and to support his rivals
against him. [G. S. Colin]
Returning now to our inscription, the date 1145 is given in the following
verse (in which the numerical values have been calculated from the Arabic
alphabetic numerals according to the Maghreb usage; see Fig. 19.11).
••
year:“the full moon of my beauty has entered
THE CHAMBER OF HAPPINESS”
Oj vJC
10 30 1 40 3 200 4 2 4 70 300 30 1 400 10 2 30 8
<
1145
Fig. 20.14.
In other words, the Qubbat al Bukhari in the Kasbah of Tangier was
constructed in the year 1145 of the Hegira, the very time when basa Ahmad
ibn ‘Ali broke away from Sultan ‘Abdallah.
We find in this chronogram, therefore, testimony to an art in which one’s
whole imagination is deployed to create a phrase which is both eloquent
and, at the same time, has a numerical value that reveals the date of an
event which one wishes to commemorate.
GNOSTICS, CABBALISTS, MAGICIANS,
AND SOOTHSAYERS
Once the letters of an alphabet have numerical values, the way is open to
some strange procedures. Take the values of the letters of a word or
phrase and make a number from these. Then this number may furnish an
interpretation of the word, or another word with the same or a related
numerical value may do so. The Jewish gematria* the Greek isopsephy
and the Muslim khisab al jumal (“calculating the total”) are examples of this
kind of activity.
* Possibly a corruption of the Greek geometrikos arilhmos, geometrical number
253
Especially among the Jews, these calculations enriched their sermons
with every kind of interpretation, and also gave rise to speculations and
divinations. They are of common occurrence in Rabbinic literature, espe-
cially the Talmud* and the Midrash. 1 But it is chiefly found in esoteric
writings, where these cabbalistic procedures yielded hidden meanings for
the purposes of religious dialectic.
Though not adept in the matter, we would here like to describe some
examples of religious, soothsaying or literary practices which derive from
such procedures.
The two Hebrew words Yayin, meaning “wine”, and Sod, meaning
“secret”, both have the number 70 in the normal Hebrew alphabetic numer-
als (Fig. 20.15), and for this reason some rabbis bring these words together:
Nichnas Yayin Yatsa Sod: “the secret comes out of the wine” (Latin: in vino
veritas, the drunken man tells all).
I" “110
Fig. 20.15.
In Pardes Rimonim, Moses Cordovero gives an example which relates
gevurah (“force”) to arieh (“lion”), which both have value 216. The lion,
traditionally, is the symbol of divine majesty, of the power of Yahweh, while
gevurah is one of the Attributes of God.
5 200 6 2 3
5 10 200 1
r. „ ZJLD ZIO
Fig. 20.16.
The Messiah is often called Shema, “seed”, or Menakhem, “consoler”
since these two words have the same value:
no 2* oma
6 40 90 40 8 50 40
Fig. 20.17.
The Rabbinic compilation of Jewish laws, customs, traditions and opinions which forms the code of Jewish
civil and canon law
t Hebrew commentaries on the Old Testament
GNOSTICS, CAB BALISTS, MAGICIANS, AND SOOTHSAYERS
The letters of Mashiyakh, “Messiah”, and of Nakhash, “serpent”, give
the same value:
Fig. 20.18.
rim rrrio
300 8 50 8 10 300 40
« <•
NAKHASH MASHIYAKH
and this gives rise to the conclusion that "When the Messiah comes upon
earth, he shall measure himself against Satan and shall overcome him.”
We may also conclude that the world was created at the beginning of the
Jewish civil year, from the fact that the two first words of the Torah ( Bereshit
Bara, “in the beginning [God] created”) have the same value as Berosh
Hashanah Nibra, “it was created at the beginning of the year”:
ana rrriina
1 200 2 400 10 300 1 200 2
«
BERESHIT BARA
1116
Fig. 20.19.
an a a mrin rima
1 200 2 50 5 50 300 5 300 1 200 2
<
BEROSH HASHANAH NIBRAtl
1116
In Genesis XXXII:4, Jacob says “I have sojourned with Laban” (in
Hebrew, ‘Im Laban Garti). According to the commentary by Rashi* on this
phrase ( Bereshit Rabbati, 145), this means that “during his sojourn with
Laban the impious, Jacob did not follow his bad example but followed the
613 commandments of the Jewish religion”; for, as he explains, Garti (“I
have sojourned”) has the value 613:
’FllJ
10 400 200 3
<■
Genesis recounts elsewhere (XIV: 12-14) how, in the battle of the kings
of the East in the Valley of Siddim, Lot of Sodom, the kinsman of Abraham,
was captured by his enemies: "When Abraham heard that his brother was
taken captive, he armed his trained servants, born in his own house, three
hundred and eighteen, and pursued them unto Dan”, where he smote his
adversaries with the help of “God Most High” (XIV:20). Then he addresses
God in these words: "Lord GOD [Yahweh], what wilt thou give me, seeing I
go childless and the steward of my house is this Eliezer of Damascus?”
(Genesis XV:2).
Rabbenu Shelomoh Yishakhi (1040-1105)
magic:, mystery, divination
The barayta of the thirty-two Haggadic rules (for the interpretation of
the Torah) gives the following interpretation (rule 29): the 318 servants are
none other than the person of Eliezer himself. In other words, Abraham
smote his enemies with the help of Eliezer alone, his trusted servant who
was to be his heir; and whose name in Hebrew means “My God is help”.
The argument put forward for this brings together the two verses
his trained servants, born in his own house, three hundred and
eighteen
and
the steward of my house is this Eliezer of Damascus
and the fact that the numerical value of the name Eliezer is 318:
Fig. 2 o. 2 i.
200 7 70 10 30 1
«
El. IE Z F. R
318
Another concordance which the exegetes have achieved brings Ahavah
(“Love”) together with Ekhad (“One”):
As well as their numerical equivalence, it is explained that these two
terms correspond to the central concept of the biblical ethic, that “God
is Love”, since on the one hand “One” represents the One God of Israel
and, on the other hand, “Love” is supposed to be at the very basis of
the conception of the Universe (Deuteronomy: V 6-7; Leviticus XIX:18). At
the same time, the sum of their values is 26, which is the number of the
name Yahweh itself:
mm
5 6 5 10
«
YHWH
Fig. zo. 23 . 26
The common Semitic word for “God” is El, but in the Old Testament this
only occurs in compounds ( Israel , Ismael, Eliezer, etc.). To refer to God, the
Torah uses Elohim (which in fact is plural), and is the word which is
supposed to express all the force and supernatural power of God. The Torah
254
refers also to the attributes of God, such as khay (“living”), Shadai (“all-
powerful”), Elllyion (“God Most High”) and so on. But YHWH, “Yahweh”,
is the only true Name of God: it is the Divine Tetragram. It is supposed to
incorporate the eternal nature of God since it embraces the three Hebrew
tenses of the verb “to be”, namely:
mn mn rrrr
HaYaH "He was” HoVVeH "He is” YiHYeH “He shall be”
Fig. 20 . 24 .
To invoke God by this name is therefore to appeal to His intervention
and His concern for all things. But this name may be neither written
nor spoken casually, and in order not to violate what is holy and incommu-
nicable, in common use it must be read as Adonai (“My Lord”).
Every kind of speculation has been founded on the numerical value of
26 which the Tetragram assumes according to the classical system of
alphabetic numerals. Some adept writers have thereby been led to point out
that in Genesis 1:26, God says: “Let us make man in our image”; that 26
generations separate Adam and Moses; that 26 descendants are listed in
the genealogy of Shem, and the number of persons named in this is a
multiple of 26; and so on. According to them, the fact that God fashioned
Eve from a rib taken from Adam is to be found in the numerical difference
(= 26) between the name of Adam (= 45) and the name of Eve (= 19):
mn Dis
5 6 8 40 4 1
« <r
KHAWAH ADAM
19 45
Fig. 20 . 25 .
The usual alphabetic numerals were not the only basis adopted by
the rabbis and Cabbalists for this kind of interpretation. A manuscript in
the Bodleian Library at Oxford (Ms. Hebr. 1822) lists more than seventy
different systems of gematria.
One of these involves assigning to each letter the number which gives
its position in the Hebrew alphabet but with reduction of numbers above
9, that is to say with the same units figure as in the usual method, but
ignoring tens and hundreds. The letter 0 (mem), for example, which tradi-
tionally has the value 40, is given the value 4 in this system.* Similarly, the
* This can be found by the alternative method of noting that Mem is in the thirteenth place, so its value is
equal to 1 + 3 = 4.
255
letter 0 (shin), whose usual value is 300, has value 3 in this system. *From
this, some have concluded that the name Yahweh can be equated to the
divine attribute Tov (“Good”):
mm a id
5 6 5 1 2 6 9
<■ <■
YHWH TOV
“Good”
17 17
Fig. 20.26.
Another method gives to the letters values equal to the squares of their
usual values, so that gimmel, for example, which usually has value 3, is here
assigned the value 9 (Fig. 20.29, column B). According to a further system,
the value 1 is assigned to the first letter, the sum (3) of the first two to the
second letter, the sum (6) of the first three to the third, and so on. The letter
yod , which is in the tenth position, therefore has a value equal to the sum
of the first ten natural numbers: 1 + 2 + 3 + .. . + 9 + 10 = 55 (Fig. 20.29,
column C).
Yet another system assigns to each letter the numerical value of the word
which is the name of the letter. Thus aleph has the value 1 + 30 + 80 = 111:
n i d
80 30 1
+
111
Fig. 20.27.
With these starting points, one can make a concordance between two
words by evaluating them numerically according to either the same numer-
ical system, or two different numerical systems. For instance, the word
Maqom (“place”), which is another of the names of God, can be equated to
Yahweh because in the traditional system the word Maqom has value 186,
and Yahweh also has value 186 if we use the system which gives each letter
the square of its usual value:
Dips mrr
40 6 100 40 5 2 6 2 5 2 10 2
^ 4
MAQOM YHWH
186 186
Fig. 20.28.
* Shin is in the twenty-first place, so its value is 2 + 1 = 3.
GNOSTICS, CABBALISTS, MAGICIANS, AND SOOTHSAYERS
Order number
and normal
values of the
letters
A
B
c
D
1
x 1
1
V
1
in
value of
^X
ALEPH
2
a 2
2
2 2
1 + 2
412
ir
rra
BET
3
1 3
3
3 2
1 + 2 + 3
73
it
*?0J
GIMMEL
4
n 4
4
4 2
1+2+3+4
434
11
npn dalet
5
n 5
5
5 2
1+2+3+4+5
6
11
xn
HE
6
1 6
6
6 2
1+2+3+4+5
+ 6
12
11
IT
VOV
7
t 7
7
7 2
1+2+3+4+5
. . + 7
67
11
r»
ZAYIN
8
n 8
8
8 2
1+2+3+4+5
. . + 8
418
11
rrn
HET
9
D 9
9
9 2
1+2+3+4+5
. . + 9
419
11
fro
TET
10
. 10
1
10 2
1+2+3+4+5
. . + 10
20
11
-rr
YOD
11
2 20
2
20 2
1 +2+3+4+5
. . + 11
100
*P
KOF
12
G, 30
3
30 2
1 +2+3+4+5
. . + 12
74
11
10*7
LAMED
13
D 40
4
40 2
1+2+3+4+5
. . + 13
90
11
O’O
MEM
14
2 50
5
50 2
1+2+3+4+5
. . + 14
no
11
T
NUN
15
O
O
6
60 2
1+2+3+4+5
. . + 15
120
11
7[00
SAMEKH
16
s 70
7
70 2
1+2+3+4+5
. . + 16
130
11
r»
AY IN
17
a so
8
80 2
1+2+3+4+5
. . + 17
85
u
ns
PE
18
90
9
90 2
1+2+3+4+5
. . + 18
104
11
'is
TSADE
19
p
1
100 2
1 +2+3+4+5
. . + 19
104
11
TP
QUF
20
1 200
2
200 2
1+2+3+4+5
. . + 20
510
"
on
RESH
21
O 300
3
300 2
1+2+3+4+5
. . + 21
360
I’D
SHIN
22
n 400
4
400 2
1 + 2 + 3 + 4 + 5
. .+22
406
»
in
TAV
Fig. 20.29. Some of the many systems for the numerical evaluation of Hebraic letters. They are
used by rabbis and Cabbalists for the interpretation of their homilies.
MAGIC, MYSTERY, DIVINATION
25S
This, it is emphasised, is confirmed by Micah 1:3.
For, behold, the LORD [Yahweh] cometh forth out of his place
[Maqom],
This selection of examples - which could easily be much extended - gives a
good idea of the complexities of Cabbalistic calculations and investigations
which the exegetes went into, not only for the purpose of interpreting
certain passages of the Torah but for all kinds of speculations.*
The Greeks also used similar procedures. Certain Greek poets, such as
Leonidas of Alexandria (who lived at the time of the Emperor Nero), used
them to create distichs and epigrams with the special characteristic of
being isopsephs. A distich (consisting of two lines or two verses) is an
isopseph if the numerical value of the first (calculated from the sum of the
values of its letters) is equal to that of the second. An epigram (a short
poem which might, for example, express an amorous idea) is an isopseph if
all of its distichs are isopsephs, with the same value for each.
More generally, isopsephy consists of determining the numerical value
of a word or a group of letters, and relating it to another word by means of
this value.
At Pergamon, isopseph inscriptions have been found which, it is
believed, were composed by the father of the great physician and mathe-
matician Galen, who, according to his son, “had mastered all there was
to know about geometry and the science of numbers.”
At Pompeii an inscription was found which can be read as “1 love her
whose number is 545”, and where a certain Amerimnus praises the mistress
of his thoughts whose “honourable name is 45.”
In the Pseudo-Callisthenes + (I, 33) it is written that the Egyptian god
Sarapis (whose worship was initiated by Ptolemy I) revealed his name to
Alexander the Great in the following words:
Take two hundred and one, then a hundred and one, four times
twenty, and ten. Then place the first of these numbers in the last place,
and you will know which god I am.
Taking the words of the god literally, we obtain
200 1 100 1 80 10 200
* We claim no competence to make the slightest commentary on these matters, neither on the delicate ques-
tions of the historical origins of Gematria in the Hebrew texts, nor on its evolution, nor on the extent to
which it was regarded (or discredited) in Rabbinic and Cabbalistic writings throughout the centuries
and in various countries. The reader who is interested in these questions may consult F. DornseifF (1925)
or G. Scholem.
f A spurious work associated with the name of Callisthenes, companion of Alexander in his Asiatic
expedition.
which corresponds to the Greek name
2APAIII2
200 1 100 1 80 10 200
Fig. 20 . 30 .
In recalling the murder of Agrippina, Suetonius (Nero, 39) relates the
name of Nero, written in Greek, to the words Idian Metera apektcinc
(“he killed his own mother”), since the two have exactly the same value
according to the Greek number-system:
NEPHN IAIAN MHTEPA AI1EKTEINE
50 5 100 800 50 10 4 10 1 50 40 8 300 5 100 1 1 80 5 20 300 5 10 50 5
^
“neho” "he killed his own mother"
1005 1005
Fig. 20 . 31 .
The Greeks apparently came rather late to the practice of speculating
with the numerical values of letters. This seems to have occurred when
Greek culture came into contact with Jewish culture. The famous passage in
the Apocalypse of Saint John clearly shows how familiar the Jews were with
these mystic calculations, long before the time of their Cabbalists and the
Gematria. Both Jews and Greeks were remarkably gifted for arithmetical
calculation and also for transcendental speculation; every form of subtlety
was apt to their taste, and number-mysticism appealed to both predilec-
tions at the same time. The Pythagorean school, the most superstitious of
the Greek philosophical sects, and the most infiltrated by Eastern influence,
was already addicted to number-mysticism. In the last age of the ancient
world, this form of mysticism experienced an astonishing expansion.
It gave rise to arithmomancy; it inspired the Sybillines, the seers and
soothsayers, the pagan Theologor, it troubled the Fathers of the Church, who
were not always immune to its fascination. Isopsephy is one of its methods.
[P. Perdrizet (1904)]
Father Theophanus Kerameus, in his Homily (XLIV) asserts the numeri-
cal equivalence between Theos (“God”), Hagios (“holy”) and Agathos
(“good”) as follows:
257
GNOSTICS, CABBA LISTS, MAGICIANS, AND SOOTHSAYERS
0EOX ATIOX
9 5 70 200 1 3 10 70 200
“god”
284
»
“holy”
284
ATA0OX
1 3 1 9 70 200
“goo d"
284
Fig. 20 . 32 .
He likewise saw in the name Rebecca (wife of Isaac and mother of the
twins Jacob and Esau) a figure of the Universal Church. According to him,
the number (153) of great fish caught in the “miraculous draught of fishes”
is the same as the numerical value of the name Rebecca in Greek ( Homily
XXXVI; John XXI).
PEBEKKA
100 5 2 5 20 20 1
153
Fig. 20 . 33 .
In another conception much exercised in the Middle Ages, numbers
were given a supernatural quality according to the graphical shape of their
symbols.
In a manuscript which is in the Bibliotheque nationale in Paris (Ms. lat.
2583, folio 30), Thibaut of Langres wrote as follows, about the number 300
represented by the Greek letter T ( tau ), which is also the sign of the Cross:
The number is a secret guarded by writing, which represents it in two
ways: by the letter and by its pronunciation. By the letter, it is
represented in three ways: shape, order, and secret. By shape, it is like
the 300 who, from the Creation of the World, were to find faith in
the image of the Crucifix since, to the Greeks, these are represented
by the letter T which has the form of a cross.
Which is why, according to Thibaut, Gideon conquered Oreb, Zeeb, Zebah,
and Zalmunna with only the three hundred men who had drunk water “as
a dog lappeth” (Judges VII:5).
A similar Christian interpretation is to be seen in the Epistle of Barnabas.
In the patriarch Abraham’s victory over his enemies with the help of 318
circumcised men, Barnabas finds a reference to the cross and to the two
first letters of the name of Jesus (It) crons)
In the New Testament, the phrase Alpha and Omega (Apocalypse
XXII: 13) is a symbolic designation of God: formed from the first and
last letters of the Greek alphabet, in the Gnostic and Christian theologies
it corresponded to the “Key of the Universe and of Knowledge” and to
“Existence and the Totality of Space and Time”. When Jesus declares that
he is the Alpha and the Omega, he therefore declares that he is the beginning
and the end of all things. He identifies himself with the “Holy Ghost” and
therefore, according to Christian doctrine, with God Himself. According
to Matthew 111:16, the Holy Ghost appeared to Jesus at the moment of
his birth in the form of a dove; the Greek word Peristera for “dove” has
the value 801; and this is also the value of the letters of the phrase “Alpha
and Omega” which, therefore, is no other than a mystical affirmation of
the Christian doctrine of the Trinity.
A and O IIEPIXTEPA
1 800 80 5 100 10 200 300 5 100 1
» *
801 801
Fig. 20 . 34 .
T + IH = 318
300 10 + 8
Fig. 20 . 35 .
He considers that the number 318 means that these men would be saved
by the crucifixion of Jesus.
In the same fashion, according to Cyprian (De pascha computus, 20),
the number 365 is sacred because it is the sum of 300 (T, the symbol of
the cross), 18 (IH, the two first letters of the name of Jesus), 31 (the number
of years Christ is supposed to have lived, in Cyprian’s opinion) and 16
(the number of years in the reign of Tiberius, within which Jesus was
crucified). This may well also explain why certain heretics believed that
the End of the World would occur in the year 365 of the Christian era.*
* "But because this sentence is in the Gospel, it is no wonder that the worshippers of the many and
false gods . . . invented I know not what Greek verses, . . . but add that Peter by enchantments brought it
about that the name of Christ should be worshipped for three hundred and sixty-five years, and, after the
completion of that number of years, should at once take end. Oh the hearts of learned men!” lAugustine,
The City of God, Book 18, Chapter 53 )
magic:, mystery, divination
258
TRANSCRIPTION
i v iiiiii v i
II III 1 1 1 1 1 1 1 1 III II
I I II II VI II II I I
II I II III III II II I II
I I II III V II III I
I II I II I V III II I
ii I IIII i III v II
I II x v
in fact it was preoccupied with the quest to know the name of God
and thence, with the aid of magic (the ancient magic of Isis), the
means to induce God to allow Man to raise himself to God’s own
level. The name, like the shadow or the breath, is a part of the
person: more, it is identical with the person, it is the person himself.
To know the name of God, therefore, was the problem which
Gnosticism addressed. At first it seems insoluble: how can we know
the Ineffable? The Gnostics did not pretend to know the name of
God, but they believed it possible to learn its formula; and for them
this was sufficient, since for them the formula of the divine name
contained its complete magical virtue: and this formula was the
number of the name of God.
Fig. 20.36. Wooden tablet found in North Africa, dating from the late fifth century CE. Note that
on each line the Roman numerals total 18 (the overline denotes a part-total). It is not known whether
this is a mathematical (indeed a teaching) document or a “ magic’’ tablet relating to speculations on
the numerical value of Greek or Hebraic letters. (See TA, act XXXIV, tabl. 3 a)
Clearly, all possible resources have been exploited for these purposes.
The Christian mystics, who wished to support the affirmation that Jesus
was the Son of God, often equated the Hebrew phrase Ab Qal which Isaiah
used to mean “the swiff cloud” on which “the Lord rideth” (XIX: 1) and the
word Bar (“son”):
bp 2 5) “13
30 100 2 70 200 2
<■ <■
202 202
Fig. 20.37.
For their part, the Gnostics* were able to draw almost miraculous
consequences from the practice of isopsephy. P. Perdrizet (1904) explains:
A text, which is probably by Hippolytus, says that in certain Gnostic
sects isopsephy was a normal form of symbolism and catechesis. It
did not serve only to wrap a revelation in a mystery: if in certain
cases it served to conceal, in others it served to reveal, throwing light
on things which otherwise would never have been understood . . .
Gnosticism seems loaded with a huge burden of Egyptian supersti-
tions. It purported to rise to knowledge of the Universal Principle;
* Gnosticism (from the Greek gnosis, “knowledge”) is a religious doctrine which appeared in the early
centuries of our era in Judaeo-Christian circles, but was violently opposed by rabbis and by the New
Testament apostles. It is based essentially on the hope that salvation may be attained through an esoteric
knowledge of the divine, as transmitted through initiation.
jnvw ,
TO \'^'w"' X
((\wukm nw' 0 "” 1 "
A\ tR w n w to wit m \ , 1
\\w\ u n u n u nuiTT""'
^unn \m n wnnu nin
~~
Fig. 20.38. One of many slates found in the region of Salamanca. This one was discovered at
Santibanez de la Sierra and dates from about the sixth century. It is a document similar to the
previous one; each line that has remained intact shows a total count of 26. (See G. Gomez-Moreno,
pp. 24, 117)
259
GNOSTICS, CABBA LISTS, MAGICIANS, AND SOOTHSAYERS
The supreme God of the Gnostics united in himself, according to
Basilides the Gnostic, the 365 minor gods who preside over the days
of the year . . . and so the Gnostics referred to God as “He whose
number is 365” (ovecmv t| 4rnaos THE). From God, on the other
hand, proceeded the magical power of the seven vowels, the seven
notes of the musical scale, the seven planets, the seven metals (gold,
silver, tin, copper, iron, lead, and mercury); and of the four weeks of
the lunar month. Whatever was the name of the Ineffable, the
Gnostic was sure it involved the magic numbers 7 and 365. We
may not know the unknowable name of God, so instead we seek a
designation which would serve as its formula, and we only have to
combine the mystic numbers 7 and 365. Thus Basilides created the
name Abrasax, which has seven letters whose values add up to 365:
ABPAHH
1 2 100 I 200 1 60
*
365
Fig. 20.39.
God, or the name of God (for they are the same) has first the
character of holiness. Ayios o 0eos ( Hagios 0 Theos) says the seraphic
hymn; “hallowed be thy name” says the Lord’s Prayer, that is “let the
holiness of God be proclaimed.”
Though the name of God remained unknown, it was known that
it had the character to be the ideal holy name. Nothing therefore
better became the designation of the Ineffable than the locution
Hagion Onoma (“Holy Name”) which the Gnostics indeed frequently
employed. But this was not only for the above metaphysical or
theological reason, nor because they had borrowed this same
appellation from the Jews, but for a more potent mystical reason
peculiar to them. By a coincidence of which Gnosticism had seen a
revelation, the biblical phrase Hagion Onoma had the same number
(365) as Abrasax.
A T I O N ONOMA
1 3 10 70 50 70 50 70 40 1
>
365
Once embarked on this path, Gnosticism made other discoveries no
less gripping.
Mingled as it was with magic, Gnosticism had a fatal tendency to
syncretism. In isopsephy it had the means to identify with its own
supreme God the national god of Egypt. The Nile, which for the
Egyptians was the same as Osiris, was a god of the year, for the regu-
larity of its floods followed the regular course of the years; and now,
the number of the name of the Nile, Neilos, is 365:
NEIAOX
50 5 10 30 70 200
>
365
Fig. 20.41.
By isopsephy, Gnosticism achieved another no less interesting
syncretism. The Mazdean cult of Mithras underwent a prodigious
spread in the second and third centuries of our era. The Gnostics
noticed that Mithras, written MEI0PAS, has the value
MEI0P AS
40 5 10 9 100 1 200
^
365
Fig. 20.42.
Therefore the Sun God of Persia was the same as the “Lord of the
365 Days”.
As Perdrizet says, the Christians often put new wine in old bottles, and they
found that this kind of practice offered ample scope for fantasy. When the
scribes and stone-carvers wished to preserve the secret of a name, they
wrote only its number instead.
In Greek and Coptic Christian inscriptions, following an imprecation or
an exhortation to praise, we sometimes come across the sign \ 0 made up
of the letters Koppa and Theta. This cryptogram remained obscure until the
end of the nineteenth century, when J. E. Wessely (1887) showed that it
was simply a mystical representation of Amen (’Apuqv), since both have
numerical value 99:
AMHN ^0
1 40 8 50 90 9
> >
99 99
Fig. 20.40.
Fig. 20.43.
MAGIC, MYSTERY, DIVINATION
Similarly, the dedication of a mosaic in the convent of Khoziba near
Jericho begins:
O A E MNH2MTI TOY AOYAOYSOY
<t>AE REMEMBER YOUR SERVANT
Fig. 20.44.
What does the group Phi-Lambda-Epsilon stand for? The problem was
solved by W. D. Smirnoff (1902). These letters correspond to the Greek
word for “Lord”, Kupie, whose numerical value is 535:
O A E K Y P I E
500 30 5 20 400 100 10 5
> >
535 535
Fig. 20.45.
Much more significant are the speculations of the Christian mystics
surrounding the number 666, which the apostle John ascribed to the Beast
of the Apocalypse, a monster identified as the Antichrist, who shortly before
the end of time would come on Earth to commit innumerable crimes, to
spread terror amongst men, and raise people up against each other. He
would be brought down by Christ himself on his return to Earth.
16 And he shall make all, both little and great, rich and poor,
freemen and bondsmen, to have a character in their right hand, or
on their foreheads.
17 And that no man might buy or sell, but he that hath the
character, or the name of the beast, or the number of his name.
18 Here is wisdom. He that hath understanding, let him count
the number of the beast. For it is the number of a man: and the
number of him is six hundred and sixty-six. [Apocalypse,
XIII: 16-18]
We clearly see an allusion to isopsephy here, but the system to be used is
not stated. This is why the name of the Beast has excited, and continues to
excite, the wits of interpreters, and many are the solutions which have been
put forward.
Taking 666 to be “the number of a man”, some have searched amongst
2 iso
the names of historical figures whose names give the number 666. Thus
Nero, the first Roman emperor to persecute the Christians, has been iden-
tified as the Beast of the Apocalypse since the number of his name,
accompanied by the title “Caesar”, makes 666 in the Hebraic system:
p “I ] “10 P
50 6 200 50 200 60 100
«
QSAR NERO
666
Fig. 20.46.
On the same lines, others have found that the name of the Emperor
Diocletian (whose religious policies included the violent persecution of
Christians), when only the letters that are Roman numerals are used, also
gives the number of the Beast:
(Diocletian Augustus)
DIoCLEs aVgVstVs
666
Fig. 20.47.
Yet others, reading the text as “the number of a type of man", saw in 666
the designation of the Latins in general since the Greek word Lateinos gives
this value:
A A T E I N O S
30 1 300 5 10 50 70 200
>
666
Fig. 20.48.
Much later, at the time of the Wars of Religion, a Catholic mystic called
Petrus Bungus, in a work published in 1584-1585 at Bergamo, claimed to
have demonstrated that the German reformer Luther was none other than
the Antichrist since his name, in Roman numerals, gives the number 666:
261
LVTHERNVC
30 200 100 8 5 80 40 200 3
>
But the disciples of Luther, who considered the Church of Rome as the
direct heir of the Empire of the Caesars, lost no time in responding. They
took the Roman numerals contained in the phrase VICARIUS FILII DEI
(“Vicar of the Son of God”) which is on the papal tiara, and drew the
conclusion that one might expect:
VICarIVs fILII DeI
5 1 100 1 5 1 50 1 1 500 1
^
666
Fig. 20.50.
The numerical evaluation of names was also used in times of war by
Muslim soothsayers, under the name of khisab al nim, to predict which side
would win. This process was described as follows by Ibn Khaldun in his
“Prolegomena” ( Muqaddimah , I):
Here is how it is done. The values of the letters in the name of each
king are added up, according to the values of the letters of the alpha-
bet; these go from one to 1,000 by units, tens, hundreds and
thousands. When this is done, the number nine is subtracted from
each as many times as required until what is left is less than nine. The
two remainders are compared: if one is greater than the other, and if
both are even numbers or both odd, the king whose name has the
smaller number will win. If one is even and the other odd, the king
with the larger number will win. If both are equal and both are even
numbers, it is the king who has been attacked who will win; if they
are equal and odd, the attacking king will win.
Since each Arabic letter is the first letter of one of the attributes of Allah
(Alif the first letter of Allah] Ba, first letter of Baqi, “He who remains”, and
so on), the use of the Arabic alphabet led to a “Most Secret” system. In this,
each letter is assigned, not its usual value, but instead the number of the
divine attribute of which it is the first letter. For instance, the letter Alif,
whose usual value is 1, is given the value 66 which is the number of the
name of Allah calculated according to the Abjad system. This is the system
used in the symbolic theology called da’wa, “invocation”, which allowed
mystics and soothsayers to make forecasts and to speculate on the past, the
present and the future.
GNOSTICS, CABBALISTS, MAGICIANS, AND SOOTHSAYERS
MAGIC, MYSTERY, DIVINATION
The same type of procedure allowed magicians to contrive their talis-
mans, and to indulge in the most varied practices. In order to give their
co-religionists the means to get rich quickly, to preserve themselves from
evil and to draw down on themselves every grace of God, some tolba of
North Africa offered their clients a kherz (“talisman”) containing:
Fig. 20 .52 a .
This is a “magic square” whose value is 66, which can be obtained as the
sum of every row, of every column, and of each diagonal:
Fig. 20.52B.
262
and is itself the number of the name of Allah according to the Abjad :
4UI
5 30 30 I
<7
ALLAH
Fig. 20.53.
We can see, therefore, to what lengths the soothsayers, seers and other
numerologists were prepared to go in applying these principles of number
to the enrichment of their dialectic.
263
CHAPTER 21
NUMBERS IN CHINESE
CIVILISATION
THE THIRTEEN FIGURES OF THE TRADITIONAL
CHINESE NUMBER-SYSTEM*
The Chinese have traditionally used a decimal number-system, with
thirteen basic signs denoting the numbers 1 to 9 and the first four
powers of 10 (10, 100, 1,000, and 10,000). Fig. 21.1 shows the simplest
representations of these, which is the one most commonly used nowadays.
1 10 -|-
2
3 ^ ioo n
4 |3
5 £ 1,000 4*
6 ft
7 A& 10,000 ft
8 A
9 X
Fig. 2i.i.
To an even greater extent than in the ancient Semitic world, this written
number-system corresponds to the true type of “hybrid” number-system,
since the tens, the thousands, and the tens of thousands are expressed
according to the multiplicative principle (Fig. 21.2).
* I wish to express here my deep gratitude to my friends Alain Briot, Louis Frederic and Leon
Vandermeersch for their valuable contributions, and for their willing labour in reading this entire chapter.
THE THIRTEEN FIGURES
Fig. 21.2. The modern Chinese notation for consecutive multiples of the first four powers of 10.
For intermediate numbers, the Chinese used a combination of addition
and multiplication, so that the number 79,564, for example, is decomposed
as:
-fc Eg *+13
Fig. 21.3.
7 x 10,000 + 9 X 1,000 + 5 X 100 + 6 x 10 + 4
79,564
NUMBERS IN CHINESE CIVILISATION
Fig. 21.4. Examples of numbers written with Chinese numerals
264
4^
n ^ *
v'X & —
i * * i
I**-* x
a
rx +
- + JL
1 *
*cA ,4
. 4fc -=• *
fetr
*■ rt
? £
* i
*• *1
A.
m-f»y
*-€)•&-
f >■/. ej
<_ — X
4 *r ^
« /N.
* .% -
** +
.% v*< -=-
H{ — f%~
t f\ A
- 4^
*r * <-
■***,«*
^ JX
A .%
*-i
$
H, yfc
* 4 $
«l 4
i*» #ti
ML $L&
^ 4 - J *'
4*?3 ^
A 7 *r ^ 4
7 T
IT 7 ♦ 9 4 4
ft & to % it
~ * it jt *
f[ A «
X "fr A
*
T X «
C9 ^ l$~
+ At*
*■
* * 4 ?
f\
- **
T 4- A*
A /Y
T »'a -f*C
I- ffi fl‘]
Fig . 21 . 5 . A / 7 ^c from a Chinese mathematical document dating from the beginning of the
fifteenth century. Cambridge University Library [Ms. Yong-le da dian, chapter 16 343, introductory
page. From Needham (1959), III, Fig. 54].
W-VSM\»t4/V<fe4 A
265
THE CHINESE ORAL NUMERAL SYSTEM
TRANSCRIPTION OF CHINESE CHARACTERS
To transcribe Chinese characters into the Latin alphabet, we shall adopt
the so-called Pinyin system in what follows. This has been the official
system of the People’s Republic of China since 1958. “This transcription”,
according to D. Lombard (1967), “was developed by Chinese linguists for
use by the Chinese people and especially to assist schoolchildren to learn
the language and its characters, and it is based mainly on phonological
principles. The majority of Western Chinese scholars nowadays tend to
abandon the older transcription systems (which sought in vain to represent
pronunciation in terms of the spelling conventions of various European
languages) in favour of this one. The reader is therefore no longer obliged
to remember any spelling conventions, but instead must try to remember
certain equivalences between sound and letter (as in beginning the study of
German or Italian).”
Since the Pinyin system was not conceived with European readers in
mind, it is natural that the values of its letters do not always coincide with
English pronunciation. Here is a list of the most important aspects from the
point of view of the English reader.
b corresponds to our letter “p”
c corresponds to our “ts”
d corresponds to our “t”
g corresponds to our “k”
u corresponds to the standard English pronunciation of “u” as
in “bull” (except after j, q or x)
I corresponds to the pronunciation of “u” as, for instance, in
Scotland or in French
z corresponds to our “dz”
zh corresponds to “j” as in “join”
ch corresponds to “ch” as in “church”
h in initial position, corresponds to the hard German “ch” (as
in “Bach”)
x in initial position, corresponds to the soft “ch” (as in German
“Ich”)
i corresponds to our “i” (as in “pin”); but, following z, c, s, sh,
sh or r it is pronounced like “e” (in “pen”) or like “u” in "fur”;
following a or u, it is pronounced like the “ei” in “reign”,
q stands for a complex sound consisting of “ts” with drawing-in
of breath
r in initial position is like the “s” in pleasure; in other cases it is
like the “el” in “channel”.
THE CHINESE ORAL NUMERAL SYSTEM
The number-signs shown above are in fact ordinary characters of Chinese
writing. They are therefore subject to the same rules as govern the other
Chinese characters. These are, in fact, “word-signs” which express in
graphical form the ideographic and phonetic values of the corresponding
numbers. In other words, they constitute one of the graphical representa-
tions of the thirteen monosyllabic words which the Chinese language
possesses to denote the numbers from 1 to 9 and the first four powers
of 10.
Having a decimal base, the oral Chinese number-system gives a separate
name to each of the first ten integers:
yi er san si wu liii qi ba jiu shi
123456789 10
The numbers from 11 to 19 are represented according to the additive principle:
11
shiyi
ten-one
= 10 + 1
12
shier
ten-two
= 10 + 2
13
shi san
ten-three
= 10 + 3
14
shi si
ten-four
= 10 + 4
The tens are represented according to the multiplicative principle:
20
er shi
two-ten
= 2x10
30
san shi
three-ten
= 3x10
40
si shi
four-ten
= 4x10
50
wu shi
five-ten
= 5x10
60
liu shi
six-ten
= 6x10
For 100 (= 10 2 ), 1,000 (= 10 3 ) and 10,000 (= 10 4 ), the words bai, qian and
wan are used; for the various
multiples of these the
multiplicative principle
is used:
100
yi bai
one-hundred
200
erbdi
two-hundred
= 2 X 100
300
san bai
three-hundred
= 3 x 100
400
si bai
four-hundred
= 4 x 100
1,000
yi qian
one-thousand
2,000
er qian
two-thousand
= 2 x 1,000
3,000
san qian
three-thousand
= 3 x 1,000
4,000
si qian
four-thousand
= 4 x 1,000
10,000
yi wan
one-myriad
20,000
erwan
two-myriad
= 2 x 10,000
30,000
san wan
three-myriad
= 3 x 10,000
40,000
si wan
four-myriad
= 4 X 10,000
NUMBERS IN CHINESE CIVILISATION
2 6 6
Starting with these, intermediate numbers can be represented very
straightforwardly:
53,781 wu wan san qian qibai basht yi
( five-myriad three-thousand seven-hundred eight-ten one )
(=5X10,000 + 3x1,000 + 7x100 + 8x10 + 1 )
Thus the Chinese number-signs are a very simple way of writing out the
corresponding numbers “word for word”.
Finally, note that such a system has no need of a zero. For the numbers
504, 1,058, or 2,003, for example, one simply writes (or says):
5.
wu
bai
a
si
(= 5 X 100 + 4)
y~ i
*
qian
wu
+
shi
A
ba
(= 1 x 1,000 + 5 x 10 + 8)
er
*
qian
san
(=2x1,000 + 3)
Fig. 21.6.
Note, however, that in current usage the word ^ , ling (which means
“zero”), is mentioned whenever any power of 10 is not represented in the
expression of the number. This is done in order to avoid any ambiguity. But
this usage was only established late in the development of the Chinese
number-system.
504 JBL W % H
5 100 0 4
wu bai ling si
(“five hundred zero four")
1,058 — T %
1 1,000 0
i + A
5 10 8
yi qian ling
(“one thousand zero
wu shi ba
five ten eight")
2,003 H ^
2 1,000 0
er qian ling san
(“two thousand zero three")
Fig. 21 . 7 .
CHINESE NUMERALS ARE DRAWN
IN MANY WAYS
Even today, the thirteen basic number-signs are drawn in several different
ways. Obviously they are spoken in the same way, but are a result of the
many different ways of writing Chinese itself.
The forms we have considered so far, which may be called “classical”, is
the one in common use nowadays, especially in printed matter. It is also the
simplest. Some of these signs are among the “keys” of Chinese writing: they
are used in the elementary teaching of Chinese, at the stage of learning the
Chinese characters.
They are part of the now standard kaishu notation, a plain style in which
the line segments making up each character are basically straight, but of
varying lengths and orientations; they are to be drawn in a strict order,
according to definite rules (Fig. 21.8).
’ - |IU+;
> * v -» u 7 C
'’inm
t
- +
lU
? 1 1 d*
u
' / &
+
' «s.
^ y* t /f # t
i§
^ ^ r ft ft s?
Fig. 21.8. The basic strokes of Chinese writing in the standard style called kaishu , and the order in
which they are to be written in composing certain characters
It is also the oldest of the common contemporary forms, having been
used as early as the fourth century CE, and it is derived from the ancient
writing called lishu* (“the writing of clerks”) which was used in the Han
Dynasty (Fig. 21.9).
* The lishu style of notation is the earliest of the modem forms: it is the first “line writing” in Chinese
history. However, “in seeking the maximum enhancement of the precision of the lishu an even more
geometrical style resulted, the inflexibly regular kaishu . ” [V. Alleton (1970)]. This regular style became
fixed as the standard for Chinese writing in the earliest centuries of the current era: administrative
documents, official and scientific writings, were usually written in this style from that time on, when most
such works were printed and the fonts for the characters had been made. When, below, we refer to “Chinese
writing” without further qualification, it is this style which is meant.
267
CHINESE NUMERALS ARE DRAWN 1 jn main i
W AIi>
—
**
tiff
or
tiff
ir
-t-
/\
ft
+
B
or
ST
?
1
2
3
4
5
6
7
8
9
10
100
1,000
10,000
Examples reconstructed from administrative documents on strips of wood or bamboo dating from the
first century CE and discovered in Central Asia.
fc 6
• | m \ 10
•fflH «
-H ’
& 3
j 1,000
n 8
W j loo
ft j »
3
*
| 100
Jp : io
1ST i 4
+* 7
63
47
3,804
397
Fig. 21 . 9 . The earliest of the modem Chinese numerical notations. This is of the lishutjye and was
in use during the Han Dynasty (206 BCE to 220 CE). The documents used for this diagram were
written by scribes of the first century CE. [See de Chavannes (1913); Maspero; Guitel]
The second form of the Chinese numerals is called guan zi (“official
writing”). It is used mainly in public documents, in bills of sale, and to write
the sums of money on cheques, receipts or bills. Although still written like
the classic kaishii, it is somewhat more complicated, having been made
more elaborate in order to avoid fraudulent amendments in financial trans-
actions (Fig. 21.10).
Classical notation
- g -s. * ir a-ms
Guan zl notation
^
yi wan san qian liii bai ba shi si
1 x 10,000 + 3 x 1,000 + 6 x 100 + 8 x 10 +4
I k;. 21.10.
The third style of writing the numerals is a cursive form of the classical
numerals, which is routinely used in handwritten letters, personal notes,
drafts, and so on. It belongs to the xingshu style of writing, a cursive style
which was developed to meet the need for abbreviation without detracting
from the structure of the characters; the changes lay in the manner of
drawing the characters more rapidly and flexibly using upward and down-
ward brushstrokes. (Fig. 21.11).
Classical notation
0 at a + - & *
Xingshu notation
IP
si wan jiu qian er bai liu shi wu
4 X 10,000 + 9 x 1,000 + 2 X 100 + 6 X 10 + 5
Fig. 2i.ii.
A combination of exaggerated abbreviation with virtuosity and imagina-
tion on the part of calligraphers rapidly brought these cursive forms, which
still resembled the classical style, into an exaggeratedly simplified style
which the Chinese call caoshu (literally, “plant-shaped”). It can only be deci-
phered by initiates, with the result that nowadays it is used only in painting
and in calligraphy* (Fig. 21.12 and 21.13).
Fig. 21 . 12 . Example of 75,696
* “Chinese writing underwent two transformations in the caoshu :
a. Lines and elements of characters were suppressed; save for characters with a small number of strokes,
almost all elements are represented by symbols, leading to a kind of “writing of writing”.
b. The strokes lose their individuality and join up: eventually a character is written in one movement; then
the characters themselves join up, and even a whole column may be written without lifting brush from
paper.” [V. G. Alleton (1970)]
-fc 7
x
K 10,000
3L 5
x
1,000
it 6
x
H 100
K ®
X
-J* 10
+
NUMBERS IN CHINESE CIVILISATION
268
lishu
kaishu
xingshu
caoshu
lr
?£ &
&
It
printed manuscript
character character
Fig. 21.13. The difference between the principal styles of modern Chinese writing, as shown in
writing the word shufa (“calligraphy”) in the styles aflishu (“official writing’’, used in the Han
period), kaishu (“standard style”, which replaced the lishu and has been used since the fourth
century CE), xingshu (the current cursive style) and caoshu (a cursive style which has been
reduced to maximum abbreviation and is now used only in calligraphy). [Alleton (1970)]
Yet another form corresponds to a curiously geometrical way of drawing
the numerals and characters, called shdngfang da zhuan, which is still
employed on seals and signatures (Fig. 21. 14).
Fig. 21.14. Example of the singular shang fang da zhuan calligraphy as used for the thirteen
basic characters of the Chinese number-system on seals and in signatures. [See Perny (1873);
Pihan (I860)]
As well as the forms already mentioned, there is the form used by traders
to display the prices of goods. This is called gdn ma zi (“secret marks”).
Anyone who travels to the interior of China should be sure of knowing
these numerals by heart, if he wishes to understand his restaurant bill
(Fig. 21.15).
There are so many different styles for writing numerals in China that we
should stop at this point, having described the important ones; to describe
them all would be self-indulgent, and little to our purpose.
gudn zi
gdn ma zi
00
0
1st form
2nd form
3rd and 4th forms
5th form
H
Cu
Pi
OO
Elaborate
Cursive forms
u
ttJ
D
Classical
augmented
Cursive forms of the
currently used in
<JT>
£
forms
forms used
classical
signs
business and
<
>
in finance
calculation
Oh
f— '
1
£ or ^
v-V
l
y-‘
2
£
•s* ° r £>
t0>
n
er
3
S or £
^ &
VI
san
4
a
&
I*
iQ
;*
si
5
£
GL
b
b
t <» v
wu
6
tc
m
H
%
*
Hu
7
-b
%
4
4
V
8
A
*
A
»
afc
bd
9
%
hs
Jy-
jiu
10
+
& » dr
h° r i
*
&
shi
100
n
is
V or 3
bai
1,000
*
if
4*
=f
qian
10,000
m
ft
h
wan
Standard
kaishu
Xingshu
Caoshu
style
style
style
Fig. 21.15. The principal graphic styles for the thirteen basic signs of the modern Chinese
number-system. [ Giles (1912); Mathews (1931); Needham (1959); Perny (1873); Pihan (I860)]
269
THE ORIGINS OF THE CHINESE NUMBER-SYSTEM
THE ORIGINS OF THE CHINESE N U M B E R- S Y S T E M
Several thousand bones and tortoise shells: these are the most ancient
evidence we have of Chinese writing and numerals. They have for the most
part been found since the end of the nineteenth century at the archaeological
site of Xiao dun;* called jiaguwen (“oracular bones”), they date from around
the Yin period (fourteenth-eleventh centuries BCE). On one side they bear
inscriptions graven with a pointed instrument, on the other the surface is a
maze of cracks due to heat. They would once have belonged to soothsayer-
priests attached to the court of the Shang kings (seventeenth-eleventh
centuries BCE) and would have been used in divination by fire.*
The writing on them is probably pictographic in origin, and seems to
have reached a well-developed stage since it is no longer purely picto-
graphic nor purely ideographic. The basis of the ancient Chinese writing in
fact consists of a few hundred basic symbols which represent ideas or
simple objects, and also of a certain number of more complicated symbols
composed of two elements, of which one relates to the spoken form of a
name and the other is visual or symbolic.* It represents a rather advanced
stage of graphical representation (Fig. 21.16). “The stylisation and the
economy of means are so far advanced in the oldest known Chinese
writings that the symbols are more letters than drawings” [J. Gernet (1970),
p. 31].
H
©
?
Ji>
*
Jl
ft
T
Divination
Sun
Day
Man
Moon
Month
Heaven
Divinity
To go up
Elk
To go down
Fig. 21.1 6 . Some archaic Chinese characters
* Village in the northwest of the An yang district in the province of Henan
f According to H. Maspero, this ritual took place as follows. Ancestor worship was of great importance in
Chinese religion, and the priests consulted the royal ancestors on a great diversity of subjects. They first
inscribed their questions on the ventral side of a tortoise shell which had been previously blessed (or on one
side of the split shoulder-blade of a stag, of an ox or of a sheep). They then brought the other side towards
the fire and the result of the divination was supposed to be decipherable from the patterns of cracks
produced by the fire.
* “The peculiarities of the Chinese language may possibly explain the creation and persistence of this very
complicated writing system. In ancient times, the language seems to have consisted of monosyllables of
great phonemic variety, which did not allow the sounds of the language to be analysed into constituents, so
Chinese writing could not evolve towards a syllabic notation, still less towards an alphabetic one. Each
written sign could correspond to a single monosyllable and a single linguistic unit.” (J. Gernet).
Gemet continues: “Moreover, this writing abounds in its very constitu-
tion with abstract elements (symbols reflected or rotated, strokes that
mark this or that part of a symbol, representations of gestures, etc.) and
with compounds of simpler signs with which new symbols are created.”
The numerals, in particular, seem to have already embarked on the
road towards abstract notation and appear to reflect a relatively advanced
intellectual perspective.
In this system, unity is represented by a horizontal line, and 10 by a
vertical line. Their origin is clear enough, since they reflect the operation of
the human mind in given conditions: we know, for instance, that the people
of the ancient Greek city of Karystos, and the Cretans, the Hittites and
the Phoenicians, all used the same kind of signs for these two numerals. A
hundred is denoted by what Joseph Needham called a “pine cone”, and a
thousand by a special character which closely resembles the character for
“man” in the corresponding writing.
The figures 2, 3 and 4 are represented each by a corresponding number
of horizontal strokes: an old ideographic system which is not used for the
figures from 5 onwards. Like all the peoples who have used a similar numer-
ical notation, the Chinese also stopped at 4; in fact few people can at a
glance (and therefore without consciously counting) recognise a series of
more than four things in a row. The Egyptians continued the series from 4
by using parallel rows, and the Babylonians and Phoenicians had a ternary
system, but the Chinese introduced five distinct symbols for each of the five
successive numbers: symbols, apparently, devoid of any intuitive sugges-
tion. The number 5 was represented by a kind of X closed above and below
by strokes; the number 6, by a kind of inverted V or by a design resembling
a pagoda; 7, by a cross; 8, by two small circular arcs back to back; and the
number 9 by a sign like a fish-hook (Fig. 21.17).
Fig. 21.17. The basic signs of archaic Chinese numerals. They have been found on divinatory
bones and shells from the Yin period (fourteenth to eleventh centuries BCE), and also on bronzes
from the Zhou period (tenth to sixth centuries BCE). [Chalfant (1906); Needham (1959); Rong
Gen (1959); Wieger (1963)]
NUMBERS IN CHINESE CIVILISATION
Now, did these number-signs evolve graphically from forms which
originally consisted of groupings of corresponding numbers of identical
elements? Or are they original creations? The history of Chinese writing
leads us to form two hypotheses about these questions, both of them plau-
sible, and not incompatible with each other.
We may in fact suppose that, for some of these numbers, their signs
were, more or less, “phonetic symbols” which were used for the sake of the
sounds they stood for, independently of their original meaning just as,
indeed, was the case for Chinese writing. Such, for example, may well be
why the number 1,000 has the same representation as “man”, since the two
words were probably pronounced in the same way at the time in question.
Another possible explanation may be of religious or magical origin, and
may have determined the choice of the other symbols. Gemet ( EPP ) writes:
“From the period of the inscriptions on bones and tortoiseshells at the end
of the Shang Dynasty until the seventh century BCE, writing remained
the preserve of colleges of scribes, adepts in the arts of divination and, by
the same token, adepts also in certain techniques which depended on
number, who served the princes in their religious ceremonies. Writing was
therefore primarily a means of communication with the world of gods
and spirits, and endowed its practitioners with the formidable power,
and the respect mingled with dread, which they enjoyed. In a society so
enthralled to ritual in behaviour and in thought, its mystical power must
have preserved writing from profane use for a very long period.”
Therefore it is by no means impossible that certain of the Chinese
number-signs may have had essentially magical or religious roots, and were
directly related to an ancient Chinese number-mysticism. Each number-
sign, according to its graphical form, would have represented the “reality”
of the corresponding number-form.
Whatever the case may be, the system of numerals which may be seen in
the divinatory inscriptions on the bones and tortoise shells from the middle
of the second millennium BCE is, intellectually speaking, already well on
the way to the modern Chinese number-notation.
Fig. 21.18A. Copy of a divinatory
inscription on the ventral surface
of a tortoise shell discovered at Xiao
dun, which dates from the Yin
period (fourteenth to eleventh
centuries BCE). [Diringer (1968)
plate 6-4: Yi 2908, translated and
interpreted by L. Vandermeersch]
270
Leaving aside the numbers 20, 30 and 40 (to which we shall shortly
return), the tens, hundreds and thousands are in fact represented according
to the multiplicative principle by combining the signs corresponding to
the units associated with them: in other words, the numbers from 50 to 90,
for instance, are represented by superpositions according to the principle:
10 10 10 10 10
x x x x x
5 6 7 8 9
271
Fic. 21.19. Principle of archaic Chinese numbering
This representation should not be confused with the one used for the
numbers 15 to 19, which was:
5 6 7 8 9
+ + + + +
10 10 10 10 10
The numbers from 100 to 900 were written by placing the symbols for
the successive units above the symbol for 100, and the thousands were
written in a similar way to the tens (Fig. 21.19). Intermediate numbers were
usually written by combining the additive and multiplicative methods.
We therefore see that, since the time of the very earliest known examples,
the Chinese system was founded on a “hybrid” principle. That the numbers
20, 30 and 40 were often written as requisite repetitions of the symbol for
10 is quite simply due to the fact that the use of the multiplicative method
would not have made the result any simpler. This kind of ideographic
notation, natural though it was, was nevertheless limited, for psychological
reasons, to a maximum of four identical elements.
THE ORIGINS OF THE CHINESE NUMBER-SVSTEM
The structure of the Chinese numerals stayed basically the same
throughout its long history, even though the arrangement of the signs
changed somewhat and their graphical forms underwent some variations
(see Fig. 21.17, 21.21, then 21.9 and finally 21.15).
Fig. 21.20. The durability of the ideographic forms of the first four numbers, as seen throughout
the history of Chinese numerals
Fig. 21.21. Variations in the graphical forms of the Chinese numerals, as found on inscriptions
from the end of the period of the warring kingdoms (fifth to third centuries BCE). [Perny ( 1873 );
Pihan (I 860 ))
NUMBERS IN CHINESE CIVILISATION
272
THE SPREAD OF WRITING THROUGHOUT
THE FAR EAST
Over all the centuries, the structure of the Chinese characters has not
fundamentally changed at all. The Chinese language is split into many
regional dialects, and the characters are pronounced differently by the
people of Manchuria, of Hunan, of Peking, of Canton, or of Singapore.
Everywhere, however, the characters have kept the same meanings
and everyone can understand them.
For example, the word for “eat” is pronounced chi in Mandarin and is
written with a character which we shall denote by “A”. In Cantonese,
this character is pronounced like hek but the Cantonese word for
“eat” is pronounced sik and itself is represented by a character which
we shall denote by “B”. Nevertheless, all educated Chinese - even if
in their dialect the word for “eat” is pronounced neither chi nor sik -
readily understand the characters “A” and “B”, which both mean “eat”.
[V. Alleton (1970)]
Chinese writing is therefore, in the words of B. Karlgren, a visual
Esperanto: “The fact that people who are unable to communicate by the
spoken word can understand each other when each writes his own language
in Chinese characters has always been seen as one of the most remarkable
features of this graphical system.” [V. Alleton (1970)] We can easily under-
stand why it is that some of China’s neighbours have adopted this writing
system for their own languages.
NUMERALS OF THE FORMER KINGDOM OF ANNAM
In the present day, and since a date usually taken as the end of the
thirteenth century CE, for most purposes (including letters, contracts,
deeds and popular literature) the numerals are made in the chii’ nom
writing which is perfectly adapted to the Annamite number-names (the so
dem annam system) (Fig. 21.23).
-*&
&
£
fa
it
It
at
m isr
mot
hai
ba
bdn
nam
sau
bdy
tarn
chin
mudi
tram nghin muon
i
2
3
4
5
6
7
8
9
10
100 1,000 10,000
Fig. 21.23. Chu’ nom numerals and the Annamite names of the numbers. [Dumouticr (1888);
Fossey (1948)J
Although they look different from their Chinese prototypes, these
numerals are in fact made up by combining a Chinese character (generally
one of the Chinese numerals) as an ideogram, with some element of a
character (or the whole character) chosen to represent the pronunciation of
the pure Annamite number which is to be written (Fig. 21.24).
figures
2
3
4
5
6
7
8
9
10
100
1,000
10,000
Chinese
_g_
*0
JL
A.
+
&
chu ' nom
A* 3
u>
£.
&
Jl
Fig. 21.24.
This last was especially the case for the literate people of Annam (now
Vietnam). They considered that the Chinese language was superior to their
own, richer and more complete, and they adopted the Chinese characters
as they stood but pronounced them in their own way (called “Sino-
Annamite”). This gave rise to the Vietnamese writing called chu’ nom
(meaning “letter writing”).
The Chinese numerals were also borrowed at the same time, and were
read as follows in the Sino-Annamite pronunciation (so dem tau ) which
derived from an ancient Chinese dialect (Fig. 21.22).
m a
A:
A
K
+
W
£
nhat nhi tam
ttr ngu
luc
that
bat
ciru
thap
bach
thien
van
12 3
4 5
6
7
8
9
10
100
1,000
10,000
Fig. 21 . 22 . The Chinese numerals and the Sino-Annamite names of the numbers [Dumoutier (1888)]
This changed nothing in the number-system itself, which continued to
follow the Chinese rule of alternating digit and decimal order of magnitude,
as in Fig. 21.25.
Fig. 21.25.
%
sau
nghin
%
bon
m
tram
it
chin
&
mudi
tam
6
x
1,000
+
4
X
100
+
9
x
10
+
8
273
JAPANESE NUMERALS
Chinese characters were however abandoned in Vietnam at the start
of the twentieth century in favour of an alphabetic system of Latin origin.
The Annamite number-names (which are the only ones in current use)
are either spelled out using Latin letters or are represented by Arabic
numerals.
JAPANESE NUMERALS
The Japanese also borrowed Chinese writing. However, according to M.
Malherbe (1995), this was ill-adapted to the multiple grammatical suffixes
of Japanese which are intrinsically incapable of ideographic representation.
Therefore the Japanese early adopted (around the ninth century) a mixed
system based on the following principle:
Whatever corresponds to an idea is rendered by one of the Chinese
kanji ideograms [the kanji system has been simplified to the point
that there now remain only 1,945 official kanji characters, plus 166
for personal names, of which 996 are considered essential and are
taught as part of primary education]. The more complicated
ideograms have fallen into disuse and have been replaced by the
hiragana characters.
Hiragana is a syllabary: there are fifty-one signs, each of which
represents a syllable, and not a letter as in the case of our alphabet.
This can represent all the grammatical inflections and endings and,
indeed, anything which cannot be written using ideograms.
Katakana is a syllabary which exactly matches the hiragana but is
used for recently imported foreign words, geographical names, foreign
proper names, and so on.
Finally, the romaji, that is to say our own Western alphabet, is used
in certain cases where using the other systems would be too compli-
cated. For example, in a dictionary it is much more convenient to
arrange the Japanese words according to the alphabetical order of
their transcriptions into Latin characters.
This writing system, which is the most complicated in the world, is
regarded as inviolable by the Japanese who would consider themselves
cut off from their culture if they gave themselves over to the use of
romaji, even though this would cause no practical difficulties nor
inconvenience. [M. Malherbe (1995)]
The traditional Japanese numerals continue to be used despite the growing
importance of Arabic numerals; they are the same as the Chinese numerals,
in all their diverse forms (classical, cursive, commercial, etc.).
However, they are not pronounced as in Chinese. There are two different
pronunciations: one is the “Sino-Japanese” which is derived from their
Chinese pronunciation at the time when these characters were borrowed
into Japanese; the other is “Pure Japanese”.
The Japanese language therefore has two completely different series of
number-names which still exist side by side.
The “Pure Japanese” system is a vestige of the ancient indigenous
number-system. It consists of an incomplete list of names, which have short
forms and complete forms (Fig. 21.26).
Short forms
Full forms
1
hi- or hito-
hitotsu 3
hitori b
2
fu- or futa-
futatsu 3
futari b
3
mi-
mitsu 3
mitari b
4
yo-
yotsu 3
yotari b
5
itsu-
itsutsu
6
mu-
mutsu
7
nana-
nanatsu
8
ya-
yatsu
9
kokono-
kokonotsu
10
to
a. The number-names ending in -tsu are only used to refer to objects
b. The number-names ending in • tari are only used to refer to persons
Fig. 21.26. The Pure Japanese names of numbers. / Frederic (1994 and 1977-87); Haguenauer
(1951); Miller (1967); Plaut (1936)1
Only the first four number-names have the ending -tari when applied
to persons. From five persons upwards the base forms are used, which
have neither inflection nor gender. This provides another instance of the
psychological phenomenon described in Chapter 1, that only four items
can be directly perceived.
The name of the number 8 also means “big number” and occurs in
numerous locutions which express great multiplicity. So, where we for
instance would say “break into a thousand pieces”, the Japanese say
A^sr*
yatsuzaki
literally: “break into 8 pieces"
Fig. 21.27.
A market greengrocer - who sells every kind of fruit and vegetable - is
likewise called
A &Jk
yaoya
literally: [the man who sells] 800 kinds of produce
Fig. 21.28.
NUMBERS IN CHINESE CIVILISATION
274
The city of Tokyo, which is of enormous extent, used to be called
Fig. 21.29.
AS AS
happyakuhakku
literally: [the town withl 808 districts
And to indicate the innumerable gods of their Shinto religion, the Japanese
say
Nowadays, however, this system has been reduced to the barest
minimum and is only now used for numbers between 1 and 10. The
words for higher numbers have mostly fallen out of use except for the
word for 20 (still used for lengths of time) and the word for 10,000
(sometimes used for the number itself, but most often simply to mean a
boundless number).
The second of the Japanese number-systems has considerably greater
capability than the one we have just looked at. It has a complete set of
names for numbers, as follows:
AS75<0&
happyakuman no kami
literally: 8 million gods
Fig. 21.30.
As C. Haguenauer (1951) points out for the Pure Japanese number-names,
there is a clear relation between the odd forms and the even forms, in
the series "one-two” [ hito-futa ] and “three-six” [mi-mu\, and an
equally clear one between the even numbers four and eight [yo-ya].
The even numbers 2 and 6 have been obtained from the corresponding
odd numbers by simple sound changes. In the latter case, a mere
change of vowel makes the difference between “four” [ yo ] and “eight”
[ya]. At first sight, only i.tsu, “five”, and to, “ten” are exceptions - as
well, of course, as the odd numbers greater than 5. (Fig. 21.31)
1
ichi
10
2
ni
100
(= 10 2 )
hyaku
3
san
1,000
(= 10 3 )
sen
4
shi
10,000
(= 10 4 )
man
5
SO
6
roku
7
shichi
8
hachi
9
ku
Fig. 21.32. The Sino-Japanese number-names. [Haguenauer (1951); Miller (1967); Plaut (1936)]
The numbers from 11 to 19 are represented according to the additive
principle:
11 ju.ichi
12 ju.ni
13 ju.san
ten-one
ten-two
ten-three
= 10+1
= 10 + 2
= 10 + 3
1
hito = hi <■---
2x1
---» 2
fata ~ fa
3
mi
2x3
— > 6
mu
4
yo <r-~~
2x4
---» 8
ya
Fig. 21.31.
This could indicate that long ago, among the indigenous peoples of
Japan, the series of numbers came to a second break at 8 (the sequence 1,
2, 3, 4 being extended up to 8 by the additive principle: 5 = 3 + 2, 6 = 3 + 3,
7 = 4 + 3, 8 = 4 + 4).
In the aboriginal Japanese number-system there were also special names
for some orders of magnitude above 10: a word for 20 (whose root is hat’)
and individual names for 100 (momo), 1,000 (chi) and 10,000 ( yorozu ).
For the tens, hundreds and thousands, and so on, it used the multiplica-
tive principle:
20
ni.ju
two-ten
= 2x10
30
san.ju
three-ten
= 3x10
100
hyaku
hundred
= 10 2
200
ni.hyaku
two-hundred
= 2 x 100
300
san. hyaku
three-hundred
= 3 x 100
1,000
sen
thousand
= 10 3
2,000
ni.sen
two-thousand
= 2 X 1,000
3,000
san. sen
three-thousand
= 3 x 1,000
10,000
ichi.man
myriad
= 10 4
20,000
ni.man
two-myriad
= 2 x 10,000
275
CUSTOM AND SUPERSTITION
£ £
H £
A W
A +
—
go. man
san.sen
roku.hyaku
hachi.ju
ichi
(“five-myriad
three-thousand
six-hundred
eight- ten
one”)
(= 5 x 10,000
+ 3 x 1,000
+ 6 x 100 +
53,681
8x10
+ 1)
Fig. 21.33.
The word for 10,000 in Sino-Japanese is man. Previously, ban was also
used but nowadays it is only used in the sense of “unlimited number” or,
rather, “maximum”. While sen.man means “a thousand myriad”, namely
10,000,000, its obsolete homologue sen.ban nowadays means “in the
highest degree” or “extremely”. The famous Japanese war-cry banzai, “long
life (to) . . .” (to the Emperor, is understood), is made up of ban, “10,000”,
and zai, a modification of sai, “life”. On its own, the word also means
“bravo”, in the sense that “for what you are doing you deserve to live ten
thousand years!”
This oral number-system is of Chinese origin and so it is called the
Sino-Japanese system. It long ago displaced the old Pure Japanese system
whose structure was rather complicated. The changeover took place under
the influence of Chinese culture and manifested itself not only in the
disappearance of the number-names for the indigenous numbers above 10,
but also by the adoption of the Chinese characters which express the names
of these numbers; these characters are, of course, pronounced in the
Japanese way. This is the reason why there are two systems in use together.
Two parallel systems are also used in Korea. In the aboriginal, true
Korean system it is only possible to count up to 99, and it is only written in
hangul (a Korean alphabet which has nothing to do with Chinese or
Japanese writing and was created in 1443 by King Sezhong of the Yi
Dynasty). The second, Sino-Korean system was derived from Chinese and
allows arbitrarily large numbers; it is written with characters of Chinese
origin or by means of Arabic numerals [see J. M. Li (1987)].
CUSTOM AND SUPERSTITION:
LINGUISTIC TABOOS
For numbers from 1 to 10, the Sino-Japanese system is only used in special
circumstances, but is used without exception for larger numbers. In conver-
sation, however, the Japanese often use both systems at the same time.
The main reason for this is the speaker’s desire to make sure that the
listener does not misunderstand. Since different words often sound alike in
Japanese, ambiguity can only be avoided by careful choice of words.
This can be seen in the following examples (Fig. 21.34 and 21.35).
The word for “evening” is ban. For “one evening” one would say hito.ban
and not ichi.ban since the latter spoken words may also mean “ordinal
number” or “first number”.
Similarly, ju.nana (combining the Sino-Japanese for 10 with the Pure
Japanese for 7) can be heard more clearly than ju.shichi (in which both
elements are Sino-Japanese) and so is more commonly used for 17; and for
the same reason 70 is pronounced nana.ju and not shichi.ju. For 4,000, the
indigenous word yon for 4 is combined with the Sino-Japanese sen for 1,000
in saying yon.sen rather than shi.sen. C. Haguenauer (1951) also gives the
following examples:
To
say:
A Japanese would never
use the form:
He would rather use the
word or expression:
4
shi
yo
7
shichi
nana
9
ku
kokono
14
jushi
ju.yon
17
jushichi
ju.nana
40
shi.ju
yon.ju
42
shi.juni
yon.ju.ni
47
shi.jushichi
yon. ju.nana
70
shichi.ju
nana.ju
400
shi.hyaku
yon.hyaku
4,000
shi.sen
yon.sen
7,000
shichi.sen
nana.sen
Fig. 21.34.
However, concern for clarity is not the whole story. Another reason is
that the Japanese have always had scrupulous respect for certain linguistic
taboos imposed by mystical fears.
In Japan, a “name” (in the widest sense of the term) has a very special
significance. The sound of the name, it is held, is produced by the action
of motive forces which, indeed, are the very essence of the name, so to
pronounce a name is not merely to utter some expression but also - and
above all - is to set in motion forces which may have malign powers. This is
an ancient and universal belief: to name a being or a thing is to assume
power over it; to pronounce a name, or even to utter a sound resembling
the name of some malevolent spirit, is to risk awakening its powers and
suffering their evil effects. We can therefore understand why the Japanese
have attached such importance to precision of utterance and why they take
such trouble to avoid using a name which might resemble the sound of a
name of evil import.
In addition to this, there are mystical reasons. Numbers, in Japan as
NUMBERS IN CHINESE CIVILISATION
276
elsewhere, have hidden meanings. The Japanese even today still have a
degree of numerical superstition, manifest as a respect or even an instinc-
tive fear for certain numbers such as 4 or 9. Try to park your car in bays 4,
9, 14, 19, or 24 of a Tokyo car park: you may locate these places, perhaps, if
the secret of perpetual motion is ever discovered. Seat number 4 in a plane
of Japanese Airlines, rooms 304 or 309 of a hotel - these can hardly ever be
found (still less in a hospital!). Simply because the number in “Renault 4”
has always been one of the most menacing, the Japanese launch of this car
failed miserably.
This superstition originates in an unfortunate coincidence of sound
(resulting from the adoption of the Chinese number-system and its
development according to the rules for reading and writing Sino-Japanese).
In the Sino-Japanese system, the word for 4 is shi which has the
same sound as the word for death. Therefore the Japanese recoil from
using the Sino-Japanese word for 4, usually using the Pure Japanese
wordyo-. For 9, the Sino-Japanese word is ku, with the same sound as the
word for pain. Throughout the Far East, including Japan, the ills of
the human race are popularly attributed to Spirits of Evil which breathe
their poisoned breath all round. Always meticulous about their health,
the Japanese therefore sought to avoid attracting the malign attention of
these spirits by avoiding the use of this word for 9, using instead the
indigenous word kokono-.
For exactly the same reason, 4,000 is spoken as yon.sen rather than shi. sen
which has the same sound as the expression for “deadly line”; for “four
men” they say yo.nin and not shi.nin which also means “death” or “corpse”.
The indigenous word nana for “seven” is preferred to the numeral shichi (7)
because the latter might be mistaken for shitou which means death, or loss.
Finally, 42 is never spoken as shi.ni (a simplified expression: “four-two”) nor
shi.ju.ni (= 4 x 10 + 2), because of the dread presence of “ death” in the
name of the number 4 as shi in each case. There is a further reason: in
the first form, the listener may hear shin.i - “occurrence of death”; in the
second form we also have the name of “42 years of age” which is held to be
an especially dangerous age for a man. This number is therefore usually
expressed asyon.ju.ni.
It is a strange paradox that a civilisation which is at the forefront of
science and technology has preserved the fears and superstitions of thou-
sands of years ago, and that there is no thought that these should be
overturned.
NUMERALS OF CHINESE ORIGIN
READ AS:
Standard
Cursive forms
Calligraphic
Commercial
Sino-
Pure Japanese
forms
forms
forms
Japanese
short
complete
1
— -
w
^9
l
ichi
hi-, hito-
hitotsu
2
T
-=T ° r S'
W0>
^-r
H
ni
fa-, fata-
fatatsu
3
3. b
1*1
san
mi-
mitsu
4
23
*5)
;*
shi
yo-
yotsu
5
£
b
b
H or ^
S»
itsu-
itsutsu
6
/r
H
&
a.
roku
mu-
mutsu
7
-e
4
4
shichi
nana-
nanatsu
8
A
A
»
hachi
ya-
yatsu
9
A
hi
h-
*
ku
kokono-
kokonotsu
10
+
or
•f
j»
to
100
6
T9 or 3
hyaku
1,000
4 *
>4
4
sen
10,000
Hit ° r >5
i
%
V
man
Fig. 21.35. Number-names and numerals in current use in Japan
WRITING LARGE NUMBERS
In everyday use, neither the Chinese nor the Japanese have need of special
signs for very large numbers. Using only the thirteen basic characters of
their present-day number-system they can write down any number, up to at
least a hundred billion (10 u ).
Although usually only used for numbers up to 10 8 , the method they use
is a simple extension of their ordinary number-system, namely introducing
ten thousand (10 4 ) as an additional counting unit. The following shows how
the Chinese represent consecutive powers of 10 (Fig. 21.36):
10,000
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
10,000,000,000
100,000,000,000
yi wan (=
shi wan (=
yi bai wan (=
yi qian wan (=
yi wan wan (=
shi wan wan (=
yi bai wan wan (=
yi qian wan wan (=
1 x 10,000)
10 x 10,000)
1 x 100 x 10,000)
1 x 1,000 x 10,000)
1 x 10,000 x 10,000)
10 x 10,000 x 10,000)
1 x 100 x 10,000 x 10,000)
1 x 1,000 x 10,000 x 10,000)
Fig. 21 . 36 A. The usual Chinese notation for the successive powers of 1 0. [ Guitel ; Menninger
(1957); Ore (1948); Tchen Yon-Sun (1958)1
277
to 4 __ 1 x 10 4
yi wan
10 e 1 X 10 4 X 10 4
yi wan wan
10 5 10 X 10 4
10 9 | ^ 10 X 10 4 x 10 4
shi wan
shi wan wan
1° 6 1 x 10 2 x 10 4
10 10 — pj 1 x 10 2 X 10 4 X 10 4
yi bai wan
yi bai wan wan
10 7 ^ 1 x 10 3 x 10 4
10 u jgr 1 x 10 3 x 10 4 x 10 4
yi qian wan
yi qian wan wan
Fig. 2i. 36 b.
For a very large number such as 487,390,629, therefore, they would write:
ts* Af tSH + iiUASl + ii
si wan ba qian qi bai san shi jiu wan liu bai er shi jiu
»
(4 x 10 4 + 8 X 10 3 + 7 X 10 2 + 3 X 10 + 9) X 10 4 + (6 X 10 2 + 2 X 10 + 9)
Fig. 21.37.
decomposing it as
(4 x 10,000 + 8 x 1,000 + 7 x 100 + 3 x 10 + 9) x 10,000 + 6x100 +
2 x 10 + 9
or 48,739 X 10,000 + 629.
The system just described is in practice the only one used for ordinary
purposes. However, though only in scientific and especially astronomical
texts, one may encounter special characters for higher orders than 10 4
which can therefore be used to express much larger numbers than are possi-
ble with the usual system. However, the signs used have meanings which
vary according to which of three value conventions is being used. Each sign
may have one of three different values depending on whether it is used
on the xia deng system (“lower degree”), the zhong deng system (“middle
degree”) or the shang deng system (“higher degree”).
The character , zhao, therefore, may represent a million (10 6 ) in
the lower degree, a thousand billion (10 12 ) in the middle degree, and 10 16
in the higher degree.
In the lower degree ( xia deng) the system is a direct continuation of the
ordinary number-system since the ten successive additional characters are
simply the ten consecutive powers of 10 following 10 4 , namely
10 5 , 10 6 , 10 7 , 10 8 . . ., 10 13 , 10 14
WRITING LARGE NUMBERS
which are represented by the characters
yi, zhao, jing, gai , . . . , zheng, zai.
So, written in the lower degree, one million and three million would be
written as follows:
-ft
yi zhao
1x10 s
or commonly:
-S*
yi bai wan
1 x 100 x 10,000
2ft
san zhao
3 x10 s
san bai wan
3 x 100 x 10,000
Fig. 21.38.
The xia deng system therefore allows any number less than 10 15
to be written down straightforwardly. For example, the number
530,010,702,000,000 would be written as
JL«H iE - * 4 ; tt n *
wu zai san zheng yi rang qi gai er zhao
5 x 10 14 + 3 x 10 13 +1 X 10 10 + 7 x 10 8 +2 x 10 6
Fig. 21.39.
In the middle system the same ten consecutive characters represent
increasing powers of 10 greater than 10 4 , but they now increase, not by a
factor of 10 each time, but by a factor of 10,000, namely
10 8 , 10 12 , 10 16 , . . . , 10 40 , 10 44 (Fig. 21.42).
With the convention that two of these characters should never occur
consecutively, this system can be used to represent all the numbers less
than 10 48 . For example:
^WE+JR-fc^H S * I + A I
san bai wu shi rang qi qian san bai zhao er shi liu yi
Fig. 21 . 40 .
NUMBERS IN CHINESE CIVILISATION
278
In the higher degree system, only the first three of these ten characters
are used, namely yi, zhdo and jing. These are given the values 10 8 , 10 16 and
10 32 respectively. With these, it is possible to represent all numbers less
than 10 64 . For example:
H S-tt- A + fi * Hit-*
sdn jing wu qidn sdn bdi yi yi er bai qi wan liu qidn yi bai ba shi wu zhdo sdn yi yi wan
(3 x 10 32 + [[5 xlO 3 + 3 x 10 2 + 1) . 10 8 + |2 x 10 2 + 7| • 10 4 + 6 x 10 3 + 1 x 10 2 + 8 x 10 + s] 10“ + 3 x 10 8 + 1 x 10 4
300 , 005 , 301 , 020 , 761 , 850 , 000 , 000 , 300 , 010,000
Fig. 21.41.
Xid deng
LOWER DEGREE
SYSTEM
Zhong deng
MIDDLE DEGREE
SYSTEM
Shang deng
HIGHER DEGREE
SYSTEM
££ wan
10 4
10 4
10 4
Cl
10 5
10 8
10 8
4k zhdo
10 6
10 12
10 16
„ .
M jing
10 7
10 16
10 32
gai
10 8
10 2 °
10 64
*8 bu h
10 9
10 24
10 128
H
Jff rang
10“
10 28
10 256 O
C *1
H
31 8 0U ‘
10 u
10 32
10 512 n
>
t-
JBJ i idn
10 12
10 36
10 1024 >
C
cn
j£ zhen g
10 13
10 4 °
2Q2048 °°
zai
10 14
10 44
2Q4096
a Graphical variant 1L b Equivalent word m c Graphical variant
Fig. 21 . 42 . Chinese scientific notation for large numbers [Giles (1912); Mathews (1931);
Needham (1959)1
Such very large numbers are, however, very infrequently used: “in math-
ematics, business or economics numbers greater than 10 14 are very rare;
only in connection with astronomy or the calendar do we sometimes find
larger numbers” [R. Schrimpf (1963-64)].
Finally, let us draw attention to a very interesting notation which
Chinese and Japanese scientists have used to express negative powers
of 10:
10-!= 1/10, lO" 2 = 1/100, 10- 3 = 1/1,000, 10^= 10,000, etc.
They especially find mention in the arithmetical treatise Jinkoki
published in 1627 by the Japanese mathematician Yoshida Mitsuyoshi
(Fig. 21.43).
fen
10- 1
m
li
10“ 2
mao
10- 3
&
mi
10^
&
hu
10- 5
ft
wei
10 6
m
xian
10- 7
&
sha
10^
m
chen
10- 9
£
ai
10-1°
Fig. 21.43. Sino-Japanese scientific notation for negative powers of 10 [Yamamoto (1985)]
THE CHINESE SCIENTIFIC POSITIONAL SYSTEM
Further evidence of advanced intellectual development in the Far East
comes from the written positional notation formerly used by Chinese,
Japanese, and Korean mathematicians.
Though we only know examples of this system dating back to the
second century BCE, it seems probable that it goes back much further.
Known by the Chinese name suan zi (literally, “calculation with rods”),
and by the Japanese name sangi, this system is similar to our modern
number-system not only by virtue of its decimal base, but also because the
279
THE CHINESE SCIENTIFIC POSITIONAL SYSTEM
values of the numerals are determined by the position they occupy. It is
therefore a strictly positional decimal number-system.
However, whereas our system uses nine numerals whose forms carry no
intrinsic suggestion of value, this system of numerals makes use of system-
atic combinations of horizontal and vertical bars to represent the first nine
units. The symbols for 1 to 5 use a corresponding number of vertical
strokes, side by side, and the symbols for 6, 7, 8, and 9 show a horizontal
bar capping 1, 2, 3, or 4 vertical strokes:
I II III llll I T I I I
12 345 678 9
Fig. 2i. 44 -
Examples of numbers written in this system are given by Cai Jiu Feng, a
Chinese philosopher of the Song era who died in 1230 [in Huang ji, in the
chapter Hong fan of his “Book of Annals”, cited by A.Vissiere (1892)].
Example:
I II II III llll T T m
1 2 2 5 4 6 6 9
> * .> ■>
12 25 46 69
Fig. 21.45.
Ingenious as it was, this system lent itself to ambiguity.
For one thing, people writing in this system tended to place the vertical
bars for the different orders of magnitude side by side. So the notation for
the number 12 could be confused with that for 3 or for 21; 25 could be
confused with 7, 34, 43, 52, 214, or 223, and so on (Fig. 21.45).
However, the Chinese found a way round the problem, by introducing a
second system for the units, analogous to the first but made up of horizon-
tal bars rather than vertical. The first five digits were represented by as
many horizontal bars, and the numbers 6, 7, 8, 9 by erecting a vertical bar
(with symbolic value 5) on top of one, two, three, or four horizontal bars:
“" = s s!_L s Li±
123456789
Then, to distinguish between one order of magnitude and the next,
they alternated figures from one series with figures from the other,
therefore alternately vertical and horizontal. The units, hundreds, tens of
thousands, millions, and so on (of odd rank) were drawn with “vertical”
symbols (Fig. 21.44), whereas the tens, thousands, hundreds of thousands,
tens of millions, etc. (of even rank) were drawn with “horizontal” symbols
(Fig. 21.46), by which means the ambiguities were elegantly resolved
(Fig. 21.48).
Numbers in scientific texts
from the Han period
( 2 nd century BCE
to 3 rd century CE)
from the end of the Song Dynasty and
from the Mongolian period (Yuan
Dynasty) ( 13 th and 14 th centuries CE).
—
1
11
1
II
—
1
2
ss
III
III
=
3
s
llll
mix
= X
4
s
III
mi -3
= ° r 6
5
1
T
T
1
6
±_
¥
¥
±
7
¥
¥
_L
8
mr
1"X
i*x
9
The value of a numeral depends on its position in
the representation of a number. Starting with the
8th century, the absence of a certain order of
magnitude is indicated by the sign O; this usage
of a ZERO sign was introduced to China under
Indian influence.
Numbers on coins of the end
of the Zhou Dynasty (6th— 5 th
centuries BCE) and of the period
of the warring kingdoms
(5th-3rd centuries BCE)
1
or |
2
= or ||
3
= 0, Ill
4
= ” 1111 “ mi
5
= [Hill]
6
1 -
0
H
7
± T ¥
8
± [-] [¥I
9
ak m m
10
"for ♦ 1 1
100
5B IK
1,000
x i s t
T S 4 i
10,000
75 ill
O *= CL
Fig. 21 . 47 . Chinese bar numerals through the ages [Needham (1959)]
Fig. 21.46.
NUMBERS IN CHINESE CIVILISATION
Fig. 21.48. Examples of numbers written in the Chinese bar notation fsuan zij
This step was taken at the time of the Han Dynasty (second century BCE
to third century CE). This did not solve all the problems there and then,
however, since the Chinese mathematicians were to remain unaware of zero
for several centuries yet. The following riddle bears witness to this, in the
words of the mathematician Mei Wen Ding (1631-1721):
The character hai has 2 for its head and 6 for its body. Lower
the head to the level of the body, and you will find the age of the
Old Man of Jiangxian.
In the above, the character playing the main role in the riddle has been
written in the kaishu style:
%
Fig. 21.49. hai
and the riddle remains obscure since the modern character is not the same
shape as it was before. According to Chinese sources, however, the riddle
dates from long before the Common Era, originating in the middle of the
280
Zhou era (seventh to sixth centuries BCE; see Needham (1959), p. 8). And
since at that time Chinese characters were drawn in the da zhuan ("great
seal”) style, we must therefore see the character in question drawn in this
style if we are to solve the riddle.
In this style, the word was written:
f«
Fig. 21.50. hai
Its “head”, therefore, is indeed the figure 2 iS , and its lower part is
a “body” consisting of three identical signs -fff each of which resembles
the “vertical” symbol for the figure 6 (Fig. 21.47). Arrange the two horizon-
tal lines of the head vertically and on the left-hand side of the body, and
you find
II _sl II T T T
head body or, nearly enough, 2 6 6 6
Fig. 21.51. Fig. 21.52.
The Chinese system being decimal and strictly positional, this represents
the number
2 X 1,000 + 6 X 100 + 6 X 10 + 6 = 2,666
so the solution of the riddle is the number 2,666. But this cannot be an age
in years, unless the Old Man of Jiangxian was a Chinese Methuselah. To
consider them as 2,666 days would give an absurd answer, since the “Old
Man” would then only be seven and a half years old. In fact, this number
system had no zero until much later, so the answer can only be one of the
numbers 26,660, 266,600, 2,666,000, etc. But since 266,600 or any higher
number is out of the question, we are left with 26,660 days. In the riddle,
the number sought does not represent days but tens of days: the Old Man
of Jiangxian had lived 2,666 tens of days, or about 73 years.
The lack of a sign to represent missing digits also gave rise to confusion.
In the first place, a blank space was left where there was no digit, but this
was inadequate since numbers like 764, 7,064, 70,640 and 76,400 could
easily be confused:
TQJII IT JJIII J JJIII
764 7064 70640
» » »
Fig. 21.53.
764
7,064
70,640
281
THE CHINESE SCIENTIFIC POSITIONAL SYSTEM
To avoid such ambiguities, some
used signs indicating different powers
of 10 from the traditional number-system, so that numbers such as 70,640
and 76,400 would be written as:
|o ||o ± 0
lony i±
IsIToooo
1 ; 0 2 ; 0 7 ; 0
1 ; 0 ; 6 ; 9 ; 2 ; 9
1;4;7;0;0;0;0
IT -L Nil w
m 1111 +
> » >
10 20 70
■>
106,929
>
1,470,000
7 6 4
“hundred’’
7 6 4
"ten thousand" “ten”
Reference: Document
reproduced in Fig. 21.59
Reference: Document
reproduced in Fig. 21.60
Reference: Chinese document of
1247 CE. Brit. Mus. Ms. S/ 930 .
76,400
70,640
[See Needham ( 1959 ), p. 10 ]
Fig. 21.54. Fig. 21.57. The use of zero in the Chinese bar numerals
Others used the traditional expression, therefore writing out in full:
76,400 7 !
x ]
H 10,000
A 6 i
1,000 i
PS *
X I
w 100 i
*
Fig. 21.55.
Yet others placed their numerals in the squares of a grid, leaving an
empty square for each missing digit:
76,400 70,064
IT
±
mi
'
¥
±
mi
7
6
4
0
0
7
0
0
6
4
Fig. 21.56.
Only since the eighth century CE did the Chinese begin to introduce
a special positional sign (drawn as a small circle) to mark a missing digit
(Fig. 21.57); this idea no doubt reached them through the influence of
Indian civilisation.
Once this had been achieved, all of the rules of arithmetic and algebra
were brought to a degree of perfection similar to ours of the present day.
m
m
ID
1 zkllll
III— T
Tsllll
-mill- 1 ±TAo
17 4
3 2 7
6 5 4
1955119680
174
327
654
1,955,119,680
Fig. 21.58. As a rule, in Chinese manuscripts or printed documents, numbers written in the bar
notation are written as monograms, i.e. in a condensed form in which the horizontal strokes are joined
to the vertical ones. (Examples taken from the document reproduced in Fig. 2160)
HD i,
4*
5
5
**
r~
"T -
1 MCP
■w
**
Fig. 2 1.59 a. Page from a text entitled Su Yuan Yu Zhian, published in 1303 by the Chinese
mathematician Zhu Shi Jie (see the commentary in the text). (Reproduced from Needham (1959), III,
p. 135, Fig. 801
NUMBERS IN CHINESE CIVILISATION
282
Fig. 21.59b.
Blaise Pascal was long believed in the West to have been the first to
discover the famous “Pascal triangle” which gives the numerical coefficients
in the expansion of (a + h) m , where m is zero or a positive integer:
BINOMIAL EXPANSIONS
PASCAL’S TRIANGLE
(«+*)»= i
1
(a+b) 1 = a + b
1 1
( a+b ) 2 = a 2 +2 ab + b 2
1 2 1
(a+b) 3 = a 3 +2a 2 b + 3 ab 2 + b 3
13 3 1
(a+b) 1 = a i +ia 3 b + 6a 2 i 2 + 4 ab 3 + b 4
1 4 6 4 1
(a+b) 3 = a 5 +5a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab A + b 5
1 5 10 10 5 1
( a+b ) 6 = a 6 +Sa 3 b + 15 a 4 b 2 + 20 a 3 b 3 + 15 a 2 b 4 + 6 ab 3 + b 6
1 6 15 20 15 6 1
>
>
In fact, as we can see from Fig. 21.59A, which is schematically redrawn
on its side in Fig. 21.59B (to be read from right to left), the Chinese had
known of this triangle long before the famous French mathematician.
T
i
&
-
W)
tit 7G
± ±
ft
*
ig
«c
@ m
14 R
2IJ
l
Wl
ip
> >
£
m -ix
-ft
ft
m
m-mi
l
s
w.
n
SC
M
M
a
ML
# Ik
M 6
1#
i
Pi
m
n
m
a mu* m m
iff
a*
in
T
m m
& m
Rill
«
SS
—
m
i
met
IS is,
ft
fpj
A
me m
—
iU
«
ft
1%
w
Z
z
m w
a
m
Z 7F.
1ft o
ng
fli
m
1
& e
;£
m
VL
me
* A
m
tMI-Sf
ft
m«f
UJl iU
HI*
ft
+
M U
95 -*
Me
Fig
21 . 60 .
Extract from Ce
ftian Hai Jing, published />
1245 fy/ //?e mathematician Li Ye.
/ Reproduced from Needham (1959), 111, page 132, Fig. 79]
o=|
0 2 1
oJ.HU
0 7 5
oooTJ_"TT=o=L
000667 308
0.21
0.75
0.00667308
Fig. 21. 6 1 . How Chinese mathematicians extended their positional notation to decimal fractions.
Reconstructed examples based on a text from the Mongol period: Biot (1839)
283
THE RODS ON THE CHECKERBOARD
EXAMPLES FROM A 13TH-CENTURY
CHINESE TREATISE (cf. Fig. 21.60)
EXAMPLES FROM
AN 18TH-CENTURY
JAPANESE TEXT
rm
IsTV
IMIA
HMHPHIPff-
654
1360
1536
152710100928
-2
-654
- 1,360
- 1,536
- 152,710,100,928
Fig. 2 1.6 2 a. Extension of scientific numerical notation to negative numbers. To indicate a
negative number, the Chinese and Japanese mathematicians often drew an oblique stroke through
the rightmost symbol of the written number. [Menninger (1957); Needham (1959)}
Polynomial P(x) = 2 x 2 + 654 x
cf. Fig. 21.60, col. V
'Wit
-2
654 a
"variable”
X 1
X
Polynomial P(x) = 2 x + 654
cf. Fig. 21.60, col. I
Hjt
nil
~ 2 ft
Character
representing
the variable
654
X
1
Polynomial
P(x) = X 4 - 654 x 3 + 106,924a: 2
cf. Fig. 21.60, col. VI
1
X 4
TWL
X 3
KIHIII 106924
X 2
°7C 0
‘ variable
X
0
1
Equation
2 x 3 + 15 x 2 + 166 x - 4460 = 0
cf. J. Needham III, p. 45
X 4
II
X 3
Him
X 2
w* »
unknown
X
-4,460
Character which means
‘‘the centre of the earth”
]
Fig. 2 i . 6 2 b . Notation for polynomials and for equations in one unknown, used by Li Ye
(1178-1265)
THE CHINESE VERSION OF THE RODS ON
THE CHECKERBOARD
Although the numerals discussed above served for writing, they were not
used for calculation. For arithmetical calculation, the Chinese used little
rods made of ivory or bamboo which were called chou (“calculating rods”)
which were placed on the squares of a tiled surface or a table ruled like a
checkerboard.
Fic. 21 . 63 . Model of a Chinese checkerboard used for calculation
The following story from the ninth century CE is evidence in point. It
tells how the Emperor Yang Sun selected his officials for their skill and
rapidity in calculation.
Once two clerks, of the same rank, in the same service, and with the
same commendations and criticisms in their records, were candidates
for the same position. Unable to decide which one to promote, the
superior officer called upon Yang Sun, who had the candidates
brought before him and announced: Junior clerks must know how
to calculate at speed. Let the two candidates listen to my question.
The one who solves it first will have the promotion. Here is the
problem:
A man walking in the woods heard thieves arguing over the
division of rolls of cloth which they had stolen. They said that, if
each took six rolls there would be five left over; but if each took
seven rolls, they would be eight short. How many thieves were
there, and how many rolls of cloth?
Yang Sun asked the candidates to perform the calculation with rods
upon the tiled floor of the vestibule. After a brief moment, one of
the clerks gave the right answer and was given the promotion, and
all then departed without complaining about the decision. (See J.
Needham in HGS 1, pp. 188-92).
NUMBERS IN CHINESE CIVILISATION
Fig. 21.64. A Chinese Master teaches the arts of calculation to two young pupils, using an
abacus with rods. Reproduced from the Suan Fa Tong Zong, published in 1593 in China:
[Needham (1959) III, p. 70]
284
Fig . 21.65. An accountant using the arithmetic checkerboard with rods. Reproduced from the
Japanese Shojutsu Sangaka Zue of Miyake Kenriyu, 1795: (D. E. Smith)
On an abacus of this kind, each column corresponds to one of the
decimal orders of magnitude: from right to left, the first is for the units,
the second for the tens, the third for the hundreds, and so on. A given
number, therefore, is represented by placing in each column, along a
chosen line, a number of rods equal to the multiplicity of the correspond-
ing decimal order of magnitude. For the number 2,645, for example, there
would be 5 rods in the first column, 4 in the second, 6 in the third and 2
in the fourth.
For the sake of simplicity, Chinese calculators adopted the following
convention (in the words of the old Chinese textbooks of arithmetic): “Let
the units lie lengthways and the tens crosswise; let the hundreds be upright
and the thousands laid down; let the thousands and the hundreds be face
to face, and let the tens of thousands and the hundreds correspond.”
The mathematician Mei Wen Ding explains that there was a fear that the
different groups might get muddled because there were so many of them.
Numbers such as 22 or 33 were therefore represented by two groups of
rods, one horizontal and the other vertical, which allowed them to be differ-
entiated. To prevent errors of interpretation, the rods were laid down
vertically in the odd-numbered columns (counting from the right), and
horizontally in the even-numbered columns (Fig. 21.67).
1
3
2
!1
3
99!
4
III!
5
III9I
6
T
7
TT
8
nr
9
1
—
=
—
—
Hill
i
1
1
10
20
30
40
50
60
70
80
90
Fig. 21.66. How the units and tens are represented by rods on the arithmetical checkerboard
285
THE RODS ON THE CHECKERBOARD
UNITS OF ODD
ORDER
(columns for even
powers of 10)
UNITS OF EVEN
ORDER
(columns for odd
powers of 10)
1
1
2
II
—
3
III
s
4
nil
aw
5
mu
1
6
T
±
7
TT
8
in
±
9
mi
•
A
Fig. 21.67. The rods are laid vertically for the units, the hundreds, the tens of thousands, and
so on; they are laid horizontally for the tens, the thousands, the hundreds of thousands, and so on.
•3
E
3 3
Si c
nr
8
1
II
2
s
2
1
1
1
1
1
-
1
1
s
3
(0)
1
(0)
±
(0)
(0)
(0)
<■ 81,221
<■ 1,111
« 3,010
<• 6,000
Fig. 21.68. How certain numbers are represented by laying rods on the checkerboard
The numbers to be added or subtracted were represented in the squares,
and rods were added or removed column by column. Multiplication was
almost as simple: the multiplier was placed at the top of the board, with
the number to be multiplied placed a few rows lower down. The partial
products were then set out on an intermediate line and added in as they
were obtained.
For example, to work out the product 736 x 247 (as set out by Yang Hui
in the thirteenth century), first of all the two numbers are set out on the
board as follows, keeping two empty squares at the right of the multiplier:
2 4 7
Multiplier »
19
s
TT
For the partial results
Multiplicand »
n
T
7 3 6
Fig. 21.69A.
Since the multiplier contains three figures, the method proceeds in three
stages.
First stage: multiplying 736 by 200
Mentally multiply the 2 of the multiplier by the 7 of the multiplicand, and
place the result 14 (in fact 140,000) in the middle line, taking care to place
the units of the result above the hundreds of the multiplicand:
1st partial result
(140,000)
2 4 7
19
n
—
Ill
n
T
7 3 6
From antiquity until recent times, the Chinese were able to perform
every kind of arithmetical operation by means of this device: addition,
subtraction, multiplication, division, raising to a power, extraction of
square and cube roots, and so on.
The methods used for addition and subtraction were straightforward.
Fig. 21.69B.
Then multiply the 2 of the multiplier by the 3 of the multiplicand, and add
the result 6 (in fact 6,000) to the partial result already obtained, placing it
on the square to the right of the 4 in 14:
NUMBERS IN CHINESE CIVILISATION
2 nd partial result
(140,000 + 6,000 =
146,000) »
Fig. 21 . 69 c.
Then multiply the 2 of 247 by the 6 of 736, and add this result 12 (in fact
1,200) to the partial result already obtained: in this case, the 2 is placed on
the square to the right of the 6 from the preceding stage, and the 1 is
placed on the next square to the left thereby being added to the number
already there:
3rd partial result
(146,000 + 1,200 =
147,200) ■»
Fig. 21 . 69 D.
Second stage: multiplying 736 by 40
The 2 of the multiplier has now done its work, so it is removed, and the
multiplicand is moved bodily one square to the right:
4 7
n
—
III! .
L 1!
tts
T
7 3 6
Fig. 21 . 69 E.
Now multiply the 4 of the multiplier by the 7 of the multiplicand, place
the result 28 (in fact 28,000) to the partial result in the middle row, and
complete the addition:
286
4 7
4th partial result
(147,200 + 28,000 =
175,200) +
Fig. 21 . 69 F.
Now multiply the 4 by the 3 of 736, and add the result 12 (in fact 1,200) to
the middle line:
4 7
5th partial result
(175,200 + 1,200 =
176,400) *
7 3 6
Fig. 21 . 69 G.
Now multiply the 4 by the 6 of 736 and add the result 24 (in fact 240) to the
middle line:
6 th partial result
(176,400 + 240 =
176,640) *
Fig. 21 . 69 H.
Third stage: multiplying 736 by 7
The 4 of the multiplier has done its work and it too is now removed, and the
multiplicand again moved bodily one square to the right:
4 7
SIT
- r
i
T
TT
T
7 3 6
287
THE RODS ON THE CHECKERBOARD
7
TT
—
TT
i
T
TT;
EE
T
7 3 6
Fig. 21.691.
The remaining 7 of the multiplier is now multiplied by the 7 of the
multiplicand, and the result 49 (in fact 4,900) is added to the middle line:
7th partial result
(176,640 + 4,900 =
181,540)
Fig. 21.69J.
7
TT
—
Ill
—
mu
1
TT
T
7 3 6
Now multiply the 7 by the 3 of 736, and add the result 21 (in fact 210) to the
middle line:
8th partial result
(181,540 + 210 =
181,750)
Fig. 21.69K.
7
TT
—
91)
—
TT
S
TT
EE
T
7 3 6
Finally multiply the 7 by the 6 of 736, and add the result 42 to the middle
line. This gives the following tableau, where the middle line shows the
result of the multiplication (736 X 247 = 181,792):
Final result
(181,750 + 42 =
181,792)
7
TT
—
ITT
—
TT
J.
n
TT
EE
T
7 3 6
Division was carried out by placing the divisor at the bottom and the
dividend on the middle line. The quotient, which was placed at the top, was
built up by successively removing partial products from the dividend.
On this numerical checkerboard it was also possible to solve equations,
and systems of algebraic equations in several unknowns. The Jiu Zhang
Suan Shu ("Art of calculation in nine chapters”), an anonymous work
compiled during the Han Dynasty (206 BCE to 220 CE), gives much detail
about the latter. Each vertical column is associated with one of the
equations, and each horizontal row is associated with one of the unknowns,
with the co-efficient of an unknown in an equation being placed in the
square where the row intersects the column. Also, for this purpose, as well
as the ordinary rods (reserved for “true” ( zheng ) numbers, i.e. positive
numbers), black rods were used for negative numbers (fit: “false” numbers).
A system of equations such as the following, for example:
2x - 3y + 8 z= 32
6x- 2 \y- z = 62
3x + 21y - 3z = 0
was therefore represented as:
01
T
an
III
ll
si
m
l
111
soo
A"
Fig. 21.70A.
The representation of a system of three
equations in three unknowns on the
arithmetical checkerboard. (From a
treatise on mathematics of the Han
period: 206 BCE to 220 CE): The
first column on the left represents
2x-3y + 8z = 32;
the second column represents
6x-2y-z = 62;
the third column represents
3x + 21y -3z = 0.
2
6
3
-3
-2
21
8
-1
-3
32
62
0
Fig. 21.70B.
It could be solved quite easily by skilful manipulation of the rods.
This system of numerals is of particular interest for the history of
numerical notation, since it is what led to the discovery of the principle
of position by the Chinese.
Their system of writing numbers with vertical and horizontal strokes
was simply the written copy of the way numbers were represented by
rods on the abacus, where the different decimal orders of magnitude
progressed in decreasing order from left to right. Once a calculation had
been completed on the abacus by manipulation of the rods, their disposi-
tion on the abacus could be copied in writing, ignoring the lines dividing
the abacus into squares. However, the rods were arranged on the abacus
according to the principle of position, for the purposes of calculation, and
so this principle was carried over into the written copy.
Fig. 21.691.
NUMBERS IN CHINESE CIVILISATION
288
REPRESENTATION OF THE NUMBER 3,764
with rods on the abacus
3
n
i
6
IHlj
-
10 3 10 2 10 1
using bar numerals combined
according to the positional principle
= TT i llll
3 7 6 4
3 X 10 3 + 7 X 10 2 + 6 X 10 + 4
Fig. 21.71. Origin of the Chinese bar numerals: how a manual calculating aid led to a written
positional number-system
The system of rods on the abacus was the practical means of performing
arithmetic calculations, and the suan z't notation was used to transcribe the
results into their mathematical texts.
The earliest known examples of the use of this abacus date from the
second century BCE, but it is very likely that it goes much further back
in time.
In any case, the characters used today for the Chinese word suan,
which means “calculation”, have a suggestive etymology. This word may be
written using three apparently quite different characters, namely:
suan (character B) suan (character C)
Fig. 21.72B. Fig. 21.72c.
Derived from the following archaic form A', the first character is an
ideogram expressing two hands, a ruled table and a bamboo rod:
lU
.~L. suan (archaic character A')
,, tY
Fig. 21.73a.
suan (character A)
Fig. 21.72A.
The second character is derived from the following archaic form B'
which expresses two hands and a ruled table:
suan (archaic character B')
and the third comes from the following ancient form C' which clearly
evokes the representation of numbers on the checkerboard by means of
rods vertically and horizontally oriented:
suan (archaic character C')
Fig. 21.73c.
THE CHINESE ABACUS:
THE CALCULATOR OF MODERN CHINA
The celebrated “Chinese abacus” is, therefore, neither the first nor the
only calculating device which has been used in China in the course of her
long history. It is in fact of relatively recent creation, the earliest known
examples being not older than the fourteenth century CE.
Amongst all the calculating devices which the Chinese have used,
however, the suan pan (meaning “calculating board”) is the only one
with which all the arithmetical procedures can be performed simply
and quickly. In fact almost everyone in China uses it: illiterate trader or
accountant, banker, hotelier, mathematician, or astronomer. The most
Westernised Chinese or Vietnamese, whether in Bangkok, Singapore,
Taiwan, Polynesia, Europe, or the United States, carry out every kind of
calculation using the abacus despite having ready access to electronic
calculators, so deeply ingrained in their culture is its use. Even the Japanese,
major world manufacturers of pocket calculators, still consider the
soroban (the Japanese word for the abacus) as the principal calculating
device and the one item that every schoolchild, businessman, peddler or
office-worker should carry with them.
Likewise in the former Soviet Union the schoty (cuerti), as the abacus is
called, may be seen alongside the cash register and will be used to calculate
the bill, in boutiques and hotels, department stores and banks.
A friend of mine, on a visit to the former Soviet Union, changed some
French francs into roubles. The cashier first worked out the amount on an
electronic calculator, and then checked the result on his abacus.
Fig. 21.73B.
2 89
THE CHINESE ABACUS
Westerners are invariably astonished at the speed and dexterity with
which the most complicated calculations can be done on an abacus. Once,
in Japan, there was even a contest between the Japanese Kiyoshi Matzusaki
(. soroban champion of the Post Office Savings Bank - a significant title,
given what it means to be champion of anything in Japan) and the
American Thomas Nathan Woods, Private Second Class in the 240th
Financial Section of US Army HQ in Japan, the acknowledged “most expert
electric calculator operator of the American forces in Japan”. It took place
in November 1945, just after the end of the Second World War, and the
men of General MacArthur’s army were eager to show the Japanese
the superiority of modern Western methods.
The match took place over five rounds involving increasingly compli-
cated calculations. And who won, four rounds out of five with numerous
mistakes on the part of the loser? Why, the Japanese with the abacus!
(Fig. 21.76)
Fie;. 21 . 74 . A Chinese shopkeeper doing his accounts with an abacus. (Reproduced from an
illustration in the Palais de la Decouverte in Paris)
RESULTS OF THE MATCH
KIYOSHI MATSUZAKI versus THOMAS NATHAN WOODS
Soroban champion of the Japanese Post Private 2nd class in the 240th financial
Office Savings Bank section of the US Forces HQ in Japan. The
“top expert with the calculator in Japan"
Contested on 12 November 1945 under the auspices of the US Army daily Stars and Stripes
1 st round
2 nd round
3rd round
4th round
Composite
round
Additions of
numbers with
3 to 6 figures
Subtractions of
numbers with
6 to 8 figures
Multiplications of
numbers with
5 to 12 figures
Divisions of
numbers with
5 to 12 figures
30 additions
3 subtractions
3 multiplications
3 divisions
(Numbers with
from 6 to 12
figures)
Matsuzaki
beat
Woods
Matsuzaki
beat
Woods
Woods
beat
Matsuzaki
Matsuzaki
beat
Woods
Matsuzaki
beat
Woods
1T4"8 / 2'00"2
ri6"0/l , 53"0
1'04 M 0 / 1'20"0
TOC'S / 1'36"0
1 ' 00"0 / 1 ' 22"0
(with mistakes)
(with mistakes
by the loser)
1'36"6 / 1'48"0
1'23"4 / 1'19"0
I'2r0/r26"6
1 ' 21"0 / 1'26"6
(with mistakes
by the loser)
Overall: Woods on the calculator is beaten 4 to 1 by Matsuzaki on the soroban
lie. 21 . 75 . A Japanese accountant working with a soroban. From an eighteenth-century
Japanese book, Kanjo Otogi Zoshi by Nakane Genjun, 1741: [Smith and Mikami (1914)1
Fig. 21 . 76 . Reader’s Digest no. 50, March 1947, p. 47
NUMBERS IN CHINESE CIVILISATION
2 ‘Ml
The match, contested on 12 November 1945 under the auspices
of the American Army daily Stars and Stripes, was a sensation. Their
reporter wrote that: “Machinery suffered a setback yester-day in the
Ernie Pyle theatre in Tokyo, when an abacus of centuries-old design
crushed the most modern electrical equipment of the United States
Government.” The Nippon Times was exultant at this modest
intellectual revenge for military defeat: “In the dawn of the atomic
age, civilisation reeled under the blows of the 2,000-year-old soroban."
An exaggeration, of course - above all concerning the age of the
soroban - but one which must be viewed in the context of a Japan
which, less than three months earlier, had seen two of its greatest cities
destroyed by unprecedented military force. But anyone who has
watched a Japanese of any competence operate the abacus would
have no doubt that the same result could be obtained even today, with
electronic instead of electrical calculators, at any rate for additions
and subtractions. The keyboard speeds of most of us would be no
match for the dexterity of the soroban operator. ( Science et Vie, no. 734,
November 1978, pp. 46-53).
The Chinese form of the instrument has a hardwood ffame which holds
a number of metal rods upon each of which slide wooden (or plastic)
beads which may be of somewhat flattened shape. The beads are on
either side of a wooden partition, two beads above and five below, and
the beads may be slid towards the partition. Each of the metal rods
corresponds to one of the decimal orders of magnitude, the value of a
bead increasing by a factor of 10 as one moves from one rod to the rod
on its left. (In theory, a base different from 10 may be used - 12 or 20 for
example - provided each rod carries a sufficient number of beads.)
The normal abacus will have between eight and twelve rods, but the
number may be fifteen, twenty, thirty, or even more, according to need.
The more rods there are, the larger the numbers that the abacus can
handle. With fifteen rods, for example, it can handle up to 10 15 -1 (a
thousand million million, minus one!)
As a rule, the first two rods on the right are reserved for decimal
fractions of first and second order, i.e. for the first two decimal places, and
it is the third rod which is used for the units, the fourth for the tens, the
fifth for the hundreds, and so on.
Fig. 21.77. The representation of numbers on the Chinese suan pan
The Russian abacus is somewhat different in design from the Chinese
suan pan (Fig. 21.78). It has ten beads on each rod, of which two (the
fifth and the sixth) are usually of a different colour, which makes it easier
for the eye to recognise the numbers from 1 to 10. To represent a number
the corresponding number of beads are slid towards the top of the frame.
Fig. 21.78. Russian abacus (schotyj. It generally has four white beads, then two black and then
four white. This type of instrument is still in use in Iran, Afghanistan, Armenia and Turkey
Fig. 21.79. French abacus used for teaching arithmetic in municipal schools in the nineteenth century
2!) 1
Fig. 21.80. Abacus marketed by Fernand Nathan at the beginning of the twentieth century as a
teaching aid
On the Chinese abacus, each of the five beads on the lower part is
worth one unit, and each of the two on the upper part is worth five.
Arithmetical operations involve sliding beads from either side towards
the central partition.
To place the number 3 on the abacus, slide three of the five beads on
the lower part of the first rod upwards towards the partition. To place the
number 9, slide four of the five lower beads upwards towards the partition,
and one of the two upper beads downwards towards the partition:
5
4
Fig. 21.81. 9
For a larger number such as 4,561,280, the same principle is adopted
for each digit: since the first digit is zero, the beads on the first rod are
not displaced (denoting absence of number in this position), giving the
result shown:
to 11 to 10 to 9 to* to 7 to 6 to 5 to 1 to 3 to 2 to 1
HI. 21.82.
4 5 6 1 2 8 0
THE CHINESE ABACUS
To place the number 57.39, which has a decimal fraction part, the same
principle is used for the hundredths, then the tenths, and then the units,
tens and hundreds (Fig. 21.83):
Fig. 21.83.
It is therefore a very simple matter to enter a number onto the Chinese
abacus. Actual arithmetic is hardly any more complicated, provided one
has learned the addition and multiplication tables by heart for the numbers
from 1 to 9.
For convenience of exposition, we shall only consider whole numbers, and
therefore we can allocate the first rod to the units, the second rod to the tens,
and so on. Now consider addition of the three numbers 234, 432 and 567.
First of all we “clear” the abacus by sliding all the beads to the top and
bottom extremities of the rods, leaving the central partition clear. To enter
the number 234, first on the third rod from the right (for the hundreds), we
slide two beads upwards; then, on the second rod (for the tens) three beads
upwards; and finally, on the first rod (for the units), four beads upwards:
2 3 4
Fig. 21.84A.
NUMBERS IN CHINESE CIVILISATION
292
Next, to add to this the number 432, we move the corresponding number
of beads towards the centre in a similar way. However, on the hundreds
rod there are already two beads touching the partition so we do not have
four beads available to slide; but we can bring down one bead (representing
5) from the top against the partition and slide one of the lower beads back
down away from the centre, since 5 - 1 = 4. On the tens rod, where three
beads have already been moved upwards leaving two, in order to add in
the 3 of 432 we again slide down one of the upper beads (for 5) and retract
two of the lower beads (since 5-2 = 3). Finally, on the units rod, we slide
down one of the upper beads (for 5) and retract three of the lower beads
(since 5-3 = 2):
Fig. 21.84B.
V
1
1
'r+f 1
Mr* to
]_
1
9
1
15
9
, 1 ,
6 6 6
As the third and final stage, to add the number 567 to this result, we start
by sliding one of the upper beads (for 5) downwards on the hundreds rod.
Then, on the tens rod, we slide down one of the upper beads (for 5)
and we slide up one of the lower beads (for 1), since 5 + 1 = 6. Finally, on
the units rod, we slide downwards one of the upper beads (for 5) and we
slide upwards two of the lower beads (for 2), since 5 + 2 = 7. Our abacus
now looks like the following:
> '
>
! 1
1
1
fil
1 !
1
1
! 1 .
1
► _J
Ji
1
, j_
' f
1
iL
;
Fig. 21.84c.
slide one lower bead (for 1) of the thousands rod towards the centre.
Next, in a similar way, the two upper beads of the tens rods are slid
upwards away from the centre and one lower bead of the hundreds rod is
slid towards the centre; and, finally, the two upper beads of the units
rod are replaced by a single bead on the tens rod. When this has been done,
the abacus looks like the following, and the result can be read off from it:
234 + 432 + 567 = 1,233.
:
n
l 1
V
j
if
1
12 3 3
Fig. 21.84D.
Subtraction is carried out by the reverse process, multiplication by
repeated addition of the multiplicand for as many times as each digit in
the multiplier, and division by repeated subtraction of the divisor from the
dividend as many times as possible, this number then being the quotient.
Suppose we want to evaluate the product 24 X 7.
We first note that the method is independent of the overall order of
magnitude of the result: technically, the procedure is identical whether
we want 24 X 7, 24,000 x 7, 24 x 700, 0.24 x 7 or 24 x 0.007, and the digits
in the result will be the same; to get the correct result it is enough to keep
the order of magnitude in mind.
To work out the above calculation, we start by placing the multiplier (7)
on a rod at the left, and the multiplicand (24) towards the right, making
sure to leave a few empty rods between them.
But it is not yet all over: what is represented on each rod is no longer a
decimal digit, and some further reduction is required before the result can
be announced. Therefore, on the third (hundreds) rod, we slide the two
upper beads away upwards: each counts for five hundreds, and so we then
Fig. 21.85A.
Now we mentally multiply 7 by 4, getting 28, and we place this result
immediately to the right of the multiplicand:
29 3
[■' i t; .
21.851).
7 2 4 2 8 « — 1st partial result
Now the 4 of the multiplicand is eliminated by sliding its four units beads
back downwards:
Next we mentally multiply 7 by 2, getting 14, and we now enter this result as
before but at one place further left. Adding it to what is already there, we
therefore slide one lower bead upwards on the hundreds rod, and on the
tens rod we slide one upper bead downwards and one lower bead upwards:
The 2 of the multiplicand is now eliminated, and the multiplier also, and all
that remains is to read off the result (168):
THE CHINESE ABACUS
So it is not very complicated to do arithmetic on the Chinese abacus.
Even square roots or cube roots, or more complicated problems still, can be
worked out by operators who know how to use it well. (Our intention here
is only to give a general idea of how to use the abacus; we therefore abstain
from describing the detailed technique for manipulating it, and we do not
discuss its general arithmetical or algebraic applications.)
Fig. 21 . 86 . Instructions for using the suan pan in the Chinese Suan Fa Tong Zong printed in 1593.
IReproducedfrom Needham (1959), III, p. 76 J
NUMBERS IN CHINESE: CIVILISATION
2!) -I
For all its convenience, this aid to calculation has a number of disad-
vantages. It takes a long time, and thorough training, to learn how to use it.
The finger-work must be extremely accurate, and the abacus must rest on a
very solid support. Moreover, if one single error is made the whole proce-
dure must be restarted from scratch, since the intermediate results (partial
products, etc.) disappear from the scene once they have been used. None
of this, however, detracts from the ingenious simplicity of the device.
After a little thought, however, we are led to ask a question touching on
the basic concept of the Chinese abacus. We have seen that on each rod nine
units are represented by one upper bead (worth five) and four lower beads
(worth one each). Therefore five beads (one upper and four lower) always
suffice to represent any number from 1 to 9. Why, therefore, do we find
seven beads, whose total value is 15? The answer lies in the fact that (as we
have seen in some of the above examples), it is often useful to represent on
one rod, temporarily, an intermediate result whose value exceeds 9.
In this connection we may note that the Japanese soroban began to do
away with the second upper bead, from around the middle of the nine-
teenth century (Fig. 21.87), and that since the end of the Second World War
it has definitively lost the fifth lower bead. This change has obliged the
Japanese abacists to undergo an even longer and more arduous training,
and it has obliged them to acquire a finger technique even more elaborate
and precise than that of the operators of the Chinese suan pan (Fig. 21.88).
The post-war Japanese abacus is therefore the fully perfected state of
the instrument and marks the close of an evolution in the techniques
of calculation which derive from arithmetical manipulations of pebbles,
an evolution which has largely been independent of the development of
written number-systems.
Fig. 21.87. Pw- war Japanese soroban with a single upper bead and five lower beads
NUMBER-GAMES AND WORD-PLAYS
We should not bid farewell to the Far Eastern civilisations without enjoying
some examples of their wit.
Both the Chinese and the Japanese have always had a great weakness
for plays on words and characters. Since their numerals correspond both to
words and to characters, they have taken every opportunity to indulge it.
Here are some examples.
The first example (noted by Mannen Veda) bears on the character for
the figure 8. For the age of a 16-year-old girl, the Chinese use the expression
pogua, which literally means “to cut the watermelon in two”:
% &
po (“cut into two") gua (“watermelon”)
This is a number-play on the form of the character gua (“watermelon")
which seems to be composed of two characters identical to the figure 8 side
by side, representing an addition:
Jtt=A+A =8 + 8 = 16
Furthermore, the pun involves the fact that “watermelon” can also mean
virginity (much as we use the word “flower”), which means that pogua is
also an erotic image of the “defloration” of the young girl.
Other examples (noted by Masahiro Yamamoto) concern the names
given to the various major anniversaries of old age in Japan.
1. The 77th birthday is the “happy anniversary”. In Japanese it is called
kiju and written
kiju
Graphically this yields the 77, since the word ki (“happy”) is written, in the
cursive style, as
u
namely as a character which can be decomposed as follows:
Fig. 21.88. Post- war Japanese soroban with a single upper bead and four lower beads
-t: + -t = 7 X 10 + 7 = 77
295
NUMBER-GAMES AND WORD-PLAYS
2. The 88th birthday is the “rice anniversary”. In Japanese it is called
beiju and written
beiju
Graphically this yields the 88, since the word bei (“rice”) is written using a
character which can be decomposed as follows:
= A + A =8x10+8=88
3. The 90th birthday is the “accomplished anniversary”. In Japanese it
is called sotsuju and is written
sotsuju
Graphically this yields the 90, since the word sotsu (“accomplished”) is
written using a character which in turn may be replaced by an abbreviation
which itself may be decomposed as follows:
3 s = ^ = 9 x 1° = 90
sotsu sotsu
4. The 99th birthday is the “white-haired anniversary”. In Japanese it is
called hakuju and is written
hakuju
Graphically this yields the 99, since the word haku (“white”) is written using
a character which is none other than the character for 100 from which one
unit (the horizontal line) has been removed:
£ = gf - — = 100 - 1 = 99
haku hyaku ichi
5. Finally, the 108th birthday is the “tea anniversary”. In Japanese it is
called chaju and is written
chaju
Graphically this yields the 108, since the word cha (“tea”) is written using a
character which can be decomposed as
* = ++ A+A = 10 + 10 + (8 x 10 + 8) = 108
10 10 8 10 8
We may also note the strange number-names used by Zen monks to express
sums of money in the Edo period (eighteenth century). For these monks,
anything to do with money was considered vulgar and not to be mentioned
directly. Therefore, to express numerical sums of money, they euphemisti-
cally made use of plays on characters (Fig. 21.89).
ZEN NUMERALS
LITERAL
MEANING
EXPLANATION OF NUMERICAL
INTERPRETATION
A*§ A
dai ni jin nashi
“size without
man”
“heaven
without man”
“king without
centre”
“fault without
evil”
“myself without
mouth”
“exchange
without man”
“cutting without
a knife”
“dividing
without a knife"
“circle without
accent”
“needle
without metal”
= without^^^^ - = 1
ten ni jin nashi
= without^^ ■» =2
6 ni chu nashi
= * without | - = 3
zai ni hi nashi
= wi, h ou.^ — * m = 4
go ni kuchi nashi
= .EL without Q — » = 5
ko ni jin nashi
= withou^^ ^ ^ =6
31*77
setsu ni to nashi
= without^J = 7
frM7J
bun ni to nashi
= without^ J » = 8
A***
gan ni chu nashi
= JlL without^^^ ■> = 9
i
shin ni kin nashi
= without > ~ ^ * = 10
Fig. 21.89. Esoteric numerals of the Zen monks (eighteenth century) (M. Yamamoto. Personal
communication from Alain Birot)
NUMBERS IN CHINESE CIVILISATION
We close with the following Japanese verses, attributed to Kobo Daishi
( 775 - 835 )*:
TRANSCRIPTION
I-ro-ha-ni-ho-he-to-
Chi-ri-nu-ru-wo
Wa-ka-yo-ta-re-so
Tsu-ne-na-ra-mu
U-i-no-o-ku-ya-ma
Ke-fu-ko-e-te
A-sa-ki-yu-me-mi-shi
E-hi-mo-se-su-n'
TRANSLATION
Though pretty be its colour.
The flower alas will fade;
What is there in this world
That can forever stay?
As I go forward from today.
To the end of the visible world,
I shall see no more dreams drift by
And l shall not befooled by them.
See L. Frederic, Encyclopaedia of Asian Civilisation, J.-M. Place, Paris, 1977-87, vol. ID.
296
This poem contains every sound of the Japanese language with no
repetitions. It is therefore often used in teaching Japanese.
However, number has never been far from poetry in the oriental
cultures: these same syllables which have been so to speak frozen into a
given order by this poem, have finally acquired numerical values. Which
is why the Japanese often count using the syllables of the poem:
I-ro-ha-ni-ho-he-to-chi-ri- . . .
123456789 ...
CHAPTER 22
THE AMAZING ACHIEVEMENTS
OF THE MAYA
The civilisation of the Maya was without question the most glorious of
all the pre-Columbian cultures of Central America. Its influence over the
others, particularly over Aztec culture, can be likened to the influence of
Greece over Rome in European antiquity.
SIX CENTURIES OF INTELLECTUAL AND
ARTISTIC CREATION
In the course of the first millennium CE the Maya people produced art,
sculpture, and architecture of the highest quality and made great strides
in education, trade, mathematics, astronomy, etc.
Maya builders discovered cement, learned how to make arches, built
roads, and, of course, they put up vast and complex cities whose buildings
were heavily decorated with sculpture and painting. Surprisingly, all this
was done with tools that had not developed since the Stone Age: the
Maya did not discover the wheel, nor use draught animals, nor any metals.
The Mayas’ true glory rests on their abstract, intellectual achievements.
They were, in the first place, astronomers of far greater precision than
their European contemporaries. As C. Gallenkamp (1979) tells us, the
Maya used measured sight-lines, or alignments of buildings that served
the same purpose, to make meticulous records of the movements of the
sun, the moon, and the planet Venus. (They may also have observed
the movements of Mars, Jupiter and Mercury.) They studied solar eclipses
in sufficient detail to be able to predict their recurrence. They were acutely
aware that apparently small errors could lead in time to major discrepan-
cies; the care they took with their observations allowed them to reduce
margins of error to almost nothing. For example, the Maya calculation of
the synodic revolution of Venus was 584 days, compared to the modern
calculation of 583.92.
Fig. 22 . 1 . The Great Jaguar Temple at Tikal, constructed in c. 702 CE. Copy by the author from
Gendrop (1978), p. 72
The Maya also made their own very accurate measurement of the solar
year, putting it at 365.242 days.* The latest computations give us the
figure of 365.242198: so the Maya were actually far nearer the true figure
than the current Western calendar of 365 days (which, with leap years,
gives a true average of 365.2425).
They were no less precise in their measurement of the lunar cycle.
Modern measuring devices of the most sophisticated kind allow us to fix
the average length of a lunar cycle at 29.53059 days. Using only their eyes
and their brains, the Mayan astronomers of Copan found that 149 new
moons occurred in 4,400 days, which gives an average for each lunar
month of 29.5302. At Palenque, the same calculation was made over 81
new moons and produced the even more accurate figure of 2,392 days, or
29.53086 per cycle.
* The Maya did not express the figure in this way of course, since they could only operate arithmetically
in integers.
T HE AMAZING A C H I E V V. M E NTS OF THE MAYA
298
Fig. 22 . 2 . Extract from a Maya manuscript (lower part of p. 93 of the Codex Tro - Cortesia n us, fro m
the American Museum, Madrid). It shows a kind of memorandum for prophet-priests, part
of a treatise on ritual magic which includes some astronomical observations.
Even more fascinating is the Mayas’ use of very high numbers for the
measurement of time. On a stela at Quirigua, for instance, there is an
inscription that mentions the last 5 alautun, a period of no less than
300,000,000 years, and gives the precise start and end of the period accord-
ing to the ritual calendar. Why did they count in terms so far beyond any
human experience of life? Perhaps that will always remain a mystery; but it
suggests that the Maya had a concept if not of infinity, then of a boundless,
unending stretch of time.
Fig. 22 . 3 . Alone in the darkness of the night, a Maya astronomer observes the stars. Detail from
the Codex Tro-Cortesianus. Copied from Gendrop (1978), p. 41, Fig. 2
It is even more puzzling that the Maya measurements were done
without any tools to speak of. They had not discovered glass, so there
were no optical instruments. They had no clockwork, no hour-glasses, no
idea of water-clocks ( clepsydras ), no means at all of measuring time in units
less than a day (such as hours, minutes, seconds, etc.); nor did the) have
any concept of fractions. It is hard to imagine how to measure time without
at least basic measuring devices.
The tool that the Maya used for measuring the true solar day was the
very simple but utterly reliable device called a gnomon. It consists of a rigid
stick or post fixed at the centre of a perfectly flat area. The stick's shadow
alters as the day progresses. When the shadow is at its shortest, then the
sun is at its meridian: that is to say, the sun has reached its highest point
above the horizon, and it is “true noon”.
As for astronomical observations, according to P. Ivanoff (1975), these
were done by means of a jadeite tube placed over a wooden cross-bar, as
shown in codices, thus:
Fig. 22 . 4 . Astronomical observations, as shown in the Mexican manuscripts, Codex NulhtU and
Codex Seldcn. Copied from Morley (1915). In the left-hand drawing, an astronomer seen in profile
watches the sky through a wooden X; the right-hand drawing shows an eye looking through the
angle of the X.
The Maya also developed an elaborate writing system, consisting ot
intricate signs known as glyphs. These include numerals (as we shall see
below) and many names or “emblem glyphs” associated with the main
299
MAYA CIVILISATION
cities in the central Mayan area. The decipherment of Maya glyphs is
currently the subject of intense and recently successful research.*
MAYA GODS
HUNAB KU AH PUCH
Great Creator-God, God of Death
supreme divinity of
the Maya pantheon
EMBLEM-GLYPHS
of some Maya cities
YUM KAX CHAC
God of maize God of rain
Piedras
Negras
m
«J!Lf
Tikal
Copan
CARDINAL POINTS
Likin Cikin
East West
OTHER GLYPHS
Kin, “day”
Stylised images of the solar
disc, suggesting the idea
of the sun and thus by
extension of a day
Uinal “month of 20 days"
This glyph is an abstract
image of the moon, the
Maya symbol for the
number 20
Fig. 22.5. Some of the Maya hieroglyphs deciphered to date
MAYA CIVILISATION
Several dozen abandoned cities buried in the tropical jungles and savannah
of Central America bear witness to one of the most mysterious episodes
of human history.
With their stately temples perched atop pyramids up to 170 feet high,
with their intricately carved pillars and altars and brightly painted earth-
enware vessels, these forgotten cities are all that is left of a sophisticated
civilisation that is thought to have begun in the jungles of Peten. At the
height of its glory, Maya civilisation covered the area shown in Fig. 22.6,
and included:
* See Michael D. Coe, Breaking the Maya Code (London: Thames and Hudson, 1992), for a fascinating
account of recent breakthroughs.
• the present-day Mexican provinces of Tabasco, Campeche,
and Yucatan, the region of Quintana Roo and a part of
Chiapas province;
• the Peten region and almost all the uplands of present-day
Guatemala;
• the whole of Belize (formerly British Honduras);
• parts of Honduras;
• the western half of Salvador;
making an area of about 325,000 km 2 .
There are reckoned to be about two million direct descendants of the
Maya alive today, most of whom are in Guatemala, and the remainder
spread around Honduras and the Mexican provinces of Yucutan, Tabasco,
and Chiapas.
Maya civilisation was fully developed at least as early as the third century
CE and reached its greatest heights of artistic and intellectual creation long
before the discovery of the New World by Christopher Columbus.
Fig. 22.6. Map by the author, after P. Ivanoff
THE AMAZING ACHIEVEMENTS OF THE MAYA
300
It is widely assumed that there was an early period of Maya civilisation
dating from about the fifth century BCE, during which the Maya differ-
entiated themselves from other Amerindian cultures; but of this era of
formation, there remain few traces apart from shards of pottery, and little
can be known of it.
The period from the third to the tenth century CE is the “classical”
period of Maya civilisation, and it is in these centuries that the Maya
developed their arts and sciences to their highest point. But at some
point in the ninth or tenth centuries there occurred an unexpected and
mysterious event which Americanists have not yet fully explained: the Maya
began to abandon their ritual centres and cities in the central area of the
“Old Empire”. Their departure was so sudden in some places that buildings
were left half-finished.
It was long thought that what had happened was an exodus of the entire
population, but recent excavations have shown this not to be true. Various
theories have been put forward to explain this resettlement of the Maya
to the north and the south - epidemics, earthquakes, climate change, inva-
sion, perhaps even their priests’ interpretation of the wishes of the gods.
The most plausible of these hypotheses are those that see the main cause of
the exodus in the exhaustion of the soil. Mayan agriculture was based on
the use of burnt clearings, which created ever more extensive infertile areas.
In addition, there may well have been a peasant revolt, provoked by the vast
inequality between the classes of Maya society.
Whatever the real cause, large sections of the Maya people left the central
area, leaving a much reduced population which gave up the traditional
rituals in the cities and allowed the religious monuments to fall into decay.
There was also an invasion of a different people, from the west. To
judge by the ruins of Chichen Itza (Yucatan), these invaders were probably
Toltecs, who came from an area north of present-day Mexico City. After the
“interregnum” (925-975 CE), the period following the fall of classical Maya
culture is called the “Mexican period”, and it lasted until 1200 CE.
The Maya accepted Toltec domination and adopted some of the Mexican
gods, including Quetzalcoatl, the plumed serpent. The Maya also became
more warlike, in line with the traditions of the Mexicans, whose gods
required countless human sacrifices. However, even if the Maya of the
Mexican period tore the hearts out of their human sacrificial victims, they
were never as bloodthirsty as their neighbours, the Aztecs, whose religious
rituals were frenetically violent.
Toltec and Maya civilisations gradually merged into one. The language,
religion and even the physical characteristics of the Maya changed so
much that it is hard to compare Maya civilisation before and after the
Mexican invasion.
Between 1200 and 1540, the course of Maya history changed completely
once again. Mexican civilisation was rejected, and the invaders adopted
Maya customs. This period is called the age of “Mexican absorption”. Maya
civilisation continued to decline, as can be seen in the art and architecture
of the period. Wars of annihilation broke out, and Maya civilisation soon
came to an end. Only a small group from Chichen Itza managed to escape
and resettle on the island of Tayasal, in Lake Peten, where they maintained
their independence until 1697.
THE DOCUMENTARY SOURCES
OF MAYA HISTORY
The first light to be shed on the civilisation of the Maya was the work of
the famous American diplomat and traveller, John Lloyd Stephens, who
explored the jungles of Guatemala and southern Mexico with the English
artist Frederick Catherwood in 1839. A more detailed survey of Maya
sites and buildings was carried out from 1881 by Alfred Maudslay, which
marked the true beginning of scholarly research on the world of the Maya.
But most of the knowledge we now have of this lost civilisation has been
gained in the last few decades.
When the Spaniards conquered Central America in the sixteenth
century, Maya civilisation had been all but extinct for several generations,
and most of its magnificent cities were but inaccessible ruins in the
midst of the jungle. This explains why the early Spanish chroniclers were
bedazzled by the Aztecs and hardly mentioned the Maya at all.
Pre-Columbian cultures, moreover, were systematically suppressed by
the conquistadors. Deeply shocked by the bloodthirstiness of Aztec and
Maya rituals, and believing that their mission was to convert the natives
to Christianity, the Spaniards sought to eradicate all traces of the devilish
practices that they came across. In order to ensure that such abominable
religions would never re-emerge, they burnt everything they could find
in autos-da-fe.
Nonetheless it is to a Spaniard that we owe a significant part of our
present knowledge of the history, customs and institutions of the Maya. In
1869, the colourful and indefatigable French monk Brasseur de Bourbourg
unearthed in the Royal Library of Madrid a manuscript entitled Relation
de las Cosas de Yucatan by the first bishop of Merida (Yucatan), Diego de
Landa. Written shortly after the Spanish conquest, the Relation is full of
priceless ethnographic information, including descriptions and drawings
of the glyphs used by the indigenous population of Yucatan in the sixteenth
century. Ironically, Landa was proud of having burned all the texts using
this writing, the better to bring the natives into the embrace of the Catholic
301
DOCUMENTARY SOURCES OF MAYA HISTORY
Church. He wrote his chronicle in order to explain why he had destroyed all
those precious painted codices - but thereby unwittingly preserved the
basic elements of one of the most important pre-Columbian civilisations
of the Americas.
The discovery of this sixteenth-century manuscript aroused great
interest, because the glyphs copied down by Landa were similar to the
carved shapes on the ruins found in the virgin jungle of Central America by
Stephens and later explorers. It provided solid evidence of the cultural
connection between the sixteenth-century population of the Yucatan penin-
sula and the builders of the lost cities of the jungle, both in Yucatan and
further south.
Landa’s manuscript is a major source for the history of the Maya, but it
is not the only one. Much was also written down by the natives themselves,
who were taught by Spanish missionaries to read and write in the Latin
alphabet, which they then also used for writing in their own tongue.
Although the teaching was intended to support the spread of Christianity,
it was also used - inevitably - to set down the fast-disappearing oral
traditions of the local populations.
A good number of anonymous accounts of this kind have survived, and
give a reflection of the history, traditions and customs of the indigenous
peoples of Spanish Central America. From the Guatemalan uplands comes
the manuscript known as Popol Vuh, which contains fragments of the
mythology, cosmology and religious beliefs of the Quiche Maya; and it
was in the same area that the Annals of the Cakchiquels were found, which
provide in addition the story of the tribe of that name during the Spanish
conquest. The Books of Chil&m Balam are a collection of native chronicles
from Yucatan, and are named after a class of “Jaguar Priests”, famed for
their prophetic powers and their mastery of the supernatural. Fourteen
of these manuscripts go a long way back in history; though they deal
mostly with traditions, calendars, astrology, and medicine, three of them
mention historical events that can be precisely situated in the year 1000 CE.
Some parts of the Childm Balam may even have been copied directly from
ancient codices.
The ancient Maya codices used parchment, tree bark, or mashed
vegetable fibres strengthened with glue to provide a writing surface. The
glyphs were written with a brush pen dipped in wood ash, and then
coloured with dyes from various animal and vegetable sources. The pages
were glued together, then folded like a concertina and bound between
wood or leather covers, much like a book. Three of them miraculously
escaped the attention of the conquistadors, and found their way back to
Europe, where they are now known by the names of the cities where they
are kept: the Dresden Codex (in the Sachsische Landesbibliothek, Dresden,
Germany) is an eleventh-century copy of an original text drafted in the
classical period, and deals with astronomy and divination; the Codex
Tro-Cortesianus (American Museum, Madrid) is less elaborate and was
probably composed no earlier than the fifteenth century; and the Paris
Codex (Bibliotheque nationale, Paris), likewise from the late period, gives
illustrations of ceremonies and prophecies.
Despite these various documentary sources, much of Maya civilisation
remains mysterious and unexplained to this day.
AZTEC CIVILISATION
The legendary homeland of the Aztecs, according to the few manuscripts
that have survived and the tales of Spanish conquerors, was called Aztlan
and was located somewhere in northwestern Mexico, maybe in Michoacan.
In a cave in Aztlan they are supposed to have found the “colibri
wizard”, Huitzilopochtli, who gave such good advice that he became the
Aztecs’ tribal god. Then began their long migration, by way of Tula and
Zumpango (on the high plateau), and the Chapultepec, where they lived
peaceably for more than a generation. Thereafter, they were defeated in
battle and exiled to the infertile lands of Tizapan, infested with poisonous
snakes and insects. A group of rebels took refuge on the islands in Lake
Texcoco, where, in 1325 CE (or 1370, according to more recent calcula-
tions), they founded the city ofTenochtitlan, which has become present-day
Mexico City.
Within a century Tenochtitlan became the centre of a vast empire. The
Aztec King Itzcoatl subdued and enslaved most of the tribes in the valley;
then under Motecuhzoma I (1440-1472) they battled on into the Puebla
region in the south. Axayacatl, son of Motecuhzoma, led the Aztec armies
even further south, as far as Oaxaca; he also attacked, but failed to conquer
the Matlazinca and Tarasques in the west.
By the time the Spaniards arrived in 1519, the Aztecs possessed most
of Mexico, and their language and religion held sway over a vast territory
stretching from the Atlantic to the Pacific Oceans and from the northern
plains to Guatemala. The name of the king, Motecuhzoma (Europeanised
as “Montezuma") struck fear from one end to another of the empire; Aztec
traders, with great caravans of porters, scoured the entire kingdom; and
taxes were levied everywhere by the king’s administrators. It was a relatively
recent civilisation, at the height of its wealth and glory.
THE AMAZING ACHIEVEMENTS OF THE MAYA
302
Fig. 22.7. Page 1 of the Codex Mendoza (post- conquest). Through a number of Aztec hieroglyphs,
this illustration sums up Aztec history and relates the founding of the city of Tenochtitldn.
It was also a very violent civilisation. The continual military campaigns
were for the most part undertaken in the service of the Aztec gods - for
every aspect of Aztec history, culture, and society can only be understood
in terms of a tyrannical religion which left no space for anything
resembling hope or even virtue in the Christian sense. The main purpose
of war-making was to seize prisoners who could be used in the ritual sacri-
fices. About 20,000 people were thus slaughtered every year in the service
of magic. The Aztecs believed that the Sun and the Earth (both considered
gods) required constant replenishment with human blood, or else the
world’s mechanism would cease to function. The slaughter also had a
straightforward nutritional use, for only the victims’ hearts were reserved
for the gods’ consumption. Human legs, arms and rumps were treated
much as we treat butcher’s meat, and sold retail at Aztec markets, for
ordinary consumption.
Beside the priestly and the warrior castes, there were also castes of
artisans and traders, organised into a set of guilds. The main market
of the empire was at Tlatelolco, Tenochtitlan’s twin town, founded in
1358, where merchandise of every sort, brought from the four corners
of the Aztec empire, was traded. The records of the taxes levied by the
imperial administrators of the Tlatelolco market have survived, and give
a good picture of the wealth and variety of trade in the Aztec empire:
gold, silver, jade, shells, feathers for ceremonial wear, ceremonial garb,
shields, raw cotton for spinning, cocoa beans, coats, blankets, embroidered
cloth, etc.
The empire and the whole of Aztec civilisation collapsed in the early
sixteenth century. “Stout” Cortez, accompanied by a mere handful of
men armed with guns, landed at Vera Cruz and marched towards the
highlands. He gained the support of tribes that were the Aztecs’ enemies
or their subjects, and from them acquired supplies and reinforcements.
After a violent struggle, Cortez seized Tenochtitlan on 13 August 1521 , and
destroyed Aztec civilisation for ever.
AZTEC WRITING
At the time of the Spanish Conquest, Mexican script was a mixture of
ideographic and phonetic representation, with some more or less “pictorial”
signs designating directly the beings, objects or ideas that they resembled,
and others (including the same ones) standing for the sound of the
thing that they represented. Names of people and places were written in
the manner of a rebus or puzzle (rather approximate ones, in fact, since the
writing took no account of case-endings). For example, the name of the city
of Coatlan (literal meaning: “snake-place”) was represented by the drawing
of a snake (= coat!) together with the sign for “teeth”, pronounced tlan.
The name of the city of Coatepec (literal meaning: “snake-mountain-place”)
was represented similarly by a snake (= coatl ) together with the sign for
“mountain” (tepetl).
303
HOW THE MAYA DID THEIR SUMS
COATLAN
COA T. :
vt-j snake
TLAN
"teeth”
Fig. 22.8. Examples of Aztec names written in the form of a rebus
Aztec script is used in a number of Mexican documents written just
before and just after the Spanish conquest. Some of these deal with matters
of religion, ritual, prophecy, and magic; others are narratives of real or
mythical history (tribal migrations, foundations of cities, the origins and
history of different dynasties, etc.); and others are registers of the vast
taxes paid in kind (goods, food supplies, and men) by the subject cities to
the lords of Tenochtitlan.
Fig. 22.9. Codex Mendoza (folio 52 r), showing the tributes to be paid by seven Mexican cities
to Tenochtitlan
The most important by far of these Aztec documents in the Codex
Mendoza, drawn up by order of Don Antonio de Mendoza, the first Viceroy
of New Spain, and sent to the court of Spain. It contains three parts, dealing
respectively with the conquests of the Aztecs, the taxes that they levied
on each of the conquered towns, and with the life-cycle of an Aztec,
from birth through education, punishment, recreation, military insignia,
battles, the genealogy of the royal family, and even the ground-plan of
Motecuhzoma’s palace ... It was written in a period of ten days (since the
fleet was about to put to sea) in the native language and script, but with a
simultaneous commentary on the meaning of every detail in Spanish. And
it is largely thanks to the Spanish commentary that we can now seek to
understand Aztec numerals . . .
HOW THE MAYA DID THEIR SUMS
Most of what could be “read” in Maya texts and inscriptions until very
recently consists of numerical, astronomical and calendrical information.
However, before we can approach Mayan arithmetic, we need to know what
their oral numbering system was.
Like all the other peoples of pre-Columbian Central America, the Maya
counted not to base 10, but to base 20. As we now know, this was due to their
ancestors’ habits of using their toes as well as their fingers as a model set.
The language of the Maya and various dialects of it are still in use nowa-
days in the Mexican states of Yucatan, Campeche, and Tabasco, in a part of
the Chiapas and the region of Quintana Roo, in most of Guatemala and in
parts of Salvador and Honduras. The names of the numbers are as follows:
1
hun
11
buluc
2
ca
12
lahca ( lahun + ca =
10 + 2)
3
ox
13
ox- lahun
(3 + 10)
4
can
14
can-lahun
(4 + 10)
5
ho
15
ho-lahun
(5 + 10)
6
uac
16
uac-lahun
(6 + 10)
7
uuc
17
uuc-lahun
(7 + 10)
8
uaxac
18
uaxac-lahun
(8 + 10)
9
bolon
19
bolon-lahun
(9 + 10)
10
lahun
Fig. 22.10A.
The units up to and including 10 thus have their own separate names,
and above that number are made of additive compounds that rely on 10
as an auxiliary base. The one exception is the name of the number 11,
buluc, which was probably invented to avoid confusion of a regular form
hun-lahun, “one + ten” with hun-lahun, in the meaning “a ten”.
Numbers from 20 to 39 are expressed as follows:
20
kun kal
score ( hun uinic, “one man”, in some dialects)
21
hun tu-kal
one (after) twentieth
22
ca tu-kal
two (after) twentieth
23
ox tu-kal
three (after) twentieth
24
can tu-kal
four (after) twentieth
25
ho tu-kal
five (after) twentieth
26
uac tu-kal
six (after) twentieth
27
uuc tu-kal
seven (after) twentieth
28
uaxac tu-kal
eight (after) twentieth
29
bolon tu-kal
nine (after) twentieth
30
lahun ca kal
ten-two-twenty
31
buluc tu-kal
eleven (after) twentieth
32
lahca tu-kal
twelve (after) twentieth
33
ox- lahun tu-kal
thirteen (after) twentieth
COATEPEC
COATL
“snake”
TEPETL
"mountain”
THE AMAZING ACHIEVEMENTS OF THE MAYA
304
34
can-lahun tu-kal
35
holhu ca kal
36
uac-lahun tu-kal
37
uuc-lahun tu-kal
38
uaxac-lahun tu-kal
39
bolon-lahun tu-kal
fourteen (after) twentieth
fi ftee n - two- twen ty
sixteen (after) twentieth
seventeen (after) twentieth
eighteen (after) twentieth
nineteen (after) twentieth
Fig. 22 . 10 B.
So, as a general rule, these numbers are formed by inserting the ordinal
prefix tu between the name of the unit and the name of the base, 20. But
there are two exceptions:
30 “ten-two-twenty”, instead often (after) twentieth”
35 “fifteen-two-twenty”, instead of “fifteen (after) twentieth”
These two anomalies cannot be explained by addition or by subtraction,
since 35 is neither 15 + (2 x 20) nor (2 x 20) - 15. Moreover, the irregular-
ity is repeated in the next sequence of numbers, which begin “one-three-
twenty”, “two-three-twenty” and so on.
40
cakal
two score
41
hurt tu-y-ox kal
one - third score
42
ca tu-y-ox kal
two - third score
43
ox tu-y-ox kal
three - third score
44
can tu-y-ox kal
four - third score
58
uaxac-lahun tu-y-ox kal
eighteen - third score
59
bolon-lahun tu-y-ox kal
nineteen - third score
60
ox kal
three score
61
hun tu-y-can kal
one - fourth score
62
ca tu-y-can kal
two - fourth score
78
uaxac-lahun tu-y-can kal
eighteen - fourth score
79
bolon-lahun tu-y-can kal
nineteen - fourth score
80
can kal
four score
81
hun tu-y-ho-kal
one - fifth score
82
ca tu-y-ho-kal
two - fifth score
98
uaxac-lahun tu-y-ho-kal
eighteen - fifth score
99
bolon-lahun tu-y-ho-kal
nineteen - fifth score
100
ho kal
five score
400
hun bak
one four-hundreder
8,000
hun pic
one eight-thousander
160.000
hun calab
one hundred-and-sixty-thousander
Fig. 22 . ioc.
To work out how such a numbering system might have come into being,
we have to imagine something like the following scene taking place several
thousand years ago somewhere in Central America.
Fig. 22.li.
As they prepare to set off to fight a skirmish, warriors line up a few men
to serve as “counting machines” or model sets, and one of the men proceeds
to check off the number of warriors in the group. As the first one files
past, the checker touches the first finger of the first “counting machine”,
then for the second he touches the second finger, and so on up to the tenth.
The “accountant” then moves on to the toes of the first model set, up to
the tenth, which matches the twentieth warrior that has filed past. For the
next man, the accountant proceeds in exactly the same way using
the second of the “counting machines”, and when he gets to the last toe
of the second man, he will have checked off forty warriors. He moves on to
the third man, which would take him up to sixty, and so on until the count
is finished.
Let us suppose that there are 53 men in the group. The accountant
will reach the third toe of the first foot of the third man, and will announce
the result of the count in something like the following manner: “There
are as many warriors as make three toes on the first foot of the third man”.
But the result could also be expressed as: “Two hands and three toes of
the third man” or even “ten-and-three of the third twenty”. If applied to
English, such a system would produce a set of number-names of the
following sort:
305
HOW THE MAYA DID THEIR SUMS
1 one
11 ten-one
2 two
12 ten-two
3 three
13 ten-three
4 four
14 ten-four
5 five
15 ten-five
6 six
16 ten-six
7 seven
17 ten-seven
8 eight
18 ten-eight
9 nine
19 ten-nine
10 ten
20 one man
Style A
Style B
one after the first man
21
one of the second man
two after the first man
22
two of the second man
three after the first man
23
three of the second man
four after the first man
24
four of the second man
five after the first man
25
five of the second man
six after the first man
26
six of the second man
seven after the first man
27
seven of the second man
eight after the first man
28
eight of the second man
nine after the first man
29
nine of the second man
ten after the first man
30
ten of the second man
ten-one after the first man
31
ten-one of the second man
ten-two after the first man
32
ten-two of the second man
ten-three after the first man
33
ten-three of the second man
ten-four after the first man
34
ten-four of the second man
ten-five after the first man
35
ten-five of the second man
ten-nine after the first man
39
ten-nine of the second man
two men
40
two men
one after the second man
41
one of the third man
two after the second man
42
two of the third man
three after the second man
43
three of the third man
ten-one after the second man
51
ten-one of the third man
ten-two after the second man
52
ten-two of the third man
ten-three after the second man
53
ten-three of the third man
ten-nine after the second man
59
ten-nine of the third man
three men
60
three men
one after the third man
61
one of the fourth man
two after the third man
62
two of the fourth man
nineteen after the third man
79
nineteen of the fourth man
four men
80
four men
Fig. 22.12.
It is now easy to see how the irregularities of the Maya number-names
arose. The numbers 21 to 39 (except 30 and 35) are expressed in terms
of Style A: 21 = hun tu-kal = “one (after) the twentieth” or “one (after the)
first twenty”, 39 = bolon-lahun tu-kal = “nine-ten (after the) twentieth” or
nine-ten (after the) first twenty”; whereas the numbers from 41 to 59, 61
to 79, etc. as well as the numbers 30 and 35, are expressed in terms of
Style B; 30 = lahun ca kal = “ten-two-twenty” or “ten of the second twenty”,
and so forth.
The Maya were not alone in counting in this way. The number 53, for
instance, is expressed as follows:
• by the Inuit of Greenland, as imp pingajugsane arkanek
pingasut, literally, “of the third man, three on the first foot”;
• by the Ainu of Japan and Sakhalin, as wan-re wan-e-re-
hotne, literally “three and ten of the third twenty” [see K. C.
Kyosuke and C. Mashio (1936)];
• by the Yoruba (Senegal and Guinea) as eeta laa din ogota,
literally “ten and three before three times twenty” [see C.
Zaslavsky (1973)];
• and other instances of similar systems can be found
amongst the Yedo (Benin) and the Tamanas of the Orinoco
(Venezuela).
THE “ordinary” NUMBERS OF THE MAYA
Now that we can see the reasons for the irregularities of the Maya number-
name system, we can try to grasp their written numerals. Or rather, we
would have been able to, had the Spanish Inquisition not stupidly destroyed
almost every trace of it. So we are forced to take a step backwards.
Amongst the cultures of pre-Columbian Central America there are four
main types of writing system: Maya, Zapotec (in the Oaxaca Valley), Mixtec
(southwest Mexico), and Aztec (around Mexico City). Zapotec is the oldest,
probably dating from the sixth century BCE, and Aztec is the most recent
(see above). Now, although these scripts served to represent languages
belonging to quite different linguistic families, they possess a number
of graphical features in common, including (as far as Aztec, Mixtec and
Zapotec are concerned) the basic features of numerical notation.
In Aztec vigesimal numerals, for instance, the unity was represented by
a dot or circle, the base by a hatchet, the square of the base (20 x 20 = 400)
by a sign resembling a feather, the cube of the base (20 x 20 x 20 = 8,000)
by a design symbolising a purse.
O or •
P
|
♦ - A
1
20
400
8,000 ®
Fig. 22.13. Aztec numerals
THE AMAZING ACHIEVEMENTS OF THE MAYA
The numeral system relied on addition: that is to say, numbers were
expressed by repeating the component figures as many times as necessary.
To express 20 shields, 100 sacks of cocoa beans, or 200 pots of honey, for
example, one, five or ten “hatchets” were attached to the pictogram for the
relevant object:
cocoa beans 200 pots of honey
Fig. 22.14.
To record 400 embroidered cloaks, 800 deerskins or 1,600 cocoa bean-
pods, one, two or four “feather” signs were similarly attached to the
respective object-sign:
Fig. 22.15.
This was the way that the scribe of the Codex Mendoza recorded the
taxes that were paid once, twice or four times a year by the subject-cities
to the Aztec lords of Tenochtitlan. The page shown in Fig. 22.9 above
gives the taxes due from seven cities in one province, and expresses them
as follows:
306
1. Left column : the names of the seven cities, expressed by combinations of
signs in the manner of a rebus:
-<!► fi k «Q £ * &
Tochpan Tlalti^apan Civateopan Papantla Ocelotepec Miaua apan Mictlan
Fig. 22.16 a.
2. Line 1, horizontally:
400 400 400 400 400
Fig. 22.16B.
• 400 cloaks of black-and-white chequered cloth
• 400 cloaks of red-and-white embroidered cloth (worn by the
lords of Tenochtitlan)
• 400 loincloths
• 2 sets of 400 white cloaks, size 4 braza (a unit of length
indicated by the finger-sign)
3. Line 2
400 400 400 400 400
Fig. 22 .i6c.
• 2 sets of 400 orange-and-white-striped cloaks, size 8 braza
• 400 white cloaks, size 8 braza
• 400 polychrome cloaks, size 2 braza
• 400 women’s skirts and tunics
:i0 7
4. Line 3
80 80 80 400 400
Fig. 22.16D.
• 3 sets of 80 coloured and embroidered cloaks (as worn by the
leading figures of the capital)
• 2 sets of 400 bundles of dried peppers (used amongst other
things to punish young people for breaking rules)
5. Line 4
Fig. 22.i6e.
• 2 ceremonial costumes, 20 sacks of down, and 2 strings of
jade pearls
6. Last line
Fig. 22.i6f.
• 2 shields, a string of turquoise, and 2 plates with turquoise
incrustation
The Codex Telleriano Remensis, another post-conquest document in Aztec
script, also provides examples of numerals:
THE “ORDINARY” NUMBERS OF THE MAYA
Fig. 22.17. Detail from a page of the Aztec Codex Telleriano Remensis
What this page says in effect is that 20,000 men from the subject
provinces were sacrificed in 1487 CE to consecrate a new building. The
number was written by the native scribe thus:
16,000 4,000
Fig. 22. i8.
The Spanish annotator, however, made a mistake in transcribing this
number: as he did not know the meaning of the two purses worth 8,000
each, he “translated” only the ten feathers, giving a total of 4,000.
Aztec numerals were identical to those of the Zapotecs and Mixtecs,
as the following painting shows. It was done in Zapotec by order of
the Spanish colonial authorities in Mexico in 1540 CE and shows the
numbering conventions common to Zapotec, Mixtec and Aztec cultures:
THE AMAZING ACHIEVEMENTS OF THE MAYA
308
Fig. 22.19. Numerical representations from a Zapotec painting made by order of the Spanish
colonial authorities in 1540 . It shows graphical conventions common to Zapotec, Mixtec, and Aztec
numeral systems.
So it seems certain that “ordinary” Maya numerals must also have been
strictly vigesimal and based on the additive principle. It can be safely
assumed that a circle or dot was used to represent the unity (the sign is
common to all Central American cultures, and derives from the use of
the cocoa bean as the unit of currency), that there was a special sign, maybe
similar to the “hatchet” used by other Central American cultures, for the
base (20), and other specific signs for the square of the base (400) and
the cube (8,000), etc.
As we shall see below, it is also quite probable that, like the Zapotecs,
the Maya introduced an additional sign for 5, in the form of a horizontal
line or bar.
Even though no trace of it remains, we can reasonably assume that the
Maya had a numeral system of this kind, and that intermediate numbers
were figured by repeating the signs as many times as was needed. But that
kind of numeral system, even if it works perfectly well as a recording device,
is of no use at all for arithmetical operations. So we must assume that
the Maya and other Central American civilisations had an instrument
similar to the abacus for carrying out their calculations.
The Inca of South America certainly did have a real abacus, as shown
in Fig. 22.20. The Spaniards were amazed at the speed with which Inca
accountants could resolve complex calculations by shifting ears of maize,
beans or pebbles around twenty “cups” (in five rows of four) in a tray or
table, which could be made of stone, earthenware or wood, or even just
laid out in the ground. Inca civilisation was obviously quite different from
the Maya world, but it did have one thing in common: a method of record-
ing numbers and tallies (the quipus, or knotted string) that was entirely
unsuitable for performing arithmetical operations. For that reason the Inca
were obliged to devise a different kind of operating tool.
Fig. 22.20. Document proving the use of the abacus amongst the Peruvian and Ecuadorian Incas.
It shows a quipucamayoc manipulating a quipu and on his right a counting table. From the
Peruvian Codex ofGuaman Poma de Ayala ( 16 th century), Royal Library, Copenhagen
THE PLACE- VALUE SYSTEM OF “LEARNED”
MAYA NUMERALS
The only numerical expressions of the Maya that have survived are in fact
not of the ordinary or practical kind, but astronomical and calendrical
calculations. They are to be found in the very few Maya manuscripts that
exist, and most notably in the Dresden Codex, an astronomical treatise
copied in the eleventh century CE from an original that must have been
three or four centuries older.
What is quite remarkable is that Maya priests and astronomers used a
numeral system with base 20 which possessed a true zero and gave a
specific value to numerical signs according to their position in the written
expression. The nineteen first-order units of this vigesimal system were
represented by very simple signs made of dots and lines: one, two, three
and four dots for the numbers 1 to 4; a line for 5, one, two, three and four
dots next to the line for 6 to 9; two lines for 10, and so on up to 19:
309
Fig. 22.21. The first nineteen units in the numeral system of the Maya priests
Numbers above 20 were laid out vertically, with as many “floors” as there
were orders of magnitude in the number represented. So for a number
involving two orders, the first order-units were expressed on the first or
“bottom floor” of the column, and the second-order units on the “second
floor”. The numbers 21 (= 1 x 20 + 1) and 79 (3 x 20 + 19) were written thus:
••• 3
••••
Fig. 22.22. Fig. 22.23.
The “third floor” should have been used for values twenty times as
great as the “second floor” in a regular vigesimal system. Just as in our
decimal system the third rank (from the right) is reserved for the hun-
dreds (10 x 10 = 100), so in Maya numbering the third level should
have counted the “four hundreds” (20 x 20 = 400). However, in a curious
irregularity that we will explain below, the third floor of Mayan astronomi-
cal numerals actually represented multiples of 360, not 400. The following
expression:
THE PLACE-VALUE SYSTEM OF LEARNED MAYA NUMERALS
corresponds to
12 x 360 + 3 x 20 + 19
Fig. 22.24.
actually meant 12 x 360 + 3 x 20 + 19 = 4,399, and not 12 x 400 + 3 x 20
+ 19 = 4,879!
Despite this, higher floors in the column of numbers were strictly
vigesimal, that is to say represented numbers twenty times as great as the
immediately preceding floor. Because of the irregularity of the third posi-
tion, the fourth position gave multiples of 7,200 (360 x 20) and the fifth
gave multiples of 144,000 (20 x 7,200) - and not of 8,000 and 160,000.
A four-place expression can thus be resolved by means of three multipli-
cations and one addition, thus:
Fig. 22.25.
1 (= 1 x 7,200)
17 (= 17 x 360)
8 (=8x20)
15 (=15x1)
= 1 x 7,200 + 17 x 360 + 8 x 20 x 15
So that each numeral would be in its right place even when there were
no units to insert in one or another of the “floors”, Mayan astronomers
invented a zero, a concept which they represented (for reasons we cannot
pierce) by a sign resembling a snail-shell or sea-shell.
For instance, a number which we write as 1,087,200 in our decimal
place-value system and which corresponds in Mayan orders of magnitude
to 7 x 144,000 + 11 x 7,200 and no units of any of the lower orders of
360, 20 or 1, would be written in Maya notation thus:
•1 7
Fig. 22.26.
1,087,200
THE AMAZING ACHIEVEMENTS OF THE MAYA
Glyphs representing sea
shells?
■gs-v
<3t>
<e>
<0
<S5>
^ <&
<^2>
♦
•
<£5>
<SB>
♦
♦
<a> <s>-
Glyphs representing snail-shells?
Another shape
U
if
Fig. 22 . 27 .
We can see the system in operation in these very interesting numerical
expressions in the Dresden Codex:
Fig. 22 . 28 . The Dresden Codex, p. 24 (part). Sdchsische Landesbibliothck, Dresden
310
Fig. 22.29. Transcriptions of the numerals on the right-hand side of Fig. 22.28
Each of these expressions in Mayan astronomical notation refers to a
number of days (we know this from the context) and gives the following set
of equivalences:
A =
[8;
2; 0] =
2,920 =
1 x 2,920 =
5x584
B =
[16;
4; 0] =
5,840 =
2 x 2,920 =
10 x 584
C =
[l;
4;
6; 0] =
8,760 =
3 x 2,920 =
15 x 584
D =
[l;
12;
8; 0] =
11,680 =
4 x 2,920 =
20 x 584
E =
[2;
0;
10; 0] =
14,600 =
5 x2,920 =
25 x 584
F =
[2;
8;
12; 0] =
17,520 =
6 x 2,920 =
30 x 584
G =
[2:
16;
14; 0] =
20,440 =
7x2,920 =
35 x 584
H =
[3;
4;
16; 0] =
23,360 =
8x2,920 =
40 x 584
1 =
[3;
13;
0; 0] =
26,280 =
9 x2,920 =
45 x 584
J =
[4;
1;
2; 0] =
29,200 =
10 x 2,920 =
50 x 584
K =
[4;
9;
4; 0] =
32,120 =
11 x 2,920 =
55 x 584
L =
[4;
17;
6; 0] =
35,040 =
12 x 2,920 =
60 x 584
311
So this series is nothing other than a table of the synodic revolutions of
Venus, calculated by Mayan astronomers as 584 days.
This gives us two indisputable proofs of the mathematical genius of
Maya civilisation:
• it shows that they really did invent a place-value system;
• it shows that they really did invent zero.
These are two fundamental disoveries that most civilisations failed
to make, including especially Western European civilisation, which had to
wait until the Middle Ages for these ideas to reach it from the Arabic world,
which had itself acquired them from India.
One problem remains: why was this system not strictly vigesimal, like
the Mayas’ oral numbering? For instead of using the successive powers
of 20 (1, 20, 400, 8,000, etc.), it used orders of magnitude of 1, 20,
18 x 20 = 360, 18 x 20 x 20 = 7,200, etc. In short, why was the third “floor”
of the system occupied by the irregular number 360?
If Maya numerals had been strictly vigesimal, then its zero would have
acquired operational power: that is to say, adding a zero at the end of a
numerical string would have multiplied its value by the base. That is how it
works in our system, where the zero is a true operational sign. For instance,
the number 460 represents the product of 46 multiplied by the base.
For the Maya, however, [1; 0; 0] is not the product of [1; 0] multiplied by
the base, as the first floor gives units, the second floor gives twenties, but
the third floor gives 360s. [1; 0] means precisely 20; but [1; 0; 0] is not 400
(20 X 20 + 0 + ), but 360. The number 400 had to be written as [1; 2; 0] or
(1 x 360 + 2 X 20 + 0):
• 1
• 1
• 1 i
0
0
• • 2 |
0
0 i
1
*
20
360
400
20 X 20
Fig. 22.30.
THE PLACE-VALUE SYSTEM OF “LEARNED” MAYA NUMERALS
This anomaly deprived the Maya zero of any operational value, and
prevented Mayan astronomers from exploiting their discovery to the full.
We must therefore not confuse the Maya zero with our own, for it does not
fulfil the same role at all.
A SCIENCE OF THE HIGH TEMPLES
To understand the odd anomaly of the third position in the Maya place-
value system we have to delve deep into the very sources of Maya
mathematics, and make a long but fascinating detour into Maya mysticism
and its reckoning of time.
Maya learned numerals were not invented to deal with the practicalities
of everyday reckoning - the business of traders and mere mortals - but to
meet the needs of astronomical observation and the reckoning of time.
These numerals were the exclusive property of priests, for Maya civilisation
made the passing of time the central matter of the gods.
Maya science was practised in the high temples: astronomy was what the
priests did. Mayan achievements in astronomy, including the invention of
one of the best calendars the world has ever seen, were part and parcel
of their mystical and religious beliefs.
The Maya did not think of time as a purely abstract means of ordering
events into a methodical sequence. Rather, they viewed it as a super-
natural phenomenon laden with all-powerful forces of creation and
destruction, directly influenced by gods with alternately kindly and
wicked intentions. These gods were associated with specific numbers,
and took on shapes which allowed them to be represented as hieroglyphs.
Each division of the Maya calendar (days, months, years, or longer
periods) was thought of as a “burden” borne on the back of one or another
of the divine guardians of time. At the end of each cycle, the “burden” of
the next period of time was taken over by the god associated with the next
number. If the coming cycle fell to a wicked god, then things would get
worse until such time as a kindly god was due to take over. These curious
beliefs supported the popular conviction that survival was impossible
without learned mediators who could interpret the intentions of the
irascible gods of time. The astronomer-priests alone could recognise
the attributes of the gods, plot their paths across time and space, and
thus determine times that would be controlled by kindly gods, or (as was
more frequent) times when the number of kindly gods would exceed
that of evil gods. It was an obsession for calculating periods of luck
and good fortune over long time-scales, in the hope that such foreknowl-
edge would enable people to turn circumstances to their advantage. [See
C. Gallenkamp (1979)]
THE AMAZING ACHIEVEMENTS OF THE MAYA
312
Fig. 22 . 31 . The cyclical conception of events in the Mayas ' mystical thinking. The inexorable cycle
ofChac, god of rain, planting a tree, followed by Ah Puch, god of death, who destroys it, and by Yum
Kax, god of maize and of agriculture, who restores it. From the Codex Tro-Cortesianus, copy from
Girard (1972), p. 241, Fig. 61
The priests were thus the possessors of the arcana of time and of the
foretelling of the gods bearing the burden of particular times. Mysticism,
religion and astronomy formed a single, unitary sphere which gave the
priestly caste enormous power over the people, who needed priestly
mediation in order to learn of the mood of the gods at any given moment.
So despite its amazing scientific insights, Mayan astronomy was very
different from what we now imagine science to be: as Girard puts it [R.
Girard (1972)], its main purpose was to give mythical interpretations
of the magical powers that rule the Universe.
(
or
IMIX
BD-©
CHUEN
d§° r ©
CIB
(
or (3^
IK
f§"©
MANIK
EB
CABAN
(
ill (SSl
JjgfJ or kj
AKBAL
HI"©
LAMAT
(§) or (5^
BEN
ETZNAB
1
m° r ©
KAN
(300
MULUC
SI"®
(Sl"0>
CAUAC^
§) or@
CHICCHAN
oc
m 'to
MEN
IDh©
AHAU
Fig . 22.32. Hieroglyphs for the twenty days of the Maya calendar, with their names in the Yucatec
language. [See Gallenkamp (1979), Fig. 9; Peterson (1961), Fig. 55J
THE MAYA CALENDAR
The Maya had two calendars, which they used simultaneously: the Tzolkin -
the “sacred almanac” or “magical calendar” or “ritual calendar”, used for
religious purposes; and the Haab, which was a solar calendar.
The religious year of the Maya consisted of twenty cycles of thirteen days,
making 260 days in all. It had a basic sequence of twenty named days in
fixed order:
Imix
Cimi
Chuen
Cib
Ik
Manik
Eb
Caban
Akbal
Lamat
Ben
Etznab
Kan
Muluc
lx
Cauac
Chicchan
Oc
Men
Ahau
Each day had its distinct hieroglyph, which also represented directly
the corresponding deity or sacred animal or object. As J. E. Thompson
explains, prayers were addressed to the days, each of which was the incar-
nation of a divinity, such as the sun, the moon, the god of maize, the god of
death, the Jaguar, etc.
Each of the days was also associated with a number-sign, in the range
1 to 13 (itself associated with thirteen Maya gods of the “upper world” or
Oxlahuntiku).
In the first cycle, the first day was associated with the number 1, the
second day with the number 2, and so on to the thirteenth day. The
numbering then started over, so that the fourteenth day was associated
with the number 1, the fifteenth with the number 2, and the last day of
the first cycle had number 7.
The second cycle thus began with 8 and reached 13 with the sixth day, so
that the numbering began again at 1 with the seventh day of the second cycle.
Thus it took thirteen cycles for the numbering to come back to where
it started, with day one counting once again as 1. As there are 13 X 20
possible pairings of the sets 1-13 and 1-20, the whole series of cycles lasted
260 days.
Each day of the religious year therefore had a unique name consisting
of its hieroglyph together with its number resulting from the cyclical
recurrence explained above. So a day-hieroglyph plus number gives an
unambiguous identification of any day in the religious year. The following
expressions, for instance:
313
THE MAYA CALENDAR
13 CHUEN 4 IMIX
Fig. 22.33.
specify the 91st and 121st days of a religious year that begins on 1 Imix.
(Fig. 22.34 below shows the whole cycle.)
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
IMIX
1
8
2
9
3
10
4
11
5
12
6
13
7
IK
2
9
3
10
4
11
5
12
6
13
7
1
8
AKBAL
3
10
4
11
5
12
6
13
7
1
8
2
9
KAN
4
11
5
12
6
13
7
1
8
2
9
3
10
CHICCHAN
5
12
6
13
7
1
8
2
9
3
10
4
11
CIM I
6
13
7
1
8
2
9
3
10
4
11
5
12
MANIK
7
1
8
2
9
3
10
4
11
5
12
6
13
LAM AT
8
2
9
3
10
4
11
5
12
6
13
7
1
MULUC
9
3
10
4
11
5
12
6
13
7
1
8
2
oc
10
4
11
5
12
6
13
7
1
8
2
9
3
CHUEN
11
5
12
6
13
7
1
8
2
9
3
10
4
EB
12
6
13
7
1
8
2
9
3
10
4
11
5
BEN
13
7
1
8
2
9
3
10
4
11
5
12
6
IX
1
8
2
9
3
10
4
11
5
12
6
13
7
MEN
2
9
3
10
4
11
5
12
6
13
7
1
8
CIB
3
10
4
11
5
12
6
13
7
1
8
2
9
CABAN
4
11
5
12
6
13
7
1
8
2
9
3
10
ETZNAB
5
12
6
13
7
1
8
2
9
3
10
4
11
CAUAC
6
13
7
1
8
2
9
3
10
4
11
5
12
AHAU
7
1
8
2
9
3
10
4
11
5
12
6
13
Fig. 22.34. The 260 consecutive days of the Maya liturgical year
Each day of the religious year had its own specific character. Some were
propitious for marriages or military expeditions, others ruled out such
events. More generally, an individual’s character and prospects were
indissolubly linked to the character of the day of his birth, a belief that is
still held by many Central American peoples, notably in the Guatemalan
uplands.
Why did the pre-Columbian civilisations of Central America choose 260
as the number of days in their liturgical calendars? F. A. Peterson (1961)
pointed out that the difference between the religious year (260) and the
solar year (365) is 105 days. Moreover, between the tropic of Cancer and the
tropic of Capricorn, the sun is at the zenith twice in every year, at intervals
of 105 and 260 days precisely. At Copan, an ancient Maya city in Honduras,
the relevant dates for the sun passing through its zenith are 13 August
and 30 April. The rainy season begins straight after the sun passes through
its “spring” zenith; 105 days later, the sun passes through its “autumn”
zenith. So the year could be divided into a period of planting and growth
that lasted 105 days, and then a period of harvesting and religious feasts
that lasted 260.
This astronomical observation, even if it has not been accepted as
the ultimate source of the Maya calendar, is certainly very interesting.
Unfortunately the correlation of the sun’s zenith with the rainy season only
fits at Copan, which is on the fringes of the Maya area.
Other scholars have pointed out that 260 must be thought of as the
product of 13 and 20, the divine (since there are 13 divinities in the “upper
world” of the Maya) and the human (since Maya numbering is vigesimal,
and the name of the number 20, uinic, means “ a man”).
Alongside the ritual calendar, the Maya used a solar-year calendar called
the Haab, and referred to as the “secular” or “civil” or “approximate” calen-
dar. It had a year of 365 days divided into eighteen uinal (twenty-day
periods), plus a short “extra” period of five days added at the end of the
eighteenth uinal. The names of the Maya twenty-day “months” were:
Pop
Yaxkin
Mac
Uo
Mol
Kankin
Zip
Chen
Muan
Zotz
Yax
Pax
Tzec
Zac
Kayab
Xul
Ceh
Cumku
These names referred to various agricultural or religious events, and they
were represented by the hieroglyphs of the tutelary god or animal-spirit
associated with the event.
THE AMAZING ACHIEVEMENTS OF THE MAYA
ZIP MOL MAC CUMKU
ZOTZ CHEN KANKIN
TZEC YAX MUAN
UAYEB
Literally: “That which has no name”
Glyph and name of the five-day period regularly added to the
eighteenth twenty-day “month” to make
up the Haab of 365 days.
Fig. 22.35. Glyphs and names of the eighteen 20-day “months" of the Maya solar calendar.
[See Gallenkamp (1979), p. SO; Peterson (1961), p. 225 ]
The “extra” five-day period was called Uayeb, meaning "The one that has
no name”, and it was represented by a glyph associated with the idea of
chaos, disaster and corruption. They were thought of as “ghost” days, and
considered empty, sad and hostile to human life. Anyone born during
Uayeb was destined to have bad luck and to remain poor and miserable
all his life long. Peterson quotes Diego de Landa, who reported that during
Uayeb the Maya never washed, combed their hair, or picked their nits;
they did no regular or demanding work, for fear that something untoward
would happen to them.
The first day of each “month”, including Uayeb, was represented by
the glyph for the “month”, that is to say the sign for its tutelary divinity,
together with a special sign:
314
Fig. 22.36.
This sign, which is usually translated by specialists as “0”, signified that
the god who had carried the burden of time up to that point was passing it
on to the following month-god. So since Zip and Zotz are the names of two
consecutive “months” in the “approximate” Maya calendar, the hieroglyph:
Fig. 22 . 37 . 0 ZOTZ
meant that Zip was handing over the weight of time to Zotz.
As a result the remaining days of each “month”, including Uayeb, were
numbered from 1 to 19, with the second day having the number 1, the
third day the number 2, and so on (see Fig. 22.40 below). As a result,
the following “date” in the secular or civil calendar:
Fig. 22 . 38 . 4 XUL
signified not the fourth, but the fifth day in the twenty-day “month” of Xull
Each of the twenty days of the basic series (laid out in Fig. 22.32 above)
kept exactly the same rank-number in each of the eighteen “months” of the
civil or secular year. If the “zero day” of the first month of the year was Eb,
for example, then the “zero day” of the following seventeen months was
also Eb. But because of the extra five days added on in each annual cycle, the
day-names stepped back by five positions each year. So, for example, if
Ahau was day 8 in year N, it became day 3 in year N + 1, day 18 in year
N + 2, day 13 in year N + 3, and day 8 again in year N + 4. The full cycle
thus took four years to complete, and only in the fifth year did the
correspondence between the names and the numbers of the days of
the “months” return to its starting position.
Within the system, there were only four day-names from the basic series
that could correspond to the calendrical expression:
s
Fig. 22.39.
0
POP
315
Fig. 22.40. The 365 consecutive days of the Maya “civil" year
Fig. 22.41. Successive positions of the twenty basic days in the Maya “civil” calendar
UAYEB
THE SACRED CYCLE OF M E S O - A M E R I C AN CULTURES
THE SACRED CYCLE OF
MESO-AMERICAN CULTURES
The Maya, as we have seen, used two different calendars simultaneously,
the Tzolkin, or religious calendar, of 260 days, and the Haab, or civil
calendar, of 365 days. So to express a date in full, they combined the
signs of its place in the religious calendar with the signs of its place in
the civil year, thus:
Position of the day in the Position of the day in the
“ritual” year “civil” year
Fig. 22.42. 13 AHAU 18 CUMKU
Since both these cycles permuted the days in regular recurrent order,
the correspondence between the two calendars returned to its starting
positions after a fixed period of time, which elementary arithmetic shows
must be 18,980 days, or 52 “approximate” or civil years. In other words, the
amount of time required for a given date in the civil calendar to match a
given date in the religious calendar a second time round was equal to 52
years of 365 days or 72 years of 260 days.
You can imagine how this worked by thinking of a huge bicycle, with a
chain wheel of 365 numbered teeth pulling round a sprocket with 260
numbered teeth. For the same chain link at the front sitting on tooth 1 to
match the same chain link at the back also sitting on tooth 1, the pedals will
have to turn 52 times, or (which is necessarily the same thing) the back
wheel will have to go round 73 times.
The number of days in this cycle is equal to the lowest common multiple
of 260 and 365. Since both these numbers are divisible by 5 and since 5 is
moreover the highest common factor of 260 and 365, the number sought is
260 x 365
5
= 18,980 = 52 civil years = 73 religious years
That is the origin of the celebrated sacred cycle of fifty-two years, otherwise
known as the Calendar Round, which played such an important role in
Maya and Aztec religious life. (The Aztecs, for example, believed that the
end of each Round would be greeted by innumerable cataclysms and
catastrophes; so at the approach of the fateful date, they sought to appease
the gods by making huge human sacrifices to them, in the hope of being
allowed to live on through another cycle.)
We must mention, finally, that Maya astronomers also took the Venusian
calendar into consideration. They had observed that after each period of 65
THE AMAZING ACHIEVEMENTS OF THE MAYA
316
Venusian years, the start of the solar year, of the religious year and of
the Venusian year all coincided precisely with the start of a new sacred
cycle of 52 “civil” years. Such a remarkable occurrence was celebrated
with enormous festivities.
TIME AND NUMBERS ON MAYA STELAE
Alongside their two calendars, the Maya also used a third and rather
amazing way of calculating the passage of time on their stelae or ceremonial
columns. This “Long Count”, as it is called by Americanists, began at
zero at the date of 13 baktun, 4 ahau, 8 cumku, corresponding quite
precisely, according to the concordance established by J. E. Thompson
(1935), to 12 August 3113 BCE in the Gregorian calendar. It is generally
assumed that this date corresponded to the Mayas’ calculation of the
creation of the world or of the birth of their gods [S. G. Morley (1915)].
However, this kind of reckoning did not use solar years, nor lunar years,
nor even the revolutions of Venus, but multiples of recurrent cycles.
Its basic unit was the “day” and an approximate “year” of 360 days. Time
elapsed since the start of the Mayan era was reckoned in kin (“day”), uinal
(20-day “month”), tun (360-day “year”), katun (20-“year” period), baktun
(400-“year” period), pictun (8,000-“year” cycle), and so on as laid out in
Fig. 22.43.
The katun (= 20 tun) obviously did not correspond exactly to twenty
years as we reckon them, but to 20 years less 104.842 days; similarly, the
baktun (= 20 katun = 400 tun) was not exactly 400 years, but 400 years less
2,096.84 days. However, Mayan astronomers were perfectly aware of the
discrepancies and of the corrections needed to the “Long Count” to make it
correspond properly to actual solar years.
Order of
magnitude
Names and definitions
Equivalences
Number
of days
First
kin
DAY
1
Second
uinal
“MONTH” OF 20 DAYS
20 kin
20
Third
tun
“YEAR” OF 18 “MONTHS”
18 uinal
360
Fourth
katun
CYCLE OF 20 “YEARS”
20 tun
7,200
Fifth
baktun
CYCLE OF 400 “YEARS”
20 katun
144,000
Sixth
pictun
CYCLE OF 8,000 “YEARS”
20 baktun
2,880,000
Seventh
calabtun
CYCLE OF 160,000 “YEARS”
20 pictun
57,600,000
Eighth
kinchiltun
CYCLE OF 3,200,000 “YEARS”
20 calabtun
1,152,000,000
Ninth
alautun
CYCLE OF 64,000,000 “YEARS”
20 kinchiltun
23,040,000,000
Fig. 22 . 43 . The units of computation of time used in Maya calendrical inscriptions (the “ Long
Count” system)
As we have seen, when counting people, animals or objects, the Maya
used a strictly vigesimal system (see Fig. 22.10 above); but their time-
counting method had an irregularity at the level of the third order of
magnitude, which made the whole system cease to be vigesimal:
1 kin
1 =
1 day
1 uinal
= 20 kin
20 =
20 days
1 tun
= 18 uinal
18 x 20 =
360 days
1 katun
= 20 tun
20 x 18 x 20 =
7,200 days
1 baktun
= 20 katun
20 x 20 x 18 x 20 =
144,000 days
1 pictun
= 20 baktun
20 x 20 x 20 x 18 x 20 =
2,880,000 days
If they had used a tun of 20 instead of 18 uinal, that is to say, using a truly
vigesimal system, then their “year” would have had 400 days, and would
have thus been even further “out” from the true solar year than was the 360-
day tun of their calendrical computations.
Fig. 22.44. Detail of lintel 48 from Yaxchilan showing a bizarre representation of the expression
“16 kin” (“16 days"): a squatting monkey (a zoomorphic glyph sometimes associated with the word
kinj holding the head of the god 6 in his hands and, in his legs, the death's-head which represents
the number 10
Each of these units of time had a special sign, which, like most Mayan
hieroglyphs, had at least two different realisations, depending on whether
it was being written with some kind of ink or paint on a codex, or carved in
stone on a monument or ceremonial column. In other words, each of these
units of time could be figured :
• by a relatively simple graphical sign, which could be more
or less motivated by what it represented, or else an abstract
geometrical shape;
• by the head of a god, a man, or an animal - otherwise
called cephalomorphic glyphs, which were used for carved
inscriptions;
• exceptionally, at Quirigua and Palenque, by anthropo-
morphic glyphs, that is to say, by a god, man, or animal
drawn in full.
317
Fig. 22.45. Various hieroglyphs for kin, “day”
To represent the numerical coefficients of the units of time in the “Long
Count”, Maya scribes and sculptors used numerals which, like the unit-
signs themselves, had more than one visual realisation.
Kin Uinal Tun Katun Baktun
Fig. 22 . 46 . Hieroglyphs for the units of time (from theQuirigua stelae/
Method One for showing the numbers was to use the cephalomorphic
signs for the thirteen gods of the upper world (the set of gods and signs
known as the Oxlahuntiku) for numbers 1 to 13. The maize-god, for instance,
was associated with and therefore represented the number 5, and the god
of death represented number 10.
12345 6789 10
11 12 13 14 15 16 17 18 19
Fig. 22.47. Maya cephalomorphic numerals 1 to 19 (found on pieces of pottery and sculpture,
on stelae/ and Fat Quirigud, and on the “hieroglyphic staircase" at Palenque). [See Peterson (1961),
P 220, Fig. 52; Thompson (1960), p. 173, Fig. 13]
For the numbers 14 to 19, however, the system used the numbers 4 to 9
with a modification that can be seen in the following figure:
TIME AND NUMBERS ON MAYA STELAE
VARIANTS OF THE GLYPH FOR "9"
VARIANTS OF THE GLYPH FOR “19”
Fig. 22.48.
If you look closely you can see that the jawbone of the “nine-god” has
been removed to make the glyph represent the number 19. Arithmetically,
this is elementary, because, as the jawbone symbolised the god of death, it
enabled an “extra ten” to be shown in the sign:
Fig. 22.49. 10
Method One was not used very often; more frequently, even in calen-
drical inscriptions, the dot-and-line system (see above. Fig. 22.21) is found.
In any case, dates and lengths of time could be expressed fairly simply
within the systems explained so far. Americanists call these expressions
“initial series”. Our first example of an initial series comes from the
“hieroglyphic staircase” of Palenque, where the numbers are represented by
heads of the divinities, as shown in Fig. 22.50.
Fig. 22.50. Initial series on the “ hieroglyphic staircase" at Palenque. The date is given in
cephalomorphic figures [From Peterson (1961), p, 232, Fig. 58J
The inscription begins with the glyph called the “initial series start
sign”, or POP:
THE AMAZING ACHIEVEMENTS OF THE MAYA
POP
Fig. 22.51.
This sign corresponds to the name of the divinity “responsible” for the
“month” of the “civil” calendar on the day that the inscription was carved
(or, to be more precise, the name of the month in the “secular” calendar in
which the last day of the inscribed date falls).
Then, at the foot of the inscription, we can read the position of the date
with respect to the “civil” and to the “religious” year, thus:
8AHAU 13 POP
Fig. 22.52.
As for the number of days elapsed since the initial date of the Mayan era,
it is expressed in the “Long Count” as follows:
Fig. 22.53.
The date is read from top to bottom, and in descending order of magni-
tude of the counting units of the Maya calendar. It can be transcribed as
follows:
9 baktun = 9 x 144,000 days 1,296,000
8 katun = 8 x 7,200 days 57,600
9 tun = 9 x 360 days 3,240
13 uinal = 13 x 20 days 260
0 kin =0x1 day 0
Total 1,357,100 days
A fairly simple calculation reveals this to be the year 603 CE.
The Leyden Plate provides another example:
318
SIDE 1 SIDE 2
Fig. 22.54. The Leyden Plate. This thin jade pendant, 21.5 cm high, was found in Guatemala,
near Puerto Barrios, and is thought to have been carved at Tikal. On side 1 it shows a richly-clad
Maya (probably a god) trampling a prisoner, and, on side 2, a date corresponding to 320 CE.
Rijksmuseum voor Volkenkunde, Leyden, Holland
As in the illustration from Palenque, this expression begins with an
"introductory glyph”, in this case the name of the god whose “burden” it
was to carry the “month” of YAXKIN, during which the building on which
this inscription was carved was completed:
YAXKIN
Fig. 22.55.
The date of completion is also expressed in terms of its position in the
civil and religious calendars, thus:
1 EB 0 YAXKIN
Fig. 22.56.
319
TIME AND NUMBERS ON MAYA STELAE
As for the corresponding date in the “Long Count” system, it is given in
this form:
Fig. 22.57.
This date is also to be read from top to bottom in descending order of
magnitudes, and from left to right within each glyph, and produces the
following numbers:
8 baktun = 8 X 144,000 days 1,152,000
14 katun = 14 x 7,200 days 100,800
3 tun = 3 x 360 days 1,080
1 uinal = 1 x 20 days 20
12 kin = 12 x 1 day 12
Total 1,253,912 days
Once again, a simple calculation reveals that in view of the number of
days since the beginning of the Mayan era this inscription was carved in the
year 320 CE.
It was long thought that the Leyden Plate was the oldest dated artefact
from Maya civilisation. However, in 1959, archaeological excavations in the
ruins of the city of Tikal, in Guatemala, turned up an even older dated
inscription. Stela no. 29 carries an inscription which can be translated as:
& baktun = 8 x 144,000 days 1,152,000
12 katun = 12 x 7,200 days 86,400
14 tun = 14 x 360 days 5,040
8 uinal = 8 x 20 days 160
0 kin = 0 x 1 day 0
Total 1,243,600 days
which works out at the year 292 CE.
Fig. 22.58. Side 2 of stela 29 from Tikal (Guatemala), the oldest dated Mayan inscription
found so far. The date written on it - usually transcribed as 8.12.14.8.0 - matches the year 292 CE.
[See Shook (1960), p. 32]
There are many other examples of calendrical inscriptions on the
numerous stelae of the Maya, each one teeming with fantastical and
elaborate signs. To conclude this section, let us look at one date found
on stela E from Quirigua.
The date of the stela’s erection begins on the top line with two glyphs:
the first, on the left, is composed of the figure 9 with the head of the
god representing baktun, and the other of the figure 17 with the head of
the god representing katun. It then goes on, on the next line, with two
compound glyphs signifying “0 tun” and “zero uinaf respectively; and, on
the bottom line, the date ends with a sign meaning “0 kin".
Fig. 22.59.
9 BAKTUN
9 X 144,000
(= 1,296,000 days)
0 TUN
0x360
(= 0 days)
0 KIN
0x1
(= 0 days)
17 KATUN
17 x 7,200
(= 122,400 days)
0 UINAL
0x20
(= 0 days)
THE AMAZING ACHIEVEMENTS OF THE MAYA
320
So the people who put up this column expressed the number of days
elapsed since the start of the Mayan era up to the date on which they made
this inscription, which is tantamount to expressing the latter date as:
9 baktun = 9 x 144,000 days 1,296,000
17 katun = 17 x 7,200 days 122,400
0 tun = 0 x 360 days 0
0 uinal = 0 x 20 days 0
0 kin =0x1 day 0
Total 1,418,400 days
So one million four hundred and eighteen thousand and four hundred
days had passed since the “beginning of time” and, given that we know
what the start-date was, we can calculate fairly easily that stela E at Quirigua
was completed on 24 January 771 CE.
INTERPRETATION AND TRANSLATION
Glyph defining the initial series
The grotesque head at the centre stands for the name of the
tutelary divinity of the month ofCumku, in which the
last day of the initial series falls.
i
i&mai
9 baktun
9 x 1-14,000
(= 1,296,000 days)
17 katun
17x7,200
(= 122,400 days)
0 tun
0 x 360
(= 0 days)
0 uinal
0x20
(= 0 days)
Okin
0x1
(= 0 days)
Name of the divinity in
charge of the 9th da)'
in the series of 9 days
(the nine gods of the
lower world)
Phases of the moon
on the last day of the
initial series (here,
"new moon")
Undeciphered
Current lunar
month (in this
case, of 29 days)
Cndeciphered
Position of the current
lunar month in the
lunar half-year (here.
“ 2 nd position”)
Un deciphered
Fig. 22 . 60 . Detail from stela E at Quirigua, giving an initial series together with a complementary
series that provides other details on the date of the stela 's erection. The date is 9.17.0.0.0 and 13 ahau,
18 cumku, which matches 24 January 771 CE, in the Gregorian calendar. [SeeMorley (1915), Fig. 251
We should note that these stelae contain some of the most interesting
Mayan inscriptions that have been found. If we compare the oldest and
newest dates found in particular places, we can get an idea of the duration
of the great Maya cities. For example, at Tikal, the oldest date found is
292 CE and the latest is 869 CE; at Uaxactun, the limit-dates are 328 CE
and 889 CE; at Copan, the relevant inscriptions are of 469 CE and 800 CE;*
and so on. The important point in this long digression is to note that in
their calendrical inscriptions the Maya represented the “zero”, that is to say
the absence of units in any one order, by glyphs and signs of the most
diverse kinds.
Fig. 2 2 . 6 1 . Hieroglyphs for “zero" found on various Maya stelae and sculptures. Left to right: the
first six, the commonest, arc symbolic notations: the seventh and eighth are cepha/omorphic, and the
last is anthropomorphic. ISee Peterson (1961), Fig. 51; Thompson (1960). Fig. 131
Fig. 22 . 62 . Detail of a plaque found al the Palenque Palace: an unusual anthropomorphic
representation of the expression “0 kin" (“no days"). From Peterson (1961), fig. 14, p. 72.
* See M. D. Coe, op.cit., p. 68
MAYA mathematics:
A SCIENCE IN THE SERVICE OF
ASTRONOMY AND MYSTICISM
MAYA MATHEMATICS
32 1
The Maya system for counting time and for expressing the date did not
really require a zero: the date expressed in Fig. 22.60 above, for example,
could have been represented just as easily and just as unambiguously by:
9 baktun, 17 katun
as by the glyphs we actually have, which say
9 baktun, 17 katun, 0 tun, 0 uinal, 0 kin
So why did Maya calendrical computation bother to invent a zero?
The answers have to do with the religious, aesthetic and graphical ideas
and customs of the Maya.
In religious terms, each of the time-units was imagined as a burden
carried by one of the gods, the “tutelary god” of that cycle of time. At the
end of the relevant cycle, the god passed on the burden of time to the god
designated by the calendar as his successor.
On the date of “9 baktun, 11 katun, 7 tun, 5 uinal, and 2 kin”, for instance,
the god of the “days” carried the number 2, the god of the “months” carried
the number 5, the god of the “years” carried the number 7, and so on, in this
manner:
God
God
God
God
God
bearing
bearing
bearing
bearing
bearing
the
the
the
the
the
baktuns
katuns
tuns
uinals
kins
Fig. 22.63.
Fig. 22.64. Stela A at Quirigua. Erected in 775 CE, this column has gods carved on its front and
back, and calendrical, astronomical and other glyphs carved on its other two sides. From Thompson
(1960), Fig. 11, p. 163
If we were to transpose this system to our own Gregorian calendar, we
would need six gods to carry the “burden” of the date “31 December 1899”.
One god - the “day-god” - would “carry” the number 31; the second would
bear the number 12, for the months; the third would carry the number
9, for the years; the fourth would “carry” the decades; and we would need
two more, for the centuries and the millennia. At the end of the day of the
31 December 1899, these gods would have rested for a moment before
THE AMAZING ACHIEVEMENTS OF THE MAYA
322
setting off on a new cycle. The day-god would resume his burden, but with
the number 1, and similarly for the month-god. But as the decade and the
century would change (to 1900), the year-god and the decade-god would be
released from their burdens for a period of time, the century-god would
now bear the burden of the number 8, and the millennium-god would carry
on with his 1 as he had been doing for the previous 900 years.
In Maya mystical thought, the fact that the gods occasionally had a rest
from their burdens did not justify simply eradicating them from the
representation of the task of carrying the burdens of time. Failing to put
them in their right places in the inscription might have angered them! It
would also have destroyed the absolute regularity of the system, in which
calendrical expressions always ran from top to bottom in descending
orders of magnitude. The aesthetically pleasing sequence of symbols in
an unchanging order would have been altered if there had been no sign
for zero. So we can say, in conclusion, that the demands of the writing
system itself, the aesthetic appearance of inscriptions intended to be cere-
monial, and a set of religious beliefs made the invention of a “zero-count”
an absolute necessity (see Fig. 22.65).
Nonetheless, the calendrical system of the Maya is also part of a long
and slow evolution leading towards the discovery of a place-value system.
The Mayan units of time were always placed in precisely the same position
in an inscription, with the same regularity as the tokens in an abacus or
“counting table". And Mayan astronomer-priests did not fail to notice the
arithmetical potential of their system.
When writing manuscripts, as opposed to inscriptions carved in stone,
they eventually came to omit the glyphs representing the units of time (or
the gods that were responsible for them), and wrote down only the corre-
sponding numerical coefficients, since the order of the magnitudes was
firm and fixed. So dates in the manuscripts are expressed just by numbers.
For example, instead of writing the date “8 baktun, 11 katun, 0 tun, 14 uinal,
0 kin” as follows:
8 BAKTUN 11 KATUN 0 TUN 14 UINAL 0 KIN
they wrote (top to bottom, and with the numerical expressions rotated
through 90°) simply:
0
*
Fig. 22.65B.
Omitting the glyphs of the tutelary gods must have had less religious
consequence in manuscripts than in the ceremonial and sacred stelae.
Taken outside of the context of mysticism and theology, the Maya system
constitutes a remarkable written numeral system, incorporating both a true
zero and the place-value principle. However, since it had been developed
exclusively to express dates and to serve astronomical and calendrical
computation, the system retained an irregular value in its third position,
which, as we may recall, was 20 x 18 = 360, and not 20 x 20 = 400. This flaw
made the system unsuitable for arithmetical operations and blocked any
further mathematical development.
It is true that Maya scholars were concerned above all with matters
religious and prophetic; but have not astrology and religion opened the
path to philosophical and scientific developments in many places in
the world? So we must pay homage to the generations of brilliant Mayan
astronomer-priests who, without any Western influence at all, developed
concepts as sophisticated as zero and positionality, and, despite having
only the most rudimentary equipment, made astronomical calculations of
quite astounding precision.
Fig. 22.65A.
323
THE LEGEND OF SESSA
CHAPTER 23
THE FINAL STAGE OF
NUMERICAL NOTATION
THE LEGEND OF SESSA
In Arabic and Persian literature it is often written that the Indian world
may glory in three achievements:
• the positional decimal notation and methods of calculation;
• the tales of the Panchatantra (from which probably came the
well-known fable of Kalila wa Dimna);
• the Shaturanja, the ancestor of chess, about which a famous
legend (adapted into modern terms) will give us an apt intro-
duction to this very important chapter.
In order to prove to his contemporaries that a monarch, no matter how
great his power, was as nothing without his subjects, an Indian Brahmin
of the name of Sessa one day invented the game of Shaturanja.
This game is played between four players on an eight by eight chessboard,
with eight pieces (King, Elephant, Horse, Chariot, and four Soldiers), which
are moved according to points scored by rolling dice.
When the game was shown to the King of India, he was so amazed by the
ingenuity of the game and by the myriad variety of its possible plays that
he summoned the Brahmin, that he might reward him in person.
“For your extraordinary invention,” said the King, “I wish to make you a
gift. Choose your reward yourself and you shall receive it forthwith. I am so
rich and so powerful that I can fulfil your wildest desire."
The Brahmin reflected on his reply, and then astonished everyone by
the modesty of his request.
“My good Lord,” he replied, “I wish that you would grant me as many
grains of wheat as will fill the squares on the board: one grain for the first
square, two for the second, four for the third, eight for the fourth, sixteen
for the fifth, and so on, putting into each square double the number of
grains that were put in the square before.”
“Are you mad to suggest so modest a demand?” exclaimed the aston-
ished King. “You could offend me with a request so unworthy of my
generosity, and so trivial compared with all that I could offer you. But let
it be! Since that is your wish, my servant will bring you your bag of wheat
before nightfall.”
The Brahmin made the merest smile, and withdrew from the Palace.
That evening, the King remembered his promise and asked his Minister if
the madman Sessa had received his meagre reward. “Lord,” replied the
Minister, “your orders are being carried out. The mathematicians of your
august Court are at this moment working out the number of grains to give
to the Brahmin.”
The King’s brow darkened. He was not used to such delay in obeying his
orders. Before retiring to bed, the King asked once more whether the
Brahmin had received his bag of grain.
“0 King,” replied the Minister, hesitating, “Your mathematicians have
still not completed their calculations. They are working at it unceasingly,
and they hope to finish before dawn.”
The calculations proved to take far longer than had been expected. But
the King, who did not wish to hear about the details, ordered that the
problem should be solved before he awoke.
The next morning, however, his order of the night before remained
unfulfilled, and the monarch, incensed, dismissed the calculators who had
been working at the task.
“0 good Lord,” said one of his Counsellors, “you were right to dismiss
these incompetents. They were using ancient methods! They are still count-
ing on their fingers and moving counters on an abacus. I permit myself
to suggest that the calculators of the central province of your Kingdom
have for generations already been using a method far better and more
rapid than theirs. They say it is the most expeditious, and the easiest to
remember. Calculations which your mathematicians would need days of
hard work to complete would trouble those of whom I speak for no more
than a brief moment of time.”
On this advice, one of these ingenious arithmeticians was brought to
the Palace. He solved the problem in record time, and came to present his
result to the King.
“The quantity of wheat which has been asked of you is enormous,” he
said in a grave voice. But the King replied that, no matter how huge the
amount, it would not empty his granaries.
He therefore listened with amazement to the words of the sage.
“O Lord, despite all your great power and riches, it is not within your
means to provide so great a quantity of grain. This is far beyond what we
know of numbers. Know that, even if every granary in your Kingdom were
emptied, you would still only have a negligible part of this huge quantity.
Indeed, so great a quantity cannot be found in all the granaries of all
the kingdoms of the Earth. If you desire absolutely to give this reward, you
should begin by emptying all the rivers, all the lakes, all the seas and the
oceans, melting the snows which cover the mountains and all the regions of
THE FINAL STAGE OF NUMERICAL NOTATION
324
the world, and turning all this into fields of corn. And then, when you have
sown and reaped 73 times over this whole area, you will finally be quit of
this huge debt. In fact, so huge a quantity of grain would have to be stored
in a volume of twelve billion and three thousand million cubic metres, and
require a granary 5 metres wide, 10 metres long and 300 million kilometres
high (twice the distance from the Earth to the Sun)!”
The calculator revealed to the King the characteristics of the revolution-
ary method of numeration of his native region.
“The method of representing numbers traditionally used in your
Kingdom is very complicated, since it is encumbered with a panoply of
different signs for the units from 10 upwards. It is limited, since its largest
number is no greater than 100,000. It is also totally unworkable, since no
arithmetical operation can be carried out in this representation. On the
other hand, the system which we use in our province is of the utmost
simplicity and of unequalled efficacity. We use the nine figures 1, 2, 3, 4,
5, 6, 7, 8, and 9, which stand for the nine simple units, but which have
different values according to the position in which they are written in the
representation of a number, and we use also a tenth figure, 0, which means
“null” and stands for units which are not present. With this system we can
easily represent any number whatever, however large it may be. And this
same simplicity is what makes it so superior, along with the ease which it
brings to every arithmetical operation."
With these words, he then taught the King the principal methods of
the calculation of the reward, and explained his operations as follows.
According to the demand of the Brahmin, we must place
1 grain of com on the first square;
2 grains on the second square;
4 (2 x 2) on the third;
8 (2 x 2 x 2) on the fourth;
16 (2 x 2 x 2 x 2) on the fifth;
and so on, doubling each time from one square to the next. On the sixty-
fourth square, therefore, must be placed as many grains as there are units
in the result of 63 multiplications by 2 (namely 2 63 grains). So the quantity
the Brahmin demanded is equal to the sum of these 64 numbers, namely
1 + 2 + 2 2 + 2 3 + . . . + 2 63 .
“If you add one grain to the first square,” explained the calculator,
“you would have two grains there, therefore 2 x 2 in the first two squares.
By the third square you would then find a total of2x2 + 2x2 grains,
or 2 x 2 x 2 in all. By the fourth the total would be2x2x2 + 2x2x2,
or 2 x 2 x 2 x 2 in all. Proceeding in this way, you can see that by the time
you reach the last square of the board the total would be equal to the result
of 64 multiplications by 2, or 2 64 . Now, this number is equal to the six-fold
product of 10 successive multiplications by 2, further multiplied by the
number 16:
2 64 = 2 10 x 2 10 x 2 10 x 2 10 x 2 10 x 2 10 x 2 4
= 1,024 x 1,024 x 1,024 x 1,024 x 1,024 x 1,024 x 16
“And so,” he concluded, “since this number has been obtained by adding
one to the quantity sought, the total number of grains is equal to this
number diminished by one grain. By completing these calculations in
the way I have shown you, you may satisfy yourself, O Lord, that the
number of grains demanded is exactly eighteen quadrillion, four hundred
and forty-six trillion, seven hundred and forty-four billion, seventy-three
thousand seven hundred and nine million, five hundred and fifty-one
thousand, six hundred and fifteen (18,446,744,073,709,551,615)!”
“Upon my word!” replied the King, very impressed, “the game this
Brahmin has invented is as ingenious as his demand is subtle. As for his
methods of calculation, their simplicity is equalled only by their efficiency!
Tell me now, my wise man, what must I do to be quit of this huge debt?”
The Minister reflected a moment, and said:
“Catch this clever Brahmin in his own trap! Tell him to come here and
count for himself, grain by grain, the total quantity of wheat which he
has been so bold to demand. Even if he works without a break, day and
night, one grain every second, he will gather up just one cubic metre in
six months, some 20 cubic metres in ten years, and, indeed, a totally
insignificant part of the whole during the remainder of his life!”
THE MODERN NUMBER-SYSTEM:
AN IMPORTANT DISCOVERY
The legend of Sessa thus attributes to Indian civilisation the honour
of making this fundamental realisation which we may call the modern
number-system. We shall see in due course that, despite the mythical
character of the story, this fact is completely true.
But first we must weigh the importance of this written number-system,
which nowadays is so commonplace and familiar that we have come to
forget its depth and qualities.
Anyone who reflects on the universal history of written number-systems
cannot but be struck by the ingeniousness of this system, since the concept
of zero, and the positional value attached to each figure in the representa-
tion of a number, give it a huge advantage over all other systems thought
up by people through the ages.
325
THE EARLIEST NUMERICAL RULE: ADDITION
To understand this, we shall go back to the beginning of this history.
But instead of following its different stages purely chronologically, and
according to the various civilisations involved, we shall for the moment
let ourselves be guided by a kind of logic of time, the regulator of historic
data, which has made of human culture a profound unity.
THE EARLIEST NUMERICAL RULE:
ADDITION
This story begins about five thousand years ago in Mesopotamia and in
Egypt, in advanced societies in full expansion, where it was required
to determine economic operations far too varied and numerous to be
entrusted to the limited capabilities of human memory. Making use of
archaic concrete methods, and feeling the need to preserve permanently
the results of their accounts and inventories, the leaders of these societies
understood that some completely new approach was required.
To overcome the difficulty, they had the idea of representing numbers by
graphic signs, traced on the ground or on tablets of clay, on stone, on sheets
of papyrus, or on fragments of pottery. And so were born the earliest
number-systems of history.
Independently or not, several other peoples embarked on this road
during the millennia which followed. And it all worked out as though,
over the ages and across civilisations, the human race had experimented
with the different possible solutions of the problem of representing and
manipulating numbers, until finally they settled on the one which finally
appeared the most abstract, the most perfected and the most effective of all.
To begin with, written number-systems rested on the additive principle,
the rule according to which the value of a numerical representation is
obtained by adding up the values of all the figures it contains. They were
therefore very primitive. Their basic figures were totally independent of
each other (each one having only one absolute value), and had to be
duplicated as many times as required.
The Egyptian hieroglyphic number-system, for example, assigned a
special sign to unity and to each power of 10: a vertical stroke for 1, a sign
like an upside-down U for 10, a spiral for 100, a lotus flower for 1,000,
a raised finger for 10,000, a tadpole for 100,000, and a kneeling man with
arms outstretched to the sky for 1,000,000. To write the number 7,659
required 7 lotus flowers, 6 spirals, 5 signs for 10 and 9 vertical strokes of
unity, all of which required a total of 27 distinct figures (Fig. 23.1).
First appearance: c.3000 - 2900 BCE
Type: A1 (additive number-system of the first type: Fig. 23.30). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 14, p.170)
Base numbers
i n
1 10
9
100
(= 10 2 )
l
1,000
(= 10 3 )
t
10,000
(= 10 4 )
V?
100,000
(= 10 5 )
*
1,000,000
(= 10 6 )
Example: 7,659
rm
O fl Ifl)
ooo
ID ffD
000
000
7,000
600
50
9
>
Representation based on additive principle, broken down thus:
7,659 = (1,000 + 1,000 + 1,000 + 1,000 + 1,000 + 1,000 + 1,000)
+ (100 + 100 + 100 + 100 + 100 + 100 )
+ (10 + 10 + 10 + 10 + 10 )
+ U + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
Fig. 23 . 1 . Egyptian hieroglyphic number-system
The Sumerian number-system (which used base 60, with 10 as auxiliary
base) gave a separate sign to each of the following numbers, in the order of
their successive unit orders of magnitude:
1 10 60 600 3,600 36,000 216,000
= 10 X 60 = 60 2 = 10 x 60 2 = 60 3
But it too was limited to repeating the figures as many times as required
to make up the number. The number 7,659 was therefore represented
according to the following arithmetical decomposition, which twice repeats
the sign for 3,600, seven times the sign for 60, three times that for 10
and nine times the sign for unity, so that 21 distinct signs are required to
represent this number (Fig. 23.2):
7,659 = (3,600 + 3,600)
+ (60 + 60 + 60 + 60 + 60 + 60 + 60)
+ (10 + 10 + 10 )
+ U + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
THE FINAL STAGE OF NUMERICAL NOTATION
First appearance: c.3300 BCE
Type: A 2 (additive number-system of the second type: Fig. 23.31). Base 60
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 8 , p.84)
Base numbers (archaic script)
j « i ij o ©
1 10 60 600* 3,600 36,000
(= 10 x60) (=60 2 ) (= 10 x 60 2 )
* Number formed by combining the sign for 60 or 600 with that for 10 (multiplicative combination)
Example: 7,659
oao
Representation based on additive principle, broken down thus:
7,659 = (3,600 + 3,600)
+ (60 + 60 + 60 + 60 + 60 + 60 + 60)
+ (10 + 10 + 10 )
+ U + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
Fig. 23 . 2 . Sumerian number-system
Other similar notations include the Proto-Elamite, the Cretan systems
(Hieroglyphic and Linear A and Linear B), the Hittite hieroglyphic system,
and even the Aztec number-system (which differed from the others
only in that it used a base of 20) (Fig. 23.3 to 23.6).
First appearance: c.2900 BCE
Type: A1 (additive number-system of the first type: Fig. 23.30). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 11, p.120)
Base numbers
0 ° 0 E3 E3
1 10 100 1,000 10,000
Example: 7,659 Lt] CJ IK1 000 ooo 001)111)
BBB " ""
7,000 600 50 9
Representation based on additive principle, broken down thus:
7,659 = (1,000 + 1,000 + 1,000 + 1,000 + 1,000 + 1,000 + 1,000)
+ (100 + 100 + 100 + 100 + 100 + 100 )
+ (10 + 10 + 10 + 10 + 10 )
+ Q + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
Fig. 23 . 3 . Proto-Elamite number-system
326
Fig. 23 . 5 . Hittite hieroglyphic number-system
327
Fig. 23 . 6 . Aztec number-system
To avoid the encumbrance of such a multitude of symbols, certain peoples
introduced supplementary signs which corresponded to inter-mediate
units. Such was the case for the Greeks, the Shebans, the Etruscans, and
the Romans, who assigned a separate symbol to each of the numbers
5, 50, 500, 5,000, and so on, in addition to those they already had for the
different powers of 10 (Fig. 23.7 and 23.8).
First appearance: c.500 BCE
Type: A2 (additive number-system of the second type: Fig. 23.31). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 16, pp.l82ff.)
Base numbers
I P A F H P X F M
1 5 10 SO* 100 500* 1,000 5.000* 10,000
(=5x10) (= 10 2 ) (=5x102) (= 10 3 ) (= 5 x 10 3 ) (= 10<)
"Numbers formed by combining the signs for 10, 100, 1,000, etc. with the one for number 5
(multiplicative principle)
Example: 7,659
F XX P H F Pllll
5,000 2,000 500 100 50 5 4
^
Representation based on additive principle, broken down thus:
7,659 = (5,000 + (1,000 + 1,000) + 500 + 100 + 50
+ 5 + (1 + 1 + 1 + 1)
Fig. 23 . 7 . Greek acrophonic number- system
THE EARLIEST NUMERICAL RULE: ADDITION
First appearance: c.500 BCE
Type: A2 (additive number-system of the second type: Fig. 23.31). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 16, pp.l87ff.)
Base numbers (archaic script)
I A X I C B ®
1 5 10 50 100 500 1,000
(=5X 10) (= 10 2 ) (= 5 X 10 2 ) (= 10 3 )
Example: 7,659
ID ®® B C i IX
5,000 2,000 500 100 50 (10-1)
*
Representation based both on additive and subtractive principles, broken down thus:
7,659 = 5,000 + (1,000 + 1,000) + 500 + 100 + 50 + (10 - 1)
Fig. 23 . 8 . Roman number-system
LARGE ROMAN NUMBERS
To note down large numbers the Romans and the Latin peoples of the Middle Ages
developed various conventions. Here
pp. 197ff.):
are the principal ones (see Chapter 16,
1. Overline rule
This consisted in multiplying by 1,000 every number surmounted by a horizontal bar:
X = 10,000 C = 100,000
CXVII = 127 X 1,000 = 127,000
2. Framing rule
This consisted in multiplying by 100,000 every number enclosed in a sort of open |
rectangle:
1 x 1 = 1,000,000 1 ccLxr
C] = 264 x 100,000 = 26,400,000
3. Rule for multiplicative combinations
The rule is occasionally found in Latin manuscripts in the early centuries CE, but most
often in European mediaeval accounting documents. To indicate multiples of 100 and
1,000, first the number of hundreds and thousands to be entered are noted down, then
the appropriate letter (C or M) is placed as a coefficient or superscript indication:
100 C
1,000 M
200 II.C or IT
2,000 1I.M or IF
300 III.C or III'
3,000 III.M or III m
900 V1III.C or Villi'
9,000 VfflLM or VHII m
Examples taken from Pliny the Elder’s
XXXIII, 3).
Natural History, first century CE (VI, 26;
LXXXIII.M
for 83,000
CX.M
for 110,000
Fig. 23.9 a. Latin notation of targe numbers (late period)
THE FINAL STAGE OF NUMERICAL NOTATION
The same system is to be found in the Middle Ages, notably in King Philip le Bel's
Treasury Rolls, one of the oldest surviving Treasury registers. In this book, dated 1299,
we find what is reproduced here below, drawn up in Latin (from Registre du Tresor de
Philippe le Bel, BN, Paris, Ms. lat. 9783, fo. 3v, col.l, line 22):
^ 9 Wirfiirp’-
V m . IIIe.XVT.l(ibras). Vl.s(oUdos)
I. d(enarios). p(arisiensium)
Fig. 23. 9B.
But this was out of the frying pan into the fire, for such systems required
even more tedious repetitions of identical signs. In the Roman system, the
conventions for writing numbers proliferated so much that the system
finally lost coherence (Fig. 23.8 and 23.9). Furthermore, since it made use
at the same time of two logically incompatible principles (the additive and
the subtractive), this system finally represented a regression with respect to
the other historic systems of number representation.
The first notable advance in this respect is in fact due to the scribes
of Egypt who, seeking means for rapid writing, early sought to simplify
both the graphics and the structure of their basic system. Starting from
excessively complicated hieroglyphic signs, they strove to devise extremely
schematic signs which could be written in a continuous trace, without inter-
ruption, such as are obtained by small rapid movements and often by a
single stroke of the brush. Great changes thus occurred in the forms of the
hieroglyphic numbers, so that the later forms had only a vague resemblance
to their prototypes. This finally resulted in a very abbreviated numerical
notation, as in the Egyptian hieratic number-system, giving a separate sign
to each of the following numbers (Fig. 23.10):
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
200
300
400
500
600
700
800
900
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
It was a cursive notation, and was succeeded by an even more abbrevi-
ated one, known as the demotic number-system.
“5,316 livres, 6 sols &
1 denier parisis"
328
Fig. 23.10. Egyptian hieratic number-system
In both cases, there were nine special signs for the units, nine more for
the tens, nine more for the hundreds, and so on. Such systems allowed
numbers to be represented with much greater economy of symbols. The
number 7,659 now only needed four signs (as opposed to the 27 required
by the hieroglyphic system), since it only requires writing down the
symbols for 7,000, 600, 50, and 9 according to the decomposition
7,659 = 7,000 + 600 + 50 + 9.
The inconvenience of such a notation is, of course, the burden on the
memory of retaining all the different symbols of the system.
329
THE EARLIEST NUMERICAL RULE: ADDITION
The Greeks and the Jews, and later the Syriacs, the Armenians and the
Arabs, used notations which are mathematically equivalent to this system
(Fig. 23.11 to 23.13, and Fig. 19.4 above). But, instead of proceeding
as the Egyptians had done to the progressive refinement of the forms of
their figures, they constructed their systems on the basis of the letters
of their alphabets. Taking these letters in their usual order (the Phoenician
“ABC”) associates the first nine letters with the nine units, the next nine
with the nine tens, and so on.
First appearance: c. fourth century BCE
Type: A3 (additive number-system of the third type: Fig. 23.32). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 17, p.220)
Base numbers
A
B
r
A
E
r
Z
H
0
1
2
3
4
5
6
7
8
9
I
K
A
M
N
B
0
n
C
10
20
30
40
50
60
70
80
90
P
2
T
Y
O
X
*
n
n\
100
200
300
400
500
600
700
800
900
Example: 7,659 *Z X N 0
7,000 600 50 9
Representation based both on additive principle, broken down thus:
7,659 = 7,000 + 600 + 50 + 9
(The notation for the number 7,000 has been derived from that for 7, applying to this a small
distinctive sign upper left.)
Fig. 23.11. Greek alphabetic number-system
First appearance: c. second century BCE
Type: A3 (additive number-system of the third type: Fig. 23.32). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 17, p.215)
Base numbers
K
3
a
1
n
1
T
n
a
l
2
3
4
5
6
7
8
9
•»
3
b
D
3
D
V
s
to
20
30
40
50
60
70
80
90
P
“i
0
n
100
200
300
400
Example: 7,659
CD
3
1
n
T
<r
9
50
200
400
7,000
Representation based both on additive principle, broken down thus:
7,659 = 7,000 + 400 + 200 + 50 + 9
(The notation for the number 7,000 has been derived from that for 7, placing two dots above this.)
Fig. 23.12. Hebraic alphabetic number- system
First appearance: c.400 CE
Type A3 (additive number-system of the third type: Fig. 23.32). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 17, pp.224ff.)
Base numbers
(Line 1, lower case; line 2, upper case)
LU
P
*
9
b
1
P
P
a
P
9-
9*
b
5
k
c
i
2
3
4
5
6
7
8
9
/
b
/
b*
A
k
<
d
9
A
L
hi
XT
u
4
2
10
20
30
40
50
60
70
80
90
£
if
y
b
l
n
t
•H
l
zf
ir
8
b
b
n
9
V
2
100
200
300
400
500
600
700
800
900
n.
u
4
lit
P
9
4 .
•b
P
fh
u
4
s
n
3
A
0
*
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
Example: 7,659
Lower case:
L
n
A
P
Upper
case:
A
7,000
n
600
XT
50
—
fb
»
9
Representation based on additive principle, broken down thus:
7,659 = 7,000 + 600 + 50 + 9
Fig. 23.13. Armenian alphabetic number-system
Such procedures allow the words of the language to be converted into
numbers, which provides ample raw material for every kind of speculation,
occultist fantasy or magical imagining, and for superstitious beliefs and
practices. But, leaving aside this inconvenient by-product, the procedure
gives a more or less acceptable solution to the problem according to the
needs of the time. As with the Egyptian hieratic and demotic systems,
the number 7,659 requires only four signs to be written down.
THE FINAL STAGE OF NUMERICAL NOTATION
330
THE DISCOVERY OF THE
MULTIPLICATIVE PRINCIPLE
There was still a long road ahead before people could arrive at a system so
well perfected as our own. Means for a numeric notation were still limited.
Various peoples, it must be said, remained deeply attached to the old
additive principle and were therefore in a blind alley. One major reason
for this blockage concerned the problem of representing large numbers,
which lie beyond the capability of the imagination when one is restricted
solely to the additive principle. For this reason, some peoples made a
radical change in their number-systems by adopting a hybrid principle
which involved both multiplication and addition.
This change took place in two stages. The introduction of the new prin-
ciple at first served only to extend the capabilities of number-systems which
had been very primitive (Fig. 23.14 and 23.15).
The Sumerians
From c.3300 BCE the Sumerians tended to represent the units of different orders in their number-
system by means of objects of conventional size and shape.
They had begun by using calculi to symbolise 1, 10, 60,
and 60 2 (see Chapter 10, p.100)
1 10 60 3,600
But, not wishing to duplicate the original symbols, they
invoked the multiplicative principle to represent the
order of 600 and of 36,000:
600 36,000
(= 10x60) (= 10 X 60 2 )
They had thus come up with the idea (very abstract for the time) of symbolising X 10 by making in
the soft clay a small circular impression (“written” symbol for the pebble representing 10) within the
large cone representing the value 60 or within the sphere representing the value 3,600.
And they used the same idea in representing these same numbers when they embarked on a
written number-system in archaic script as well as in cuneiform (see Chapter 8, p.84):
Curviform number-symbols
9
©
Cuneiform number-symbols
600
36,000
(= 10x60) (= 10 x 60 2 )
The Cretans (second millennium BCE):
The Cretans introduced the number for 10,000 by combining the horizontal stroke of 10 with the
sign for 1,000 (see Chapter 15, p.180):
■O’ 10,000 (= 1,000 x 10)
The Greeks (from the fifth century BCE):
The Greeks invoked the same principle, completing their acrophonic number-system by intro-
ducing a notation with its own traits for each of the numbers 5, 50, 500, and 5,000 (see Chapter 16,
p p ' 182tt > : H p 1 pi
5 50 500 5,000
(= 5 X 10) (= 5 X 10 2 ) (= 5 X 10 3 )
Fig. 23.14. First emergence of the multiplicative principle
Thus from the beginning of history, people have sometimes introduced the multipli-
cation rule into systems essentially based on the additive principle. But during this
first stage, the habit was confined to certain particular cases and the rule served only
to form a few new symbols.
But in the subsequent stage, it gradually became clear that the rule could be applied
to avoid not only the awkward repetition of identical signs, but also the unbridled
introduction of new symbols (which always ends up requiring considerable efforts of
memory).
And that is how certain notations that were rudimentary to begin with were often
found to be extensible to large numbers.
The Greeks
This idea was exploited by ancient Greek mathematicians whose "instrument” was
their alphabetic number-system: in order to set down numbers superior to 10,000, they
invoked the multiplicative rule, placing a sign over the letter M (initial of the Greek
word for 10,000, (xvptoi) to indicate the number of 10,000s (see Chapter 17, p.220):
a (3 7 i(3
M M M M
10,000 20,000 30,000 120,000
(= 1 x 10,000) (= 2 x 10,000) (= 3 x 10,000) (= 12 x 10,000)
The Arabs
Using the twenty-eight letters of their number-alphabet the Arabs proceeded likewise,
but on a smaller scale: to note down the numbers beyond 1,000, all they had to do was
to place beside the letter ghayin (worth 1,000 and corresponding to the largest base
number in their system) the one representing the corresponding number of units, tens
or hundreds (see Chapter 19, p.246):
& c* b
2,000 3,000 10,000
(= 2 X 1,000) (= 3 X 1,000) (= 10 X 1,000)
The ancient Indians
The same idea was invoked by the Indians from the time of Emperor Asoka until the
beginning of the Common Era in the numerical notation that related to Brahmi script
(see Chapter 24, pp.378ff.). To write down multiples of 100, they used the multiplica-
tive principle, placing to the right of the sign for 100 the sign for the corresponding
units. For numbers beyond 1,000 they wrote to the right of the sign for 1,000 the sign
for the corresponding units or tens:
n
Tv
Tx
400
4,000
6,000
10,000
V*
T*
T<r
To r
(= 100 X 4)
(= 1,000 X 4)
(= 1,000 x 6)
(= 1,000 x 10)
» •
e
50,000
(= 50 X 1,000)
Fig. 23.15A. First extension of the multiplicative principle
331
The Egyptian hieroglyphic system (late period)
In Egyptian monumental inscriptions we find (at least from the beginning of the New
Kingdom) a remarkable diversion from the “classical" system: when a tadpole (hiero-
glyphic sign for 100,000) was placed over a lower number-sign, it behaved as a
multiplicator. In other words, by placing a tadpole over the sign for 18, for instance,
the number 100,018 (= 100,000 + 18) was no longer being expressed, but rather the
number 100,000 x 18 = 1,800,000 (a number which in the classical system would have
been expressed by setting eight tadpoles adjacent to the hieroglyphic for 1,000,000).
Example, 27,000,000
Expressed in the form:
100,000 x 270
£> 100,000
M
Taken from a Ptolemaic
nn
hieroglyphic inscription
Rn
(third - first century BCE)
The Egyptian hieratic system
But the preceding irregularity was actually the result of the way the hieroglyphic system
was influenced by hieratic notation: this used a more systematic method to note
down numbers above 10,000 according to the rule in question. (See Chapter 14,
pp. 171ff.)
Early Middle .
, New Kingdom
Kingdom Kingdom
10,000,000
Example: The number 494,800
(From the Great Harris Papyrus: 73, line 3.
New Kingdom)
'- aV *-
The Assyro-Babylonians and the Aramaeans provide a case in point.
They had a separate symbol for each of the numbers 1, 10, 100 and 1,000,
but instead of representing the hundreds or thousands by separate signs
or by repeating the 100 or 1,000 symbol as often as required, they had
the idea of placing the signs for 100 or 1,000 side by side with the symbols
for the units, thereby arriving at a multiplicative principle representing
arithmetical combinations such as
THE DISCOVERY OF THE MULTIPLICATIVE PRINCIPLE
lx 100
1 x 1,000
2x100
2 x 1,000
3x100
3 x 1,000
4x100
4 x 1,000
5x100
5 x 1,000
9x100
9 X 1,000
However, they continued to write numbers below 100 according to the old
additive method, repeating the sign for 1 or for 10 as often as required. The
number 7,659, for example, was written according to the following decom-
position (Fig. 23.16 and 23.17):
7,659 = (l + l + l + l + l + l + l)x 1,000
+ ( 1 + 1 + 1 + 1 + 1 + 1) X 100
+ (10 + 10 + 10 + 10 + 10 )
+ (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 )
First appearance: c.2350 BCE
Type: B1 (hybrid number-system of the first type: Fig. 23.33). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 13, pp.l37ff: Chapter 18, p.230)
Base numbers
*Symbol made up of that for
100 and that for 10
Representation based (in part) on
hybrid principle, broken down thus:
7,659 = (l + l + l + l + l + l + l)x 1,000
+ (l + l + l + l + l + l)x 100
+ (10 + 10 + 10 + 10 + 10 )
+ U + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
NOTATION FOR LARGE NUMBERS
This notation has succeeded in extending to the thousands by virtue of considering 1,000 as a fresh
unit of number and using the multiplicative rule:
•4. -C]*—
10,000
(= 10 X 1,000)
Example: 305,412
|k — Ik-
100,000
(=100x1,000)
1,000,000
(=1,000X1,000)
rrri-Vf it
= (3 x 100 + 5) x 1,000 + 4 x 100 + 10 + 2
(From Assyrian tablets dating from King Sargon II)
+ No doubt influenced by the structure of their oral number-system, the Mesopotamian Semites were the first to
consider extending the multiplicative rule to the notion of other orders of units, thus creating the first hybrid
number-system in history.
Fig. 23.16. Common Assyro-Babylonian number-system t
THE FINAL STAGE OF NUMERICAL NOTATION
First appearance: c.750 BCE
Type: B1 (hybrid number-system of the first type: Fig.23.33). Base 10
Need for zero sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 18, pp.228ff.)
Base numbers (Elephantine papyrus script)
1 ^ X 4
*Sign deriving from a multiplicative superimposition of two variants of the sign for 10
**Sign deriving from a multiplicative combination of two variants of the sign for 10 with that for 100
Example: 7,659
fit Ml Ml i Ml ill
Representation based (in part) on hybrid principle, broken down thus:
7,659 »(l + l + l + l + l + l + l)x 1,000
+ (l + l + l + l + l + l)x 100
+ (10 + 10 + 10 + 10 + 10)
+ (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
Fig. 23.17. Aramaean number-system
By such partial use of the multiplicative principle, the Assyro-Babylonian
number-system was therefore of the "partial hybrid” type.
At a later period, the inhabitants of Ceylon went through the same
change, but starting from a much better system than those above. They
assigned a separate sign not only to every power of 10, but also to each of
the nine units and to each of the nine tens, and then applied the same
principle as above. In this way, the number 7,659 can be broken down
(Fig. 23.18) as
7 x 1,000 + 6 x 100 + 50 + 9.
First appearance: c.600
-900 CE
Type: B2 (hybrid number-system of the second type: Fig. 23.34). Base 10
Need for zero
sign: No. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 24, p.374)
Base numbers (modem script)
61
ev>
GY©
00 Shv>
0
tn,
© 1
1
2
3
4 5
6
7
8
9
ea
a
g
«*) £!
V
lea
6
6
10
20
30
40 50
60
70
80
90
<35
©
100 (= 10 2 )
1,000 (=
10 3 )
Example: 7,659
3
©
<35
<8
©1
7
1,000
6
100
50
9
Representation based (in part) on hybrid principle, broken down thus: 7 x 1,000 + 6 x 100 + 5 x 10 + 9
Fig. 23 . 18 . Singhalese number-system
332
However, it was the Chinese, and the Tamils and Malayalams of
southern India, who made the best use of this approach. They too had
special signs for the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1,000, 10,000
but, instead of representing the tens by special signs, they had the idea
of extending the multiplicative principle to all the orders of magnitude of
their system, from the unit upwards. For intermediate numbers, they
placed the sign for 10 between the sign for the number of units and the sign
for the number of hundreds, the sign for 100 between the sign for the
number of hundreds and the sign for the number of thousands, and so on.
For the number 7,659 this gave rise to a decomposition of the type
7,659 = 7 x 1,000 + 6 x 100 + 5 x 10 + 9.
Such systems are of “complete hybrid" type, in which the representation
of a number resembles a polynomial whose variable is the base of the
number-system (Fig. 23.19 to 23.21).
First appearance: c.1450 BCE
Type: B5 (hybrid number-system of the fifth type: Fig. 23.37). Base 10
Need for zero sign: No, when the hybrid principle is rigorously applied. Yes, when the simplified rule below is
applied. Existence of zero sign: Yes, at a later date
Capacity for representation: Limited in the case of the unsimplified system (see Chapter 21, pp.263ff.)
Base numbers (modern script)
-IHHiA-bAJl + I T H
12 3 4 5 6 7 8 9 10 100 1,000 10,000
(= 10 2 ) (= 10 3 ) (= 10 4 )
Example: 7,659
Normal script
-t; A U £. + K
Representation based entirely on hybrid principle, broken down thus: 7 x 1,000 + 6 x 100 + 5 X 10 + 9
Abridged script in use since modem times
The above representation was sometimes produced in the simplified form below, thus tending towards
an application of the positional principle with base 10:
■fc i /i
NOTATION FOR LARGE NUMBERS
With the thirteen basic characters of this number-system, considering 10,000 as a fresh unit of number, the
Chinese were able to give a rational expression to all the powers of 10 right up to 100,000,000,000 (and
hence of all numbers from 999,999,999,999,999).
10,000 =
100,000 =
1 x 10,000
10 x 10,000, etc
Example: 487,390,629
(4 x 10 4 + 8 x 10 3 + 7 x 10 2 + 3 x 10 + 9) x 10 4 + (6 x 10 2 + 2 x 10 + 9)
Fig. 23.19. Common Chinese number-system
333
THE DIFFICULTIES OF THE PRECEDING SYSTEMS
The discovery of such hybrid principles was a great step forward, in
the context of the needs of the time, since it not only avoided tedious
repetitions of identical signs but also lightened the burden on the memory,
no longer required to retain a large number of different signs.
By the same token, the written representation of numbers could be
brought into line with their verbal expression (the linguistic structure of the
majority of spoken numbers had conformed, since the earliest times, to this
kind of mixed rule).
The principal benefit, however, of this procedure was greatly to extend
the range of numbers that could be represented (Fig. 23.15,16 and 19).
THE DIFFICULTIES OF THE PRECEDING SYSTEMS
Despite the considerable advance which these changes represent, the
capabilities of numerical notation remained very limited.
By making use of certain conventions of writing, the Greek mathemati-
cians managed to extend their alphabetic notation to cope with larger
numbers. Archimedes provides an important example. In his short arith-
metical treatise The Psammites, he conceived a rule which would allow him
to express very large numbers by means of the numeric letters of the Greek
alphabet, such as the number of grains of sand which would be contained
in the Sphere of the World (whose diameter is the distance from the earth
to the nearest fixed stars). In our modem notation, this number would be
expressed as a 1 followed by 64 zeros.
Chinese mathematicians also succeeded in extending their number-
system to accommodate numbers which could exceed 10 4096 , a number
which is far beyond any quantity that could be physically realised.
None of these systems, however, succeeded in achieving a rational
notation for all numbers, since they did not have the unlimited capacity
for representation which our own system has. The greater the order of
magnitude required, the more special symbols must be invented, or further
conventions of writing imposed.
We can therefore appreciate the undoubted superiority of our modern
system of numerical notation, which is one of the foundations of the intel-
lectual equipment of modem humankind. With the aid of a very small
number of basic symbols, any number whatever, no matter how large, may
be represented in a simple, unified and rational manner without the need
for any further artifice.
Yet another reason for the superiority of our system is that it is directly
adapted to the written performance of arithmetic.
It is precisely this fact which underlies the difficulty, or even impossibil-
ity, of doing arithmetic with the ancient number-systems, which remained
blocked in this respect for as long as they were in use.
For example, let us try to perform an addition using Roman numerals:
CCLXVI
+ DCL
+ MLXXX
+ MDCCCVII
=??????
Clearly, unless we translate this into our modem notation, this would be
very hard:
266
+ 650
+ 1,080
+ 1,807
= 3,803
But this is a mere additionl What about multiplication or division?
In systems of this kind, we are barely able to do arithmetic. This is due
to the static nature of the number-signs, which have no operational signifi-
cance but are more like abbreviations which can be used to write down the
results of calculations performed by other means.
In order to do arithmetical calculations, the ancients generally made use
of auxiliary aids such as the abacus or a table with counters. This requires
long and difficult training and practice, and remains beyond the reach of
ordinary mortals. It therefore remained the preserve of a privileged caste
of specialist professional calculators. This is not to say, however, that such
systems did not allow any written calculation.
The above addition can be carried out in the Roman system. This
involves proceeding by stages, by counting and then reducing the results
from each order of magnitude (five "I” replaced by one “V”, two “V” by one
“X”, five “X” by one “L”, two “L” by one “C”, and so on):
CC
L
X
V
I
+
M
D
CCC
V
II
+
D
C
L
+
M
L
XXX
MMM D CCC III
The Romans probably did use such a method. But since it is at bottom
a reduction to written form of operations performed on an abacus, they
THE FINAL STAGE OF NUMERICAL NOTATION
334
probably preferred to continue to use that instrument whose counters,
for all their inconvenience, were nonetheless easier to manipulate than the
symbols in their primitive representation of numbers.
We know also that, despite its very primitive character, the Egyptian
number-system allowed arithmetical calculations. The methods certainly
had the advantage of not obliging calculators to rely on memory. To multi-
ply or to divide, it was in fact enough to know how to multiply or to divide
by 2. Their methods, however, were slow and complicated compared with
our modem ones. Worse, though, they lacked flexibility, unification and
coherence.
On the other hand, the Graeco-Byzantine mathematicians certainly
succeeded in devising various rules for multiplication and division in terms
of the number-letters of their alphabet. There again, however, their proce-
dures were much more complicated, and above all far more artificial and
less coherent than ours.
These are all, therefore, mere attempts to invent rules of calculation
during the ancient times. But, “The fact is that the difficulties encountered
in former times were inherent in the very nature of the number-systems
themselves, which did not lend themselves to simple straightforward rules”
[T. Dantzig (1967)].
Therefore it was the discovery of our modern number-system, and above
all its popularisation, which allowed the human race to overcome the
obstacles and to dispense with all auxiliary aids to calculation such as we
have been considering.
DECISIVE FIRST STEP:
THE PRINCIPLE OF POSITION
In order to achieve a system as ingenious as our own, it is first necessary to
discover the principle of position. According to this, the value of a figure
varies according to the position in which it occurs, in the representation of
a number. In our modern decimal notation, a “3” has value 3 units, 3 tens
or 3 hundreds depending on whether it is in the first, second or third posi-
tion. To write seven thousand, six hundred and fifty-nine, all we have to do
is to write down the figures 7, 6, 5, and 9 in that order, since according to
the rule the representation 7,659 denotes the value
7 x 1,000 + 6 x 100 + 5 x 10 + 9.
Because of this fundamental convention, only the coefficients of the
powers of the base, into which the number has been decomposed, need
appear.
This, therefore, is the principle of position. Apparently as simple as
Columbus’s egg; but it had to be thought of in the first placel
Nowadays, this principle seems to us to have such an obvious simplicity
that we forget how the human race has stammered, hesitated and groped
through thousands of years before discovering it, and that civilisations as
advanced as the Greek and the Egyptian completely failed to notice it.
SYSTEMS WHICH COULD HAVE BEEN POSITIONAL
For all that, even in the earliest times a goodly number of different number-
systems could have led on to the discovery of the principle of position.
Consider for example the Tamil and Malayalam systems from south
India. According to the hybrid principle, the figure representing the
number of tens was placed to the left of the symbol for 10, the one repre-
senting the number of hundreds to the left of the symbols for 100, and so
on (Fig. 23.20 and 23.21).
First appearance: c.600 - 900 CE
Type: B5 (hybrid number-system of the fifth type: Fig.23.37). Base 10
Need for zero sign: No, when the hybrid principle is rigorously applied. Yes, when the simplified rule
below is applied.
Existence of zero sign: Not before the modem era
Capacity for representation: Limited in the case of the unsimplified system (see Chapter 24,
p.372)
System used among the Tamils (southern India)
Si
i
Base numbers (modern script)
e.nhff’&ShersiSi)
234567 89
U) m Zs
10 100 1,000
(= 10 2 ) (= 10 3 )
Example: 7,659
Normal script
cr & ffh m @ uo dm
>
7 1,000 6 100 5 10 9
Representation based entirely on hybrid principle, broken down thus:
7 x 1,000 + 6 x 100 + 5 x 10 + 9
Abridged script in use since modern times
The above representation was sometimes produced in the simplified form below, thus tending towards
an application of the positional principle with base 10 :
( oT fin @ fin
>
7 6 5 9
Fig. 23 . 20 . Tamil number-system
335
First appearance: c.600 - 900 CE
Type: B5 (hybrid number-system of the fifth type: Fig. 23.37). Base 10
Need for zero sign: No, when the hybrid principle is rigorously applied. Yes, when the simplified rule
below is applied.
Existence of zero sign: Not before the modern era
Capacity for representation: Limited in the case of the unsimplified system (see Chapter 24,
p.373)
System used among the Malayalam (southern India, Malabar coast)
Base numbers (modern script)
ca W <3) "3 9 OJ orb
Example: 7,659
Normal script
9 <T&° *D 'O ® JJJ rib
Representation based entirely on hybrid principle, broken down thus:
7 x 1,000 + 6 x 100 + 5 x 10 + 9
Abridged script in use since modern times
The above representation was sometimes produced in the simplified form below, thus tending towards
an application of the positional principle with base 10:
9 "3 ( 3 ) nrt>
7 6 5 9
Fig. 23.21. Malayalam number- system
In this way, the number 6,657, for example, would usually be written as
follows:
Bn Bn m @ U) <sr nr j fl&° mj fjj (§) JJJ 9
6 1,000 6 100 5 10 7
6 1,000 6 100 5 10 7
which corresponded to the decomposition
6 x 1,000 + 6 x 100 + 5 x 10 + 7.
Malayalam
SYSTEMS WHICH COULD HAVE BEEN POSITIONAL
Now, when we look at certain Tamil or Malayalam writings, we find that
the symbols for 10, 100, and 1,000 have in many cases been suppressed
[L. Renou and J. Filliozat (1953)]. The number 6,657 would then appear in
the abbreviated notation
Bn Bn @ CT mj mj Q) 9
6657 6657
> ■>
Tamil Malayalam
The result of this simplification is that the figures 6, 6, 5, and 7 have been
assigned values as follows:
• seven units to the figure 7 in the first place;
• five tens to the figure 5 in the second place;
• six hundreds to the figure 6 in the third place;
• six thousands to the figure 6 in the fourth place.
Thus the Tamil and Malayalam figures could be assigned values which
depended on where they occurred in the representation of a number.
This remarkable potential for evolution towards a positional number-
system is characteristic of hybrid numbering systems.
In such systems, in fact, the signs which indicate the powers of the base
(10, 100, 1,000) are always written in the same order, either increasing or
decreasing. Therefore it is natural that the people who used these systems
would be led, for the sake of abbreviation, to suppress these signs leaving
only the figures representing their coefficients.
This is what led certain Aramaic stone-cutters of the beginning of our era
to sometimes leave out the sign for 100 in their numeric inscriptions.
The inscription of Sa’ddiyat is a remarkable piece of evidence for this. We
know that in this region a hybrid system was used, whose basic signs had
the following forms and values:
» > -■ 3 >
1 5 10 20 100
But we see in this inscription (which dates from the 436th year of the
Seleucid era, or 124-125 CE) that the number 436 is written in the form
[B. Aggoula (1972), plate II]
THE FINAL STAGE OF NUMERICAL NOTATION
i >-i3ihi
1 + 5+10+20 4
instead of
1 + 5 + 10 + 20 + 100 x 4
36 + 100 X 4
For the same reason, the scribes of Mari often left out the cuneiform
figure for 100. This is all the more remarkable since the Mari system,
uniquely among Mesopotamian systems, was in use around the nineteenth
century BCE, therefore earlier than the period in which the Babylonian
positional system appeared (J.-M. Durand).
First appearance: c.2000 BCE
Type: B3 (hybrid number-system of the third type: Fig.23.35). Base 100
Need for zero sign: No, when the hybrid principle is rigorously applied. Yes, when the simplified rule
below is applied. Existence of zero sign: No
Capacity for representation: Limited (see Chapter 13, p.143)
T < V
1 10 100
Base numbers
(= 10 x 100) (=1002)
‘Number spelt out in letters
“Symbol derived by allocating a multiplicative function to the combination of the middle symbol with that for 10
Example: 7,659
Normal script
w < y
Representation based entirely on hybrid principle, broken down thus:
7,659 = (l + l + l + l + l + l + l)x 1,000
+ (l + l + l + l + l + l)x 100
+ (10 + 10 + 10+10+10)
+ C1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)
Abridged script
The above representation was sometimes produced in the simplified form below, with the number 100
4(ff 4f
/ 10 + 10 + 10 + 10 + 10 +10 + 10 \ / 10 + 10 + 10 + 10+10 \
Vi+l+l+l+i+l /^Vi+i+i+i+i+l+l+l+l/
Put differently, the notation thus tends towards a partial application of the positional principle with
base 100:
7,659 + [76 ; 59] = 76 X 100 + 59
Fig. 23 . 22 . Mari number-system
336
At Mari they used a hybrid system whose basic signs had the following
forms and values (Fig. 23.22):
r < t- <h &-
1 10 100 1,000 10,000
The number 476 would therefore be represented as:
4 X 100 + 76
At any rate, that is the normal representation of this number. But, as we
have only recently discovered, the Mari gave an abbreviated representation
to this number [D. Soubeyran (1984), tablet ARM, XXII 26]:
7
4 ; 76
This simplification was, nevertheless, only made for the hundreds figure,
not for all the powers of the base. For this reason, the Mari system never
became positional in the full sense. This system of notation remained
strongly bound to the methods of the old additive principle, and was there-
fore held back from taking the one vital further step forward from this
significant advance.
A similar simplification can be found in certain Chinese writers, who
also simplified their writing of numbers by suppressing the signs indicating
the tens, hundreds, thousands, etc. (see Fig. 23.19 above). For the number
67,859 we therefore find [E. Biot (1839); K. Menninger (1969)]:
instead of TYiUTi'f'ASjL'f*
> »
6 7 8 5 9 6 x 10,000 + 7 x 1,000 + 8xl00 + 5x 10+ 9
337
SYSTEMS WHICH COULD HAVE BEEN POSITIONAL
Finally, consider the Maya priests and astronomers. In order to simplify
the “Long Count” of their representation of dates, they too were led to
suppress all indications of the glyphs for their units of time, leaving only
the series of corresponding coefficients.
Let us take, for example, the Maya period of time expressed, in days, as
5 X 144,000 + 17 X 7,200 + 6 X 360 + 11 x 20 + 19. This would usually be
shows on the stelae as:
.
5 baktun 17 katun 6 tun
(= 5 x 144,000) (= 17 x 7,200) (= 6 x 360)
11 uinal 19 kin
(= ll x 20) (= 19 x i)
these were belated and therefore of no consequence for the universal
history); apart from these marginal exceptions it must be said that none of
these earlier systems arrived at the level of a truly positional numbering
system.
We therefore see yet again how people who have been widely separated
in time or space have, by their tentative researches, been led to very similar
if not identical results.
In some cases, the explanation for this may be found in contacts and
influences between different groups of people. But it would not be correct
to suppose that the Maya were in a position to copy the ideas of the people
of the Ancient World. The true explanation lies in what we have previously
referred to as the profound unity of human culture: the intelligence of homo
sapiens is universal, and its potential is remarkably uniform in all parts of
the world. The Maya simply found themselves in favourable conditions,
strictly identical to those of others who obtained the same results.
But, in their manuscripts, these astronomer-priests often preferred the
following form in which appear only the numerical coefficients associated
with the different time periods kin (days), uinal (periods of 20 days), tun
(periods of 360 days), katun (periods of 7,200 days), etc. This gives a strictly
positional representation:
5 (= 5 X 144,000)
17 (= 17 x 7,200)
6 (= 6 X 360)
11 (= 11 x 20)
19 (= 19 X 1)
This proves clearly that hybrid numbering systems had the potential to
lead to the discovery of the principle of position. However, a simplification
of a partial hybrid system could only lead to an incomplete implementation
of the rule of position, whereas the simplification of a fully hybrid system
was capable of leading to its complete implementation.
The simplification of the Maya notation for “Long Count” dates did give
rise to a positional system, as also did changes in certain other systems (but
THE EARLIEST POSITIONAL N U M B E R- S Y S T E M S
OF HISTORY
The civilisation which developed the basis of our modern number system
was therefore neither the first nor the only one to discover the principle
of position.
In fact, three peoples came to its full discovery earlier, and indepen-
dently. The numerical rule which is the basis of the positional system
was created:
• for the first time, some 2,000 years BCE, by the Babylonians;
• for the second time, slightly before the Common Era, by
Chinese mathematicians;
• for the third time, between the fourth and the ninth century
CE, by the Mayan astronomer-priests.
The Babylonian sexagesimal system represented a number such as 392
by writing the number 6 in the second (sixties) place, and the number 32 in
the first place, corresponding to a notation which might be transcribed
(Fig. 23.23) as [6; 32] (= 6 x 60 + 32).
THE FINAL STAGE OF NUMERICAL NOTATION
First appearance: c. 1800 BCE
Type: Cl (positional number-system of the first type: Fig. 23.38). Base 60
Need for zero sign: Yes. Existence of zero sign: Yes, but only later on (from the fourth century BCE)
Capacity for representation: Unlimited (see Chapter 13, pp.l46ff.)
Example: 7,659
TT ^
2 ; 7 ; 39
(7,659 = 2 x 60 2 + 7 X 60 + 39)
Representation based on positional principle, broken down thus:
11 + 1:1 + 1 + 1 + 1 + 1 + 1 + 1 ; 10 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ]
Fig. 23.23. Learned Babylonian number- system (the first positional number-system in history)
338
First appearance: c. 200 BCE
Type: Cl (positional number-system of the first type: Fig. 23.38). Base 10
Need for zero sign: Yes. Existence of zero sign: Yes, but only later on (from the eighth century, under
Indian influence)
Capacity for representation: Unlimited
Significant numbers
I II III Mil Mill T IT tit mr
(Symbols formed according to the additive principle, starting from two basic symbols, one
representing the number 1, the other the number 5)
Positional values
Example: 7,659
1 st rank:
1
2 nd rank:
10
3rd rank:
102 =
4th rank:
10 3 =
5th rank:
10 “ =
1
T
7
6
100
1,000
10 ,000, etc.
7 6 5 9
(7,659 = 7 x 10 J + 6 x 10 2 + 5 x 10 + 9)
Representation based on positional principle, broken down thus:
(5 + 1 + 1:5 + 1:1 + 1 + 1 + 1 + 1:5 + 1 + 1 + 1 + 11
Fig. 23.24. Learned Chinese number-system
339
SYSTEMS WHICH DID NOT SUCCEED
This is very much as we might today write 392' = 6 x 60' + 32' in the
form 6° 32' (6 degrees, 32 minutes).
The Chinese system was based on the same principle, with the difference
that the base of the number-system was decimal instead of being equal to
60. To write 392 in this system, we therefore place the figures 3, 9 and 2
in this order in a notation which we may (Fig. 23.24) transcribe as [3; 9; 2]
(= 3 x 100 + 9 x 10 + 2).
In the Maya system with base 20, we may write (Fig. 23.25)
[19; 12] (= 19 X 20 + 12). These Babylonian, Chinese and Maya systems
were, therefore, the earliest positional number-systems of history.
First appearance: c. fourth - ninth centuries CE
Type: Cl (positional number-system of the first type: Fig. 23.38). Base 20 (with an irregularity after the
units of the third order)
Need for zero sign: Yes. Existence of zero sign: Yes
Capacity for representation: Unlimited (see Chapter 22, pp.308ff., 316ff.)
Significant numbers
6 . . . tm
• •• • • • ^ * 25 22 SS
1 2 3 4 5 6 ... 10 11 12 ... 19
(Symbols formed according to the additive principle, starting from two basic symbols, one
representing the number 1, the other the number 5)
Positional values
1 st rank
1
2 nd rank
20
3rd rank
18x20 = 360
4 th rank
18 x 20 2 = 7,200
5 th rank
18 x 20 3 = 144,000, etc.
Example: 7,659
• 1 i
* 1 I
4 i
•— •
SB 19 +
(7,659 = 1 x 7,200 + 1 X 360 + 4 x 20 + 19)
Representation based on positional principle, broken down thus:
ll:l;l + l + l + l;5 + 5 + S + l + l + l + U
Fig. 23 . 25 . Learned Maya number-system
SYSTEMS WHICH DID NOT SUCCEED
Having made this fundamental and essential discovery, the way was in fact
open to each of these three peoples to represent any number whatever, no
matter how large, by means of a small set of basic signs. But none of these
three succeeded in taking advantage of their discovery.
The Babylonians indeed discovered the principle of position and applied
it to base 60. But it never occurred to them, for more than two thousand
years, to attach a particular symbol to each unit in their sexagesimal
system. Instead of fifty-nine different figures, they in reality had only two:
one for unity, and one for 10. All the rest had to be composed by duplicat-
ing these as many times as necessary up to 59 (Fig. 23.23).
The Chinese also discovered the principle of position and applied it to
base 10. But they did no better, for, instead of assigning a different sign
to each of the nine units, they preserved their ideographic system, in which
the number 8 was represented by the symbol for 5 with three copies of the
symbol for unity (Fig. 23.24).
Likewise the Maya system used the principle of position applied to base
20. But they again had only two distinct figures, one for unity and the other
for 5, instead of the nineteen which are required for full dynamic notation
in base 20 (Fig. 23.25).
For each of these three, it is somewhat as if the Romans had applied the
rule of position to their first few number-signs, for example writing 324 in
the form III II IIII, which would surely have led to confusion with:
I IIII IIII
(144)
II III IIII
(234)
II IIII III
(243)
III III III
(333)
III IIII II
(342)
IIII I IIII
(414)
IIII II III
(423) etc.
The Maya system had another source of difficulty inherent in its
very structure. The rule of position was not applied to the powers of
the base, but to values which were in fact adapted to the requirements
of the calendar and of astronomy.
Each number greater than 20 was written in a vertical column with as
many levels as there were orders of magnitude: the units were at the bottom
level, the twenties on the second level, and so on.
This system therefore became irregular from the third level onwards,
and was not rigorously founded on base 20. Instead of giving the multiples
of 20 2 = 400, 20 3 = 8,000, and so on, the different levels from the third
upwards in fact indicated multiples of 360 = 18 x 20, 7,200 = 18 x 20 2 ,
and so on.
But there was no such problem with the Babylonian and Chinese
systems, whose positional values corresponded exactly to the progression
of the values of their base:
THE FINAL STAGE OF NUMERICAL NOTATION
340
Units of
Learned
Babylonian
system (base 60)
Learned
Chinese system
(base 10)
Regular
positional
system (base 20)
Learned Maya
system (irregular
use of base 20)
1st order
i
i
i
i
2nd order
60
10
20
20
3rd order
60 2
10 2
20 2
18x20
4th order
60 3
10 3
20 3
18 X 20 2
5th order
60 4
10 4
20 4
18 X 20 3
6th order
60 s
10 5
20 s
18 x 20 4
If the Maya positional system had been constructed regularly on base 20,
the expression [7; 9; 3] would surely have signified
7 X 20 2 + 9 X 20 + 3 = 7 x 400 + 9 x 20 + 3.
But for the Maya priests this corresponded to 7 x 360 + 9 x 20 + 3.
This is one of the reasons why their system remained unsuited to practi-
cal written calculation.
A MAJOR SECOND STEP: DEVELOPMENT OF A
DYNAMIC NOTATION FOR THE UNITS
From what we have seen so far, it is dear that for a numerical notation to
be well adapted to written calculation, it must not only be based on the
principle of position but must also have distinct symbols corresponding
to graphic characters which have no intuitive visual meaning.
Otherwise put, the graphical structure of the number-signs must be like
that of our modern written numbers, in that “9”, for example, is not
composed of nine points nor of nine bars, but is a purely conventional
symbol with no ideographic significance (Fig. 23.26):
123456789
First appearance: c. fourth century CE
Type: C2 (positional number-system of the second type: Fig. 23.28). Base 10
Need for zero sign: Yes. Existence of zero sign: Yes
Capacity for representation: Unlimited (see Chapter 24, pp.356ff.)
Base numbers (present-day script)
123456789
(Symbols devoid of all direct visual intuition)
Positional values
1st rank: 1 3rd rank: 10 2 = 100
2nd rank: 10 4th rank: 10 J = 1.000. etc.
Example: 7,659 7 6 5 9
>
(7,659 = 7 x 10 3 + 6 X 10 2 + 5 x 10 + 9)
Fig. 23.26. Af odern number-system
THE FINAL FUNDAMENTAL DISCOVERY: ZERO
A no less fundamental condition for any number-system to be as well
developed and as effective as our own is that it must possess a zero.
For so long as people used non-positional notations, the necessity of this
concept did not make itself felt. The fact that there were signs for values
greater than the base of the system meant that these systems could avoid
the stumbling block which occurs whenever units of a certain order of
magnitude are absent. To write, for instance, 2,004 in Egyptian hieroglyph-
ics, it was sufficient to put two lotus flowers (for the thousands) and four
vertical bars (for the units), the total of the values thus being
1,000 + 1,000 + 1 + 1 + 1 + 1 = 2,004.
In the Roman numerals, this number would be written MMIIII, and there
was no need to have a special symbol to show that there were no hundreds
and no tens. In the Chinese system, they would represent this number in
the hybrid system, as a “2” followed by the symbol for 1,000 followed by
a “4”, corresponding to the decomposition 2,004 = 2 x 1,000 + 4.
On the other hand, once one has begun to apply place values on a regular
basis, it is not long before one faces the requirement to indicate that tens,
or hundreds, etc., may be missing. The discovery of zero was therefore a
necessity for the strict and regular use of the rule of position, and it was
therefore a decisive stage in an evolution without which the progress of
modern mathematics, science and technology would be unimaginable.
Consider our decimal system. To write thirty, we have to place “3” in the
second position, to have the value of three tens. But how do we show that
it is in the second position if there is nothing at all in the first position?
Therefore it is essential to have a special sign whose purpose is to indicate
the absence of anything in a particular position. This thing which signifies
nothing, or rather empty space, is in fact the zero. To arrive at the realisa-
tion that empty space may and must be replaced by a sign whose purpose
is precisely to indicate that it is empty space: this is the ultimate abstrac-
tion, which required much time, much imagination, and beyond doubt
great maturity of mind.
In the beginning, this concept was simply synonymous with empty space
thus filled. But it was gradually perceived that “empty” and “nothing", orig-
inally thought of as distinct, are in reality two aspects of one and the same
thing. Thus it is that the zero sign originally introduced to mark empty
space finally symbolises in our eyes the value of the null number, a concept
at the heart of algebra and modern mathematics.
Nowadays this is so familiar that we are no longer aware of the difficul-
ties which its lack caused to the early users of positional number-systems.
3 4 1
Its discovery was far from a foregone conclusion, for apart from India,
Mesopotamia and the Maya civilisation, no other culture throughout
history came to it by itself. We can gain some idea of its importance
when we recall that it escaped the eyes of the Chinese mathematicians,
who nonetheless succeeded in discovering the principle of position. Only
since the eighth century of our era, under the influence of our modern
number-system, did this concept finally appear in Chinese scientific
writings.
The Babylonians themselves were unaware of it for more than a thou-
sand years, leading as one can imagine to numerous errors and confusions.
They certainly tried to get round the difficulty by leaving empty space
where the missing units of a certain order would normally be found.
Therefore they wrote much as if we wrote the number one hundred and
six as 1. .6. But this was not enough to solve the problem in practice,
since scribes could easily overlook it in copying, through fatigue or care-
lessness. Moreover it was difficult to indicate precisely the absence of
two or more consecutive orders of magnitude, since one empty space
beside another empty space is not easily distinguished from a single
empty space.
It was therefore necessary to await the fourth century BCE to see the
introduction of a special sign dedicated to this purpose. This was a
cuneiform sign, which looked like a double oblique chevron, which was
used not only in the medial and final positions but also in the initial posi-
tion to indicate sexagesimal fractions of unity.
Medial: [3; 0; 9; 2] = 3 x 60 3 + 0 x 60 2 + 9 x 60 + 2
[3; 0; 0; 2] = 3 X 60 3 + 0 X 60 2 + 0 X 60 + 2
Final: [3; 1; 5; 0] = 3 x 60 3 + 1 x 60 2 + 5 X 60 + 0
[3; 1; 0; 0] = 3 X 60 3 + 1 X 60 2 + 0 x 60 + 0
Initial: [0; 3; 4; 2] = 0 + 3 x— + 4 x + 2 X i
60 60 2 60 3
This epoch, late in the history of Mesopotamia, saw the emergence of an
eminently abstract concept, the Babylonian zero, the first zero of all time,
to be followed some centuries later by the Maya zero.
ZEROS AND SYSTEMS
IMPERFECT ZEROS
Fig. 23.27. Classification of zeros
THE FINAL STAGE OF NUMERICAL NOTATION
34 2
The Maya of course understood that it was a genuine zero sign, since
they used it in medial as well as in final position. But, because of the
anomalous progression they introduced at the third position of their
positional system, this concept lost all operational usability.
The Babylonian zero not only had this possibility, it even filled the
role of an arithmetical operator, at least in the hands of the astronomers
(adjoining the zero sign at the end of a representation multiplied the
number represented by sixty, i.e. by the value of the base). But it was
never understood as a number synonymous with “empty”, and never corre-
sponded to the meaning of “null quantity” (Fig.23.27).
Despite these fundamental discoveries, therefore, none of these peoples
was able to take the decisive step which would result in the ultimate
perfection of numerical notation. Because of these imperfections, neither
the Babylonian nor the Chinese nor the Maya positional system ever
became adapted to arithmetical calculation, nor could ever give rise to
mathematical developments such as our own.
NUMBER-SYSTEMS WHICH COULD HAVE
BECOME DYNAMIC
We saw above how the complete adaptation of modern numerical notation
to practical arithmetic comes not only from the principle of position and
from the zero, but also from the fact that its figures correspond to graphic
signs which have no direct intuitive visual meaning.
Once again, the inventors of this system have neither the privilege nor
the honour of priority, since certain other systems had already enjoyed this
property since the earliest times.
With the Egyptians, as we have seen, the transition from hieroglyphic to
hieratic, and then to demotic script, radically changed the notation for the
first whole numbers. Starting with groupings of identical strokes repre-
senting the nine units, in the end we find cursive signs, independent of each
other, with no apparent intuitive meaning [G. Moller (1911-12); R. W.
Erichsen]:
123456789
Hieroglyphic
1
II
ill
ii
ill
ill
llll
llll
III
III
III
notation
ii
ii
ill
III
llll
Hieratic
notation
1
u
ui
Mil
L
—
Demotic
notation
1
M
b
r~
1
i
"1
Sim
1
The Egyptian cursive notations could therefore have risen to the status
of a number-system mathematically equivalent to our modern one if they
had only eliminated all the signs for numbers greater than or equal to 10,
replacing their additive principle by a principle of position which would
then have been applied to the signs for the first nine units. However,
this did not take place, since the Egyptian scribes remained profoundly
attached to their old and traditional method.
The same characteristic was present in yet another number-system, the
Singhalese, whose first nine number-signs certainly correspond to indepen-
dent graphics stripped of any capacity to directly and visually evoke the
corresponding units (Fig. 23.18):
123456789
Singhalese
notation
61 && GY® (Q 6Vv> 0 0 1 2? @1
But this system too preserved its initial hybrid principle, and therefore
remained stuck throughout its existence.
Why therefore did not well-conceived systems like the Tamil or the
Malayalam take this decisive step, and why did they not become positional
number-systems worthy of the name?
This is all the more surprising since both underwent simplification
conducive to such an end, since they had distinct signs for the nine
units which had no immediate visual associations as we have seen
(Fig. 23.20 and 23.21):
1
2
3
4
5
6
7
8
9
Tamil
notation
<55
lb
(5
ffn
sr
do
Malayalam
notation
CL
<rx
(V
(3)
9
0=1
orb
The reason is that this simplification was not extended to all the numbers.
The largest order of magnitude represented in these systems was 1,000.
Numbers greater than or equal to 10,000 were either spelled out in full,
or else they used the hybrid principle with the signs for 10, 100 and
1,000. These systems therefore remained firmly attached to their original
principle, and for this reason they too remained blocked.
Furthermore, because there was no zero, the rule of simplification would
only work on condition that every missing power of the base was followed
by the sign for the order of magnitude immediately below.
In order to avoid confusion between the abbreviated Tamil notation for
343
NUMBER-SYSTEMS WHICH COULD HAVE BECOME DYNAMIC
3,605, and the number 365, it was necessary to keep the indicator for the
hundreds in the representation of the former:
ffo ffn (3
3 6 5
»
365
ffh 5o m (5
3 6 x 100 5
>
3,605
TRANSLATION
— . O fit 3069
g 3 l «
* ^ ^ _35L 15345
— g ir-fc y? 12276
— HA - o i 138105
These systems were surely capable of rising to the level of our own if
only they had eliminated the signs for the numbers greater than or equal
to 10, and if the principle of position had been rigorously applied to
the remaining figures. For a while there would have been difficulties due
to the absence of zero, but, as necessity is the mother of invention, these
would have been overcome by the invention of zero.
The common Chinese system of numeration (which, as we have seen, is
in the same category as the two above) indeed went through this change.
In a table of logarithms, which is part of a collection of mathematical
works put together on the orders of the Emperor Kangshi (1662-1722 CE)
and published in 1713, we see the number 9,420,279,060 written in the
form [K. Menninger (1957), II, pp. 278-279]:
It, Rz:0 zl-kK o yr o
9420279060
By fully suppressing the classical signs for 10, 100, 1,000, and 10,000, by
systematising the rule of position for all numbers, and by introducing
a sign in the form of a circle to signify absence of an order of magnitude,
the ordinary Chinese notation has been transformed into a number
system equipped with a structure which is strictly identical to our own
(Fig. 23.19). These number representations are perfectly adapted to arith-
metical calculation.
The following example is taken from a work entitled Ding zhu suan fa
(“Ding zhu’s Method of Calculation"), published in 1335. It gives a table
showing the multiplication of 3,069 by 45 laid out in a way which no one
will have any difficulty in recognising [K. Menninger (1957), II, p. 300]:
This change only took place very late in the history of number-systems,
however; the “push in the right direction” to the traditional Chinese system
in fact came from the influence of the modern number-system.
THE “INVENTION” OF THE MODERN SYSTEM:
AN IMPROBABLE CONJUNCTION OF
THREE GREAT IDEAS
This fundamental “discovery” did not, therefore, appear all at once like
the fully formed act of a god or a hero, or single act of an imaginative
genius. These pages show clearly that it had an origin and a very long
history. Fruit of a veritable cascade of inventions and innovations, it
emerged little by little, following thousands of years during which an
extraordinary profusion of trials and errors, of sudden breakthroughs and
of standstills, regressions and revolutions occurred.
The discovery is the “fruit of slow maturation of primitive systems,
initially well conceived, and patiently perfected through long ages. With the
passage of time, some scholars succeeded when the circumstances were
right in perfecting the primitive instrument they had inherited from their
ancestors. Their motive for this effort was the passion they had to be
able to express large numbers. Other scholars, coming after them, realistic
and persistent, managed to get this revolutionary novelty accepted by the
calculators of their time. We inherit from both” (G. Guitel).
Finally it all came to pass as though, across the ages and the civilisations,
the human mind had tried all the possible solutions to the problem of
writing numbers, before universally adopting the one which seemed the
most abstract, the most perfected, and the most effective of all (Fig. 23.26,
23.27, 23.28, and 23.29).
THE FINAL STAGE OF NUMERICAL NOTATION
344
Base
m- 10
Base
m
FUNDAMENTAL MATHEMATICAL PROPERTIES OF
POSITIONAL NUMBER-SYSTEMS WITH BASE m
1. The number of digits (including zero) is equal to m.
2. Every integer x may be broken down in one single manner in the form of a polynomial in the degree
k- 1, with base m as a variable, and with coefficients all smaller than m. In other words, any number
x may be written in one way in the form:
+ + . . . + u 4 rr^ + u 3 n? + u 2 m + u,
where the integers u k , u k _ lt . . . u 2 , u lt all inferior to m, are symbolised by numbers in the system
under consideration. One may agree to write the number x in the following manner (where the hori-
zontal dash serves to avoid any confusion with the product u k u k _, . . . u 4 u 3 u 2 u
X = U k U k _ 1 ...U 4 U 3 U 2 U 1
3. The four fundamental arithmetical operations (addition, subtraction, multiplication and division)
are easily carried out in such a system, according to simple rules entirely independent of the base m
envisaged.
4. This positional notation may be extended easily to fractions with a base power for denominator,
and thus to a simple and coherent notation for all the other numbers, rational and irrational, by dint
of a point, following developments in positive and negative powers of m, thus analogous to decimal
numbers.
EGYPTIAN
hieroglyphic
PROTO-ELAMITE
CRETAN
HITTITE
hieroglyphic
GREEK
archaic
AZTEC
ZAPOTEC
GREEK
acrophonic
ETRUSCAN
ROMAN
SOUTHERN ARABIC
SUMERIAN
HEBRAIC
GREEK
COPT
SYRIAC
ARMENIAN
GEORGIAN
GOTHIC
ARABIC
Abjad
EGYPTIAN
hieratic and demotic
INDIAN
Brihmi
ASSYRO-
BABYLONIAN
current system
ARAMAEAN
PHOENICIAN
NABATAEAN
PALMYRENEAN
KHATREAN
Fig. 23.28. Classification of positional number-system (Type C2)
Fig. 23.29A. Classification of written number-systems
345
THE “INVENTION” OF THE MODERN SYSTEM
SINGHALESE
MARI
CATEGORY B2
Base m systems
Q
Figures for
1 2 3 4 ... (m - 1)
X
HYBRID
10
m 2m 3 m Am . . . (m - l)/n
-
-
NUMBER-
Additive notation for
<
SYSTEMS
numbers below m 2
h
(continued)
Multiplicative notation for
<
multiples of m 2 m 3 m 4 . . .
CATEGORY B3
Base m systems
Figures for
1 m 2 m 3 m 4 m 5 . . .
Additive notation for numbers
below m 2
Multiplicative notation for
multiples of m 2 m 3 m 4 . . .
CATEGORY B4
Base m systems
Figures for
1 2 3 4 . . . (m - 1)
m 2m 3 m Am ... {m - 1 )m
Additive notation for
numbers below m 2
Multiplicative notation for
multiples of m 2 m 3 m 4 . . .
CHINESE
current system
TAMIL
10
10
'
CATEGORY B5
Base m systems
Figures for
1 2 3 A .. .(m-1) m m 2 m 3
MALAYALAM
MAYA
10
20
_
Multiplicative notation for
multiples of m 2 m 3 m 4 . . .
expressing length
of time (stelae)
CATEGORY Cl
BABYLONIAN
learned system
60
- i =10 — '
Base m systems
Figures for / and k (privileged
divisor for base m).
CHINESE
10
- * = 5 —
Additive notation for numbers
learned system
(number-bars)
below the base m (i.e. the significant
units of each order [m - 1 J are
MAYA
learned system
(Dresden codex)
20
-
= 5 —
denoted by repeating each of the
two base numbers as many times
as needed).
Systems based
(at least after a
certain order) on
a mixed principle
(both additive
and multi-
plicative) that
invokes the
multiplication
rule to represent
consecutive
orders of units.
MODERN
NUMERATION
CATEGORY C2
Base m systems
Figures for
1 2 3 4 ... (m - 1)
These figures are distinct and
unconnected with any direct
visual intuition.
TYPE C
POSITIONAL }
NUMBER-
SYSTEMS
The value of
number-symbols |
is determined
by their position |
in the writing
of the numbers. I
|Such numerations]
require the use
of the zero.
Fig. 23 . 29 B. Classification of written number-systems
The story begins with primitive systems whose structure was based on
the realities encountered in the course of accounting operations in ancient
times. A certain amount of progress in the right direction was made, result-
ing in the creation of number-systems distinctly superior to the incoherent
Roman numerals. But the paths which were taken led to dead ends, because
these procedures incorporated only addition.
The awkwardnesses of these representations, together with the need for
rapid writing, then brought about the development of hybrid systems, very
conveniently mirroring spoken language, of which they can be seen as a
more or less faithful transcription, sometimes showing a polynomial struc-
ture identical to that of the counting table, and at the same time extending
considerably the power to express large numbers. Here too, however, the
road was blocked. The principle they incorporated was inappropriate for
arithmetical calculation, allowing addition and subtraction at best though
at the cost of complicated manoeuvres, but useless for multiplication or
division. In short, these systems were really only adequate for noting and
recording numbers.
The decisive step in the adoption of systems of numerical notation with
unlimited capacity, simple, rational, and immediately useable for calcula-
tion, could only be taken by inventing a well- conceived positional notation.
This step was finally taken by simplifying hybrid notation, or by abbreviat-
ing systems for transferring numbers to the abacus, by the suppression of
the signs indicating the powers of the base or by eliminating the columns
of the abacus itself.
On the other hand this progress demanded a much higher level of
abstraction, and the most delicate concept of the whole story: the zero. This
was the supreme and belated discovery of the mathematicians who soon
would come to extend it, from its first role of representing empty space, to
embrace the truly numeric meaning of a null quantity (Fig. 23.27).
THE KEYSTONE OF OUR MODERN
NUMBER-SYSTEM
Number and culture are one, for “to know how a people counts is to know
what kind of people it is” (to adapt Charles Moraze). At least from this
point of view, the degree of civilisation of a people becomes something
measurable.
Thus it now appears to us indisputable that the Babylonians, the Chinese
and the Maya were superior to the Egyptians, the Hebrews and the
Greeks. For, while the former took the lead with their fundamental dis-
coveries of the principle of position and the zero, the others remained
locked up for centuries with number-systems which were primitive,
THE FINAL STAGE OF NUMERICAL NOTATION
incoherent, and unuseable for practically any purpose save writing
numbers down.
The measure of the genius of Indian civilisation, to which we owe our
modern system, is all the greater in that it was the only one in all history to
have achieved this triumph.
Some cultures succeeded, earlier than the Indian, in discovering one or
at best two of the characteristics of this intellectual feat. But none of them
managed to bring together into a complete and coherent system the neces-
sary and sufficient conditions for a number-system with the same potential
as our own.
We shall see in Chapter 24 that this system began in India more than
fifteen centuries ago, with the improbable conjunction of three great ideas
(Fig. 23.26), namely:
• the idea of attaching to each basic figure graphical signs
which were removed from all intuitive associations, and did
not visually evoke the units they represented;
• the idea of adopting the principle according to which the
basic figures have a value which depends on the position they
occupy in the representation of a number;
• finally, the idea of a fully operational zero, filling the empty
spaces of missing units and at the same time having the
meaning of a null number.
This fundamental realisation therefore profoundly changed human exis-
tence, by bringing a simple and perfectly coherent notation for all numbers
and allowing anyone, even those most resistant to elementary arithmetic,
the means to easily perform all sorts of calculations; also by henceforth
making it possible to carry out operations which previously, since the dawn
of time, had been inconceivable; and opening up thereby the path which led
to the development of mathematics, science and technology.
It is also the ultimate perfection of numerical notation, as we shall see
in the classification of the numerical notations of history to follow. In
other words, no further improvement of numerical notation is necessary,
or even possible, once this perfect number-system has been invented. Once
this discovery had been made, the only possible changes remaining could
only affect
• the choice of base (which could be 2, 8, 12, or any other
number greater than 2);
• the graphical form of the figures.
But no further change is possible in the essential structure of the system,
now once and for all unchangeable by virtue of its mathematical perfection.
346
Apart from the base (which is only a matter of how things are to be
grouped, and therefore of the number of different basic figures for the
units), a number-system structurally identical to ours is completely
independent of its symbolism. It does not matter if the symbols are
conventional graphic signs, letters of the alphabet, or even spoken words,
provided it rests strictly and rigorously on the principle of position and
it incorporates the full concept of the symbol for zero.
Here is an instructive example. It concerns the great Jewish scholar
Rabbi Abraham Ben Meir ibn Ezra of Spain, better known as Rabbi Ben
Ezra. He was born at Toledo around 1092, and in 1139 undertook a long
journey to the East, which he completed after passing some years in Italy.
Then he lived in the South of France, before emigrating to England where
he died in 1167. No doubt influenced by his encounters while travelling, he
instructed himself in the methods of calculation which had come out of
India (precursors of our own). He then set out the principal rules of these
in a work in Hebrew entitled Sefer ha mispar (“The Book of Number”)
[M. Silberberg (1895); M. Steinschneider (1893)].
Instead of conforming strictly to the graphics of the original Indian
figures, he preferred to represent the first nine whole numbers by the first
nine letters of the Hebrew alphabet (which, of course, he knew well since
childhood). And, instead of adopting the old additive principle, on which
the alphabetic Hebrew number-system had always been based (Fig. 23.12),
he eliminated from his own system every letter which had a value greater
than or equal to 10. He kept only the following nine, to which he applied the
principle of position, and he augmented the series with a supplementary
sign in the shape of a circle, which he called either sifra (from the Arab word
for “empty”)
or galgal (the Hebrew word for "wheel”):
X
2 2 *T H 1 T
n
D
1
2 3 4 5 6 7
8
9
aleph
bet gimmel dalet he vov zayin
het
tet
Thus, instead of representing the number 200,733 in the traditional
Hebrew form (below, on the right), he wrote it as follows (below, left):
2 2 T 0 0 3 instead of ) 0 H "1
3 3 7 0 0 2 3 30 300 400 200.000
Thus it was that the Hebrew number-system, in his hands, changed from
a very primitive static decimal notation, by becoming adapted to the prin-
ciple of position and the concept of zero, into a system with a structure
rigorously identical to our own and, therefore, infinitely more dynamic.
However, this remarkable transformation seems not to have been
347
followed by anyone other than Rabbi Ben Ezra himself, a unique case, it
would seem, in the history of this system.
This isolated case, nonetheless, provides us with a model for a situation
which must have come about many times following the invention and prop-
agation of the positional system originating in India, mother of the modern
system and of all those influenced by it. This is the situation in which schol-
ars and calculators making contact, individually or in groups, with Indian
civilisation and then, becoming aware of the ingenuity and many merits of
their positional number-system, decide either to adopt it (individually or
collectively) in its entirety or else to borrow its structure in order to perfect
their own traditional systems.
Now that we can stand back from the story, the birth of our modem
number-system seems a colossal event in the history of humanity, as
momentous as the mastery of fire, the development of agriculture, or the
invention of writing, of the wheel, or of the steam engine.
THE CLASSIFICATION OF THE WRITTEN NUMBER-SYSTEMS
THE CLASSIFICATION OF THE WRITTEN
NUMBER-SYSTEMS OF HISTORY
With this survey we shall close our chapter. Its aim is to systematise the
various comparisons we have made up to this point in a more formal and
mathematical manner.
Before I enter into the heart of the matter, I wish at this point to render
special homage to Genevieve Guitel, whose remarkable Classification hierar-
chisee des numerations ecrites has, for the first time, permitted me to bring
together, intellectually speaking, systems which distance and time have
separated almost totally.
This classification was published in her monumental Histoire comparee
des numerations ecrites, which has been an essential contribution to my
understanding of this field.
Prior to her, as Charles Moraze has emphasised, there were certainly
other histories of the number-systems, but none has attributed such
importance to the comparisons which she has established on the basis of
a principle of classification “which has the double merit of being both
mathematically rigorous and remarkably relevant to the historical data
which were to be put in order”.
This classification, which I take up in my turn (while presenting it under
a new light and amending certain details, resulting especially from the most
recent archaeological discoveries), reveals that the numerical notations
devised over five thousand years of history and evolution were not of unlim-
ited variety. They may in fact be divided into three main types, of which
each may be subdivided into various categories (Fig. 23.29):
• additive systems, which fundamentally are simply transcrip-
tions of even more ancient concrete methods of counting
(Fig. 23.30 to 23.32);
• hybrid systems, which were merely written transcriptions of
more or less organised verbal expressions of number (Fig.
23.33 to 23.37);
• positional systems, which exhibit the ultimate degree of
abstraction and therefore represent the ultimate perfection
of numerical notation (Fig. 23.28 and 23.38).
NUMBER-SYSTEMS OF THE ADDITIVE TYPE
These are the ones based on the principle of addition, where each figure has
a characteristic value independent of its position in a representation.
Number-systems of this type in turn fall into three categories.
THE FINAL STAGE OF NUMERICAL NOTATION
348
Additive number-systems of the first kind
Our model for this is the Egyptian hieroglyphic system, which assigns a
separate symbol to unity and to each power of 10, and uses repetitions of
these signs to denote other numbers (Fig. 23.1).
CLASSIFICATION OF ADDITIVE NUMBER-SYSTEMS
This is exactly what also happens in the Cretan number-system and in
the Hittite hieroglyphic and archaic Greek systems. All of these systems are
therefore strictly identical, and they differ only in the written forms of their
respective figures (Fig. 23.3 to 23.5).
When they are in base 10, the additive systems of the first kind are
therefore characterised by a notation which is based on arithmetical
decompositions of the type:
Table 1
1st decimal order
(units)
2nd decimal order
(tens)
3rd decimal order
(hundreds)
4th decimal order
(thousands)
1
10
10 2
10 3
1 + 1
10 + 10
10 2 + 10 2
10 3 + 10 3
1 + 1 + 1
10 + 10 + 10
10 2 + 10 2 + 10 2
10 3 + 10 3 + 10 3
Special notation for 1, 10, 10 2 , 10 3 , etc.
Additive notation for all other numbers.
Now if we consider the Aztec number-system, we find that even though
it uses a different base (base 20), still like the others it assigns a special
symbol only to unity and to the powers of the base (Fig. 23.6):
Base Aztec system
20
1
20
20 2
20 3
20 4
m
1
m
m 2
m 3
m 4
10
1
10
10 2
10 3
10 4
Egyptian hieroglyphic system
Since this is an additive system and proceeds by repetition of identical
signs, it is characterised by a notation which depends on arithmetical
decompositions of the type:
Table 2
1st vigesimal order
(units)
2nd vigesimal order
(twenties)
3rd vigesimal order
(four hundreds)
4th vigesimal order
(eight thousands)
i
20
20 2
20 3
1 + 1
l + i + i
20 + 20 + 20
20 2 + 20 2 + 20 2
20 3 + 20 3 + 20 3
Special notation for 1, 20, 20 2 , 20 3 , etc.
Additive notation for all other numbers.
Fig. 23.30. Classification of additive number-systems (Type Al)
349
NUMBER-SYSTEMS OF THE ADDITIVE TYPE
The Aztec system, therefore, is intellectually related to the preceding
ones, and differs only in having base 20 instead of base 10.
All of these notations therefore belong to the same type (Fig. 23.30).
CLASSIFICATION OF ADDITIVE NUMBER-SYSTEMS (continued)
Fig. 23 . 31 . Classification of additive number-systems (type A2)
Additive systems of the second kind
A characteristic example is the Greek acrophonic system. It is in base
10, and adopts the principle of addition assigning a special symbol to each
of the numbers 1, 10, 100, 1,000, etc., as well as to each of the following:
5, 50, 500, 5,000, and so on (Fig. 23.7). Intellectually, therefore, it is of
the same kind as the Southern Arabic system, the Etruscan, and the Roman,
characterised by arithmetical decompositions of the type (Fig. 16.18, 16.35
and 23.8):
Table 3
1st decimal order
(units)
2nd decimal order
(tens)
3rd decimal order
(hundreds)
4th decimal order
(thousands)
1
10
10 2
10 3
10 + 10
10 2 + 10 2
10 3 + 10 3
10 + 10 + 10
10 2 + 10 2 + 10 2
10 3 + 10 3 + 10 3
5
5x10
5 X 10 2
5 X 10 3
5 + 1
5 X 10 + 10
5 X 10 2 + 10 2
5 X 10 3 + 10 3
5 + 1 + 1
5 X 10 + 10 +10
5 x 10 2 + 10 2 + 10 2
5 x 10 3 + 10 3 + 10 3
Special notation for 1, 5, 10, 5 x 10, 10 2 , 5 x 10 2 , etc.
Additive notation for all other numbers.
Denoting by k the divisor of the base m which thus acts as auxiliary base
(here, m = 10 and k = 5), we see that these systems assign a special symbol
not only to each power of the base (1, m, m 2 , m 3 , . . .) but also to the
product of each of these with k (k, km, km 2 , km 3 , . . . ). As the following table
shows, this is exactly the structure which can be discerned in the regular
progression of the Sumerian number-system (Fig. 23.2):
Sumerian system (where m=60 and £=10)
1
10
60
10X60
60 2
10 X 60 2
60 3
10 x 60 3
1
k
m
km
m 2
km 2
m 3
km 3
1
5
10
5x10
10 2
5 x 10 2
10 3
5 X 10 3
Greek acrophonic numerals (where m = 10 and k = 5)
THE FINAL STAGE OF NUMERICAL NOTATION
Looking at it from another point of view, the succession of numbers receiv-
ing a particular sign in the Sumerian system may be expressed as:
1st order 1 <
10 <
2nd order 60 <
10 x 60 <
3rd order 60 2 <
10 X 60 2 <
4th order 60 3 <
10 x 60 3 <
> 1
> 10
> 10.6
> 10 . 6.10
> 10 . 6 . 10.6
> 10 . 6 . 10 . 6.10
•> 10 . 6 . 10 . 6 . 10.6
> 10 . 6 . 10 . 6 . 10 . 6.10
and so on, alternating the numbers 10 and 6.
Let a and b denote the divisors of m which act as alternating auxiliary
bases (where, in the Sumerian case, we have m = 60, a = 10 and b = 6). This
succession therefore exactly corresponds to that of the Greek acrophonic
system (where m = 10, a = 5 and b = 2):
Table 4
Sumerian
1
<• •
Mathematical Characterisation
••> 1 <• •
* >
Greek
1
10
<• *
* >
a
<• •
• >
5
10.6
<• ■
• *>
a.b
<• *
• •>
5.2
10.6.10
<• •
• *>
a 2 b =
a.b.a
<* *
• >
5.2.5
10.6.10.6
<♦ ♦
• >
a 2 b 2 =
a.b.a.b
<• •
♦ >
5.2.5.2
10.6.10.6.10
<• •
• >
a 3 b 2 =
a.b.a.b.a
<• *
• >
5.2.5.2.5
10.6.10.6.10.6
<• •
■ •>
a 3 b 3 =
a.b.a. b.a.b
<• ‘
■ >
5.2. 5.2. 5.2
a = 10 a = 5
b = 6 b = 2
The Greek structure is thus mathematically identical to that of the
Sumerian, corresponding to arithmetical decompositions of the type:
1st sexagesimal order 2nd sexagesimal order 3rd sexagesimal order 4th sexagesimal order
(units) (sixties) (multiples of 60) (multiples of 60)
1 60 60 2 60 3
1 + 1 60 + 60 602 + 60 2 60 3 + 60 3
1 + 1 + 1 60 + 60 + 60 60 2 + 60 2 + 60 2 60 3 + 60 3 + 60 3
10 x 60 10 x 60 2 10 x 60 3
10 X 60 + 60 10 x 60 2 + 60 2 10 x 60 3 + 60 3
10 x 60 + 10 x 60 10 x 60 2 + 10 x 60 2 10 x 60 3 + 10 x 60 3
Special notation for 1, 10, 60, 10 x 60, 60 2 , 10 x 60 2 , etc.
Additive notation for all other numbers.
350
All these systems therefore belong to the same category (Fig. 23.31).
CLASSIFICATION OF ADDITIVE NUMBER-SYSTEMS (concluded)
Systems of Type A 3
(additive number-systems of the third type)
Number-systems derived from
additive systems of the first type
Notations whose numbers are simply the letters of the
alphabet, considered in the original Phoenician order
Fig. 23.32. Classification of additive number-systems (Type A 3 )
Additive systems of the third kind
The Egyptian hieratic system and the Greek alphabetic system are typical
examples of this type. Intellectually, they correspond to the following
characterisation (Fig. 23.10 to 23.13, and 23.32):
351
SYSTEMS OF HYBRID TYPE
1st decimal order
(units)
2nd decimal order
(tens)
3rd decimal order
(hundreds)
4th decimal order
(thousands)
1
10
100
1,000
2
20
200
2,000
3
30
300
3,000 etc.
Special notation for each unit of each number.
1
2
3
4
5
6
7
8
9
10
2.10
3.10
4.10
5.10
6.10
7.10
8.10
9.10
10 2
2.10 2
3.10 2
4.10 2
5.10 2
6.10 2
7.10 2
8.10 2
9.10 2
10 3
2.10 3
3.10 3
4.10 3
5.10 3
6.10 3
7.10 3
8.10 3
9.10 3
10 4
2.10 4
3.10 4
4.10 4
5.10 4
6.10 4
7.10 4
8.10 4
9.10 4
Additive notation for all other numbers.
SYSTEMS OF HYBRID TYPE
These are founded on a mixed system in which both addition and multipli-
cation are involved. On this basis, the multiples of the powers of the base
are, from a certain order of magnitude onwards, expressed multiplicatively.
This type of system can be divided into five categories.
CLASSIFICATION OF HYBRID NUMBER-SYSTEMS
Fig. 23.33. Classification of hybrid number-systems (Type Bl, partial hybrid)
Hybrid systems of the first kind
The common Assyro-Babylonian system and that of the western Semitic
peoples (Aramaeans, Phoenicians, etc.) are typical examples of this type.
They have base 10, and assign a special symbol to each of the numbers 1,
10, 100, 1,000, etc., and use multiplicative notation for consecutive multi-
ples of each of these powers of 10. At the same time, the units and the tens
are still represented according to the old principle of additive juxtaposition.
When in base 10, hybrid systems of the first kind are characterised by
arithmetical decompositions of the type (Fig. 23.16, 23.17, and 23.33):
Table 5
1st order
2nd order
3rd order
4th order
(units)
(tens)
(hundreds)
(thousands)
1
10
lXlO 2
lxlO 3
1 + 1
10 + 10
(1 +' 1) x 10 2
(1 + 1) x 10 3
1 + 1 + 1
10 + 10 + 10
(1 + 1 + 1) x 10 2
(1 + 1 + 1) x 10 3
Special notation for 1, 10, 10 2 , 10 3 , etc. Additive notation for the numbers 1 to 99.
Multiplicative notation for the multiples of the powers of 10, starting with 100.
A notation involving both addition and multiplication for other numbers
CLASSIFICATION OF HYBRID NUMBER-SYSTEMS (continued)
Fig. 23.34. Classification of hybrid number-systems (TypeB 3 , complete hybrid)
THE FINAL STAGE OF NUMERICAL NOTATION
352
Hybrid systems of the second kind
The model for this type is the Singhalese system. It has base 10, and assigns
a special symbol to each unit, to each of the tens, and to each of the
powers of 10. The notation for the hundreds, thousands, etc. follows
the multiplicative rule (Fig. 23.18).
When in base 10, hybrid systems of this kind are characterised by a nota-
tion which is based on arithmetical decompositions of the type (Fig. 23.34):
1st order
2nd order
3rd order
4th order
(units)
(tens)
(hundreds)
(thousands)
1
10
lxlO 2
1 x 10 3
2
20
2 X 10 2
2 X 10 3
3
30
3 x 10 2
3 x 10 3
Special notation for units, tens, 10 2 , 10 3 , etc. Additive notation for the numbers 1 to 99.
Multiplicative notation for the multiples of the powers of 10, starting with 100.
A notation involving both addition and multiplication for other numbers
CLASSIFICATION OF HYBRID NUMBER-SYSTEMS (continued)
Fig. 23.35.
Hybrid systems of the third kind
The model for this type is the Mari system. It uses base 100, and gives
a special symbol for each unit, for 10, and for each power of 100. The
notation for the hundreds, the ten thousands, etc. uses the multiplicative
rule. The system is characterised by a notation based on arithmetical
decompositions of the type (Fig. 23.22 and 23.35):
1st centennial order
2nd centennial order
units
tens
hundreds
thousands
1
10
1X10 2
lxlO 3
1 + 1
10 + 10
(1 + 1) x 10 3
1 + 1 + 1
10 + 10 + 10
(1 + 1 + 1) x 10 3
Special notation for 1, 10, 10 2 , 10 3 , etc.
Additive notation for the numbers from 1 to 99.
Additive notation for the numbers 1 to 99.
Multiplicative notation for multiples of the powers of 10 2 , starting with the first (100).
A notation involving both addition and multiplication for other numbers.
CLASSIFICATION OF HYBRID NUMBER-SYSTEMS (continued)
Fig. 23.36. Classification of hybrid number-systems (Type B4, complete hybrid)
353
SYSTEMS OF HYBRID TYPE
Hybrid systems of the fourth kind
The model for this type is the Ethiopian system. It has base 100, and assigns
a special sign to each unit and to each of the tens, and also to each power of
100. The notation for the hundreds, the ten thousands, etc. uses a multi-
plicative rule applied to these figures. The system is characterised by a
notation based on arithmetical decompositions of the type (Fig. 23.36):
1st centennial order
2nd centennial order
units
tens
hundreds
thousands
1
10
lxlO 2
1X10 3
2
20
2 x 10 2
2 x 10 3
3
30
3 x 10 2
3 x 10 3
4
40
4 x 10 2
4 x 10 3
5
50
5 x 10 2
5 x 10 3
6
60
6 x 10 2
6 x 10 3
7
70
7 x 10 2
7 x 10 3
8
80
8 x 10 2
8 x 10 3
9
90
9 x 10 2
9 x 10 3
Special notation for each unit, each ten and for each of 10 2 , 10 3 , etc. Additive notation for the numbers from 1 to 99.
Multiplicative notation for multiples of the powers of 10 2 , starting with the first (100).
A notation involving both addition and multiplication for other numbers.
CLASSIFICATION OF HYBRID NUMBER-SYSTEMS (concluded)
Fig. 23.37. Classification of hybrid number-systems (Type B 5 , complete hybrid)
Hybrid systems of the fifth kind
The model for this type is the common Chinese system, as well as the Tamil
and Malayalam systems. These systems have base 10, and assign a special
symbol to each unit and to each power of 10. The notation for the tens,
the hundreds, the thousands, etc. uses the multiplicative principle.
When in base 10, hybrid systems of the fifth kind are characterised by
a notation based on arithmetical decompositions of the following type
(Fig. 23.37):
1st order
(units)
2nd order
(tens)
3rd order
(hundreds)
4th order
(thousands)
1
1x10
lxlO 2
lxlO 3
2
2x10
2 x 10 2
2 x 10 3
3
3x10
3 x 10 2
3 x 10 3
4
4x10
4 X 10 2
4 x 10 3
5
5x10
5 x 10 2
5 x 10 3
6
6x10
6 x 10 2
6 x 10 3
7
7x10
7 x 10 2
7 x 10 3
8
8x10
8 x 10 2
8 x 10 3
9
9x10
9 x 10 2
9 x 10 3
Special notation for each unit of the first order and for each of the numbers 10, 10 2 , 10 3 .
Multiplicative notation for multiples of powers of the base, starting with 10. Notation involving both
addition and multiplication for other numbers.
Unlike hybrid systems of the first kind, which only partially use the
multiplicative principle, those of types 3, 4 and 5 bring the principle into
play in the notation for all the orders of magnitude greater than or equal to
the base. Additionally, the representation of other numbers is based on the
coefficients of a polynomial whose variable is the base. For these reasons
systems of this type are also called complete hybrid systems.
POSITIONAL SYSTEMS
The systems are based on the principle that the value of the figures is
determined by their position in the representation of a number.
Historically, there have been only four originally created positional
systems:
• the system of the Babylonian scholars;
• the system of the Chinese scholars;
• the system of the Mayan astronomer-priests;
• and finally our modem system which, as we shall see in the
next chapter, originated in India.
THE FINAL STAGE OF NUMERICAL NOTATION
354
These systems (which require the use of a zero) may be divided into two
categories.
CLASSIFICATION OF POSITIONAL NUMBER-SYSTEMS
1 1 +1
10 10 + 10
10 + 10 + 10 +
1 + 1 + 1 l + l + l + l...
10 + 10+10 10 + 10 + 10 + 10 . . .
10+ 10+1 + 1 + 1 + 1+ l + l + l + l +1
Fig. 23.38. Classification of positional number-systems (Type Cl)
Positional systems of the first kind
This type includes:
1. - The system of the Babylonian scholars
This has base 60. The notation for the units of the first order (from 1
to 59) corresponds to arithmetical decompositions of the following type
for the two basic figures, of which one represents unity and the other
10 (Fig. 23.23):
2. - The system of the Chinese scholars
This has base 10. The notation for the units of the first order (from 1 to 9)
corresponds to arithmetical decompositions of the following type for the
two basic figures, of which one represents unity and the other 5 (Fig.
23.24):
1
5 + 1
1 + 1
5 + 1 + 1
1 + 1 + 1
5 + 1 + 1+1
l+l+l+l
5 + 1 + 1 + 1+1
3. - The system of the Maya scholars
This has base 20. The notation for the units of the first order (from 1 to 19)
corresponds to arithmetical decompositions of the following type using
two base figures, one representing unity and the other the number 5. In
addition there is an irregularity starting with the third order in the succes-
sion of positional values (Fig. 23.25):
1
5 + 1
5 + 5+1
5+5+5+1
1 + 1
5 + 1 + 1
5+5+1+1
5+5+5+1+1
1 + 1 + 1
5+1+1+1
5+5+1+1+1
5+5+5+1+1+1
1+1+1+1 5
5+1+1+1+1 5+5
5 + 5 + 1 + 1 + 1+1 5 + 5+5
5 + 5 + 5 + 1 + 1 + 1+1
Systems of this type with base m use the principle of position, but they only
possess two digits in the strict sense: one for unity, and the other for a
particular divisor of the base, here denoted by k. The m-l units are repre-
sented according to the additive principle (Fig. 23.38).
All of these systems clearly require a zero, and in the end have come to
possess one, independently or not of outside influence.
Positional systems of the second kind
This category includes our own modern decimal notation, whose nine units
are represented by figures (Fig. 23.26):
123456789
355
augmented by a tenth sign, written 0. Known as zero, this is used to mark
the absence of units of a given rank, and at the same time enjoys a true
numerical meaning, that of null number.
The fundamental characteristic of this system is that its conventions
can be extended into a notation both simple and completely consistent
for all numbers: integers, fractions, and irrationals (whether these be
transcendental or not). In other words, the discovery of this system enables
us to write down, in a simple and rational way, and using a completely
natural extension of the principle of position and of the zero, not only
fractions but entities such as ^2 , V3 or II.
A decimal fraction is a fraction of which the denominator is equal to
10 or to a power of 10. 3/10, 1/100, 251/10,000 are therefore decimal
fractions.
Now, the sequence of decimal fractions of unity (those which have
numerator 1 and denominator a power of 10) has its terms called succes-
sively one tenth (or decimal unit of the first order), one hundredth (or
decimal unit of the second order), one thousandth (or decimal unit of
the third order), and so on:
J_ J_ J_ 1_ i_ ±
io io 2 io 3 io 4 io 5 io G etc '
Thus we have a sequence where each term is the product of its predeces-
sor by 1/10, which means that the convention of our decimal notation
applies here also, ten units of any order being equal to one unity of the
order immediately above. These decimal units may therefore be unambigu-
ously represented by a convention which extends the convention which
applies to the integers, so that we may represent them in the form:
0.1 0.01 0.001 0.0001, etc
^lO- 1 ) (= 10~ 2 ) (=10- 3 ) (=10-*)
If we now consider any decimal fraction, for example 39,654/1,000, we
find its arithmetical decomposition according to the positive and negative
powers of 10:
39,654 _ 39,000 + 600 + _50_ + 4
1,000 1,000 1,000 1,000 1,000
We observe therefore that this may be written in the form:
POSITIONAL SYSTEMS
39,654 _ 39 + _600 + _50_ + _4_
1,000 1,000 1,000 1,000
or, in accordance with the preceding convention:
= 39 + 0.6 + 0.05 + 0.004
1,000
= 39 + 6xl0-‘ + 5xl0 - 2 + 4xl0 - 3
This number is therefore composed of 39 units, 6 tenths, 5 hundredths
and 4 thousandths. Adopting the convention for the representation of the
integers, one may make the convention of separating the integer units from
the decimal units by a point, so that the fraction in question may now be
put in the form:
39,654
1,000
= 39.654
It is therefore expressed as a decimal number which can be read as 39 units
and 654 thousandths.
Thus we see how the principle of position allows us to extend its appli-
cation to decimal numbers.
One can also show that any number whatever can be expressed as a
decimal number whose development may be finite or infinite (i.e. having a
finite or an infinite number of figures following the decimal point).
One can therefore see the many mathematical advantages which flow
from the discovery of our number-system.
But, clearly, this system is only a special case of the systems in this cate-
gory. These are nowadays known as systems with base m, the number m
being at least equal to 2 (m>l). Historically speaking, these are simply posi-
tional systems with base m furnished with a fully operational zero, whose
(m-1) figures are independent of each other and without any direct visual
significance (Fig. 23.28).
The written positional systems of the second kind are therefore the most
advanced of all history. They allow the simple and completely rational
representation of any number, no matter how large. Above all, they bring
within the reach of everyone a simple method for arithmetical operations.
And all this is independent of the choice of base (Fig. 23.29). It is precisely
in these respects that our modern written number-system (or any one of its
equivalents) is one of the foundations of the intellectual equipment of the
modern human being.
INDIAN CIVILISATION
CHAPTER 24
INDIAN CIVILISATION
THE CRADLE OF MODERN NUMERALS
As G. Beaujouan (1950) has said, “the origin of the so-called ‘Arabic’
numerals has been written about so often that every view on the question
seems plausible, and the only way of choosing between them is by personal
conviction.” Most of the literature (much of which is indeed of great value
and has been used in the following pages) deals with one particular disci-
pline from the many that are relevant to this tricky question of the origin
of Arabic numerals. The few comprehensive works on the subject (Cajori,
Datta and Singh, Guitel, Menninger, Pihan, Smith and Karpinski, or
Woepcke) are now several decades old, and many discoveries have been
made in more recent years. Since the beginning of the twentieth century, a
wealth of reliable information has been compiled from the various spe-
cialised fields, and the findings all point to the fact that the number-system
that we use today is of Indian origin. But no collective work has been
produced that contained rigorous reasoning or an entirely satisfactory
methodology. Moreover, the problem has been tackled in a somewhat loose
manner in the past and was seen from a more limited and biased perspec-
tive than it is today. So it is well worth while going back to square one and
looking at the question from a completely new angle, not only in the light
of the results seen in the previous chapter and those of certain recent devel-
opments, but also, and most importantly, using a multidisciplinary process
which takes into account the events of Indian civilisation.*
First, however, it is necessary (in order to eliminate them once and for
all) to remember some of the main and rather unlikely theories which are
still in circulation today on this subject.
FANCIFUL EXPLANATIONS FOR THE ORIGIN OF
“ARABIC” NUMERALS
According to a popular tradition that still persists in Egypt and northern
Africa, the “Arabic” numerals were the invention of a glassmaker-geometer
from the Maghreb who came up with the idea of giving each of the nine
* Due to the complex nature of this civilisation, a “Dictionary of the numerical symbols of Indian
Civilisation” has been compiled (see the end of the present chapter), which acts as both a thematic index
and a glossary of the many notions which it is necessary to understand in order to grasp the ideas intro-
duced in the following pages. In the present chapter, each word (whether in Sanskrit or in English) which is
also found in the dictionary, is accompanied by an asterisk. (Examples: *anka, *ankakramena, *Ashvin,
* Indian Astronomy, * Infinity, * Numeral, *sthdna, * Symbols, * yuga, *zero, etc.)
356
numerals a shape, the number of angles each one possessed being equal to
the number it denoted: one angle represented the number 1, two angles the
number 2, three angles the number 3, and so on (Fig. 24. 1A).
1ZIT5E7BB
1234 56789
Fig. 24.1A. The first unlikely hypothesis on the origin of our numerals: the number of angles each
numeral possesses
At the end of the nineteenth century, P. Voizot, a Frenchman, put forward
the same theory, apparently influenced by a Genoese author. But he also
thought that it was “equally probable" that the numerals were formed by
certain numbers of lines (Fig. 24. IB).
| 7
a-.;'
t S
6
•L
y\ ,
NX Q.
\X IjJ
1 2
3
4 5
6
7
8 9
Fig. 24. ib. Second unlikely hypothesis: the number of lines each numeral possesses
Another similar hypothesis was put forward in 1642 by the Italian Jesuit
Mario Bettini, which was taken up in 1651 by the German Georg Philip
Harsdorffer. This time the idea was that the ideographical representation of
the nine units would have been based on a number of points which were
joined up to form the nine signs (Fig. 24.1C). In 1890, the Frenchman
Georges Dumesnil also adopted this theory, believing the system to be of
Greek invention: attributing the form of our present-day numerals to the
Pythagoreans, his argument stated that the joining of the points to form
the geometrical representations of the whole numbers played an important
role for the members of this group.
i 2 5 * 5 & 1 8 $
12345 6 789
Fig. 24.1c. Third unlikely hypothesis: the number of points
A corresponding theory was put forward by Wiedler in 1737 which he
attributed to the tenth-century astrologer Abenragel: according to him, the
invention of numerals was the result of the division into parts of the shape
which is formed by a circle and two of its diameters. In other words,
according to Wiedler, all the figures could be made from this one geometri-
cal shape “as if they were inside a shell”: thus the vertical diameter would
357
FANCIFUL EXPLANATIONS FOR THE ORIGIN OF “ARABIC" NUMERALS
have formed 1; the same diameter plus two arcs at either end formed 2; a
semi-circle plus a median horizontal radius made 3, and so on until zero,
which was said to be formed by the complete circle (Fig. 24.1D).
I 1/ J 4- !r M 4 1 O
123456 78 90
Fiu. 24 . 1 H. Fourth unlikely hypothesis: the shapes formed by a circle and its diameters
It is also worthwhile mentioning the theories of the Spaniard Carlos Le
Maur (1778), who believed that the signs in question acquired their shape
from a particular arrangement of counting stones (Fig. 24. 1C) or from the
number of angles that can be obtained from certain shapes formed by a rec-
tangle, its diagonals, its medians, etc. (Fig. 24.1E).
1ZIX5E?I9o
123 4567890
Fig. 24 . 1 E. Fifth unlikely hypothesis: a variation of Fig. 24.1A
Finally, Jacob Leupold, in 1727, offered an “explanation” which goes by the
name of the legend of Solomon’s ring. According to this theory, the numer-
als were formed successively by the ring inscribing a square and its
diagonals (Fig. 24. IF).
I ZZ4Z47X70
123 4567890
Fig. 24 . if. Sixth unlikely hypothesis: the numerals come from a square (legend of Solomon’s ring)
If we were to believe any of these theories, it would mean that the appear-
ance of the numerals that we use today would have to have been the fruit of
one isolated individual’s imagination. An individual who would have given
each number a specific shape through a system based either on the use of
different numbers of lines, angles or dots to add up to the amount of units
the sign represents, or through the use of geometrical representations such
as a triangle, rectangle, square or circle, which would mean that the signs
were created according to a simple process of geometrical ordering.
These theories, then, all have one thing in common: their “explanation”
for the appearance of our numerals is that these figures were the result of
some kind of spontaneous generation; their shape, right from the outset,
being perfectly logical. In fact, as F. Cajori (1928) explains, “the validity of
any hypothesis depends upon the way in which the established facts are
presented and the extent to which it opens the door to new research." In
other words, a hypothesis can only acquire “scientific” value if it has the
potential to broaden our knowledge of a given subject.
The hypotheses that have been mentioned so far in this chapter are basi-
cally sterile. None of them offers any explanation for the fact that the nine
figures have appeared in an immense variety of shapes and forms over the
centuries and in different parts of the world. Their approach is merely to
consider the final product, in other words the numerals that we use today
(as they appear in print), which fails to take account of the fact that these
figures appear at the end of a very long story and have slowly evolved over
several millennia.
These a posteriori hypotheses are flawed because they are the fruit of the
pseudo-scientific imaginations of men who are fooled by appearances and
who jump to conclusions which completely contradict both historical facts
and the results of epigraphic and palaeographic research.”
It is still widely believed that the number-system that we use today was
invented by the Arabs.
However, it definitely was not the Arabs who invented what we know as
“Arabic” numerals. Historians have known for some time now that the name
was coined as the result of a serious historical error. Significantly, and curi-
ously, no trace of this belief is to be found in actual Arabic documents.
In fact, many Arabic works that concern mathematics and arithmetic
reveal that Muslim Arabic authors, without the slightest hint of prejudice
or complex, have always acknowledged that they were not the ones that
made the discovery. But whilst it is incorrect, the name which was given to
our numerals is not totally unfounded. There is always some basis for an
historical error, no matter how widespread or long-standing it is, and this
one is no exception, especially considering the fact that we are dealing with
a broad geographical area and a duration of many centuries.
The belief that our numerals were invented by the Arabs is only found in
Europe and probably originated in the late Middle Ages. This theory was
only really voiced by mathematicians or arithmeticians who, in order to
* Moreover, this book demonstrates quite dearly that despite the significance and vast number of inven-
tions that have punctuated the history of numerals as a whole, the findings have always been anonymous.
Men would work for and in groups and gain no qualifications for their work. Certain documents made of
stone, papyrus, paper and fabric immortalising the names of men who are sometimes associated with num-
bers mean nothing to us. Names of those who made use of and who reported numbers and counting
systems are also known. But those of the inventors themselves are irretrievably lost, perhaps because their
discoveries were made so long ago, or even because these brilliant inventions belonged to relatively humble
men whose names were not deemed worthy of recording. It is also possible that the discoveries could be a
result of the work of a team of men and so they could not really be attributed to a specific person. The
"inventor" of zero, a meticulous scribe and arithmetician, whose main concern was to define a specific point
in a series of numbers ruled by the place-value system, was probably never aware of the revolution that he
had made possible. All of this proves the absurdity of the preceding hypotheses.
INDIAN CIVILISATION
358
distinguish themselves from the masses, wanted to fill what they perceived
as a void with random hypotheses based on preconceived ideas, and thus
sacrificing historical truth to satisfy the whims of their own individual
inspiration. To the uninitiated, the writings of these mathematicians would
have seemed to constitute the linchpin of a doctrine that was sure to sur-
vive for many centuries. This is due to the fact that numerals and
calculation have always been considered (rightly or wrongly) to be the very
essence of mathematical science. The cause of the error is more easily
understood now that it is known that the numerals in question arrived in
the West at the end of the tenth century via the Arabs. At that time, the
Arabs were relatively superior to Western civilisations in terms of both cul-
ture and science. Therefore the figures were given the name “Arabic”.
This theory, however, was just one of the many explanations offered.
As the following evidence shows, European Renaissance authors offered
many similar and equally unreliable theories, attributing the invention of
our numerals to the Egyptians, the Phoenicians, the Chaldaeans and the
Hebrews alike, all of whom are totally unconnected to this discovery.
It is interesting to note that even in the twentieth century certain
authors, whilst being known for the quality of their work, have fallen
into the trap of supporting unsatisfactory explanations and taking things
solely at face value. At the turn of the century, historical scientists
(G. R. Kaye, N. Bubnov, and B. Carra de Vaux, etc., who strongly opposed
the idea that our number-system could be of Indian origin) alleged that our
numerals were developed in Ancient Greece [see JPAS 8 (1907), pp. 475 ff.;
N. Bubnov (1908); SC 21 (1917), pp. 273 ff.].
These men believed that the system originated in Neo-Pythagorean cir-
cles shortly before the birth of Christianity. They claimed that the system
came to Rome from the port of Alexandria and soon after made its way to
India via the trade route; it also travelled from Rome to Spain and the
North African provinces, where it was discovered some centuries later by
the Muslim Arabic conquerors. As for Middle Eastern Arabs, they picked
up the system from Indian merchants. According to this view of things,
European and North African numerals were formed by the “Western”
transmission, and the radically different Indian and Eastern Arabic figures
emerged by the “Eastern" route.
This tempting explanation is in fact an amalgam of the speculations of
the early humanists, as we can see from the list of quotations that follows:
1. Kobel, Rechenbiechlin, first published in 1514: Vom welchen
Arabischen auch disz Kunst entsprungen ist: “This art was also invented
by the Arabs”. [Kobel (1531), f 13]
2. N. Tartaglia, General trattato di numeri etmisuri (“General treatise
of numbers and measures") first published in 1556: ... & que estofu
trouato di fare da gli Arabi con diece figure: “ . . . and this is what the
Arabs did with ten figures [ten numerals].” [Tartaglia (1592), P 9]
3. Robert Recorde, The Grounde ofArtes: In that thinge all men do agree,
that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei
set all their letters. For they wryte backwarde as you tearme it, and so doo they
reade. And that may appeare in all Hebrew, Chaldaye and Arabike bookes . . .
where as the Greekes, Latines, and all nations of Europe, do wryte and reade
from the lefte hand towarde the ryghte. [Recorde (1558), P C, 5]
4. Peletarius, Commentaire sur I'Arithmetique de Gemma Frisius
(“Commentary on Arithmetic by Gemma Frisius") first published in
1563: La valeur des Figures commence au coste dextre vers le coste senestre:
au rebours de notre maniere d'escrire par ce que la premiere prattique est
venue des Chaldees: ou des Pheniciens, qui ont ete les premiers trajfiquers de
marchandise: “The figures read in ascending order from right to left
which is the opposite of our way of writing. This is because the former
practice comes from the Chaldaeans: or the Phoenicians, who were the
first to trade their merchandise.” [Peletarius, P 77]
5. Ramus, Arithmetic, published in 1569: Alii referunt ad Phoenices
inventores arithmeticae, propter eandem commerciorum caussam: Alii
ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Lutetiae . . . , refert
ad Arabes: “Others attribute the invention of arithmetic to the
Phoenicians, for the same commercial reasons; others credit
the Indians. Jean de Sacrobosco, whose tomb is in Paris . . . attributes
the discovery to the Arabs.” [Ramus (1569), p. 112]
6. Conrad Dasypodius, Institutionum Mathematicarum, published in
1593-1596: Qui est harum Cyphrarum auctor? A quibus hae usitatae
syphrarum notae sint inventae: hactenus incertum fiuit: meo tamen iudicio,
quod exiguum esse fateor: a Graecis librarijs (quorum olim magna fuit copia)
literae Graecorum quibus veteres Graeci tamquam numerorum notis usunt
usu fiuerunt corruptae, vt ex his licet videre. Graecorum Literae corruptae.
« P r s t 5 -z am
i pr • j 4 v < v 9
1 2 3 4 5 6 7 8 9
Sed qua ratione graecorum literae ita fuerunt corruptae? Finxerunt has cor-
ruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri,
vel inuersione vt in septenarij numeri nota nostrae notae, quibus hodie
utimur, ab his sola differunt elegantia, vt apparet: “Who invented these
signs that are used as numerals? Until now no one has really known;
9
FANCIFUL EXPLANATIONS FOR THE ORIGIN OF “ARABIC” NUMERALS
however, as far as I know (and I admit I know little), the letters that
Ancient Greeks used to denote numbers were distorted and trans-
formed through their use by Greek scribes (of which there were many),
as one can see below.This is how the distorted letters look:
123456789
(as shown above).
“But how were these letters corrupted? The sign for number two has
been reversed, the sign for the number seven has been inverted. The
only difference between the signs that we use today and the Greek
signs is that our signs are more elegant in appearance.” [Dasypodius,
quoted in Bayer]
7. Erpenius, Grammatica Arabica, published in 1613: “Arabic"
numerals are “actually the figures used by Toledo’s men of law”, which
he believes would have been transmitted to them by the Pythagoreans
of Ancient Greece. But Golius, who published the book after the death
of the author, realised that Erpenius had been mistaken, and sup-
pressed that particular passage in the 1636 edition. [Erpenius]
8. Laurembergus, The Mathematical Institution, first published in
1636: Supersunt volgares illi characteres Barbari, quibus hodie utitur uni-
versus fere orbis. Suntque universum novem: 1, 2, 3, 4, 5, 6, 7, 8, 9, queis
additur o cyphra: seufigura nihili, Nulla, Zero Arabibus. Nonnullorum sen-
tentia est, primos harumfigurarum inventores faisse Arabes (alii Phoenices
malunt; alii Indos ) quae sane opinio non est a veritate aliena. Nam sicut
Arabes olim totiusfere orbis potiti sunt, ita credibile est, scientiarum quoque
fuisse propagatores. Quicunque sit Inventor maxima sane illi debetur
gratia : “These ordinary, barbaric characters have survived the ages and
are used throughout most of the modern world. There are nine alto-
gether: 12345678 9, to which the figure 0 can be added, which
denotes “nothing”, the Arabic zero. Some think that it was the Arabs
who originally invented these signs (whilst others believe it was the
Phoenicians or even the Indians), and this is highly probable; the
Arabs once dominated most of the world and it is likely that they
invented the sciences. Whoever is to thank for the existence of our
numerals deserves the highest recognition.” [Laurembergus, p. 20, 1.
14; p. 21, 1. 2]
9. 1. Vossius, De Universae mathesos Natura et constitutione (c.1604):
“Arabic” numerals passed from “the Hindus or Persians to the Arabs,
then to the Moors in Spain, then finally to the Spanish and the rest of
Europe”. His theory that the series was originally passed from the Greeks
to the Hindus is without foundation. [Vossius (1660), pp. 39-40]
10. Nottnagelus, The Mathematical Institution, first published in
1645: Computatores autem ob majorem supputandi commoditatem pecu-
liares sibi finxerunt notas (quarum quidem inventionem nonnulli
Phoenicibus adscribunt, quidam, ut Valla et Cardanus, Indis assignant,
plerique vero Arabibus et Saracenis acceptam referunt) quas tamen alii ab
antiqua vel potius corrupta Graecarum literarum forma, nonnulli vero ali-
unde derivatas autumant. Atque his posterioribus hodierni quoque utuntur
Arithmetici : “To facilitate calculation, arithmeticians invented their
own unique signs (some believe it was the Phoenicians who invented
them, others, such as Valla and Cardanus, believe it was the Indians;
most people attribute the invention to the Arabs or Saracens); how-
ever, others claim that the numerals originated from the ancient, or
rather, distorted shape of Greek letters; some even suggest another
origin. The signs are still used by arithmeticians today.” [Nottnagelus,
p. 185]
11. Theophanes, Chronicle, first published in 1655: Hinc numerorum
notas et characteres, cifras vulgo dictos, Arabicum inventum aut Arabicos
nulla ratione vocandos, qui haec legerit, mecum contendet . . .: “The reader
can appreciate that I can find no reason why the signs and characters
that express numbers - which we vulgarly refer to as figures - are an
Arabic invention ...”
The following is an extract from a note written by Father Goar,
which comments on the above passage: Notas itaque characteresque,
quibus numeros summatim exaramus, 1, 2, 3, 4, 5, 6, 7, 8, 9, ab Indis et
Chaldaeis usque ad nos venisse scite magis advocat Glareanus in
Arithmaticae praeludiis. “In his Preludes to Arithmetic, Glareanus claims
that the signs and characters which we use to write the numbers in an
abbreviated form (1, 2, 3, etc.) actually came from the Indians and the
Chaldaeans.” [ Theophanis Chronographia, p. 616, 2nd col., and p. 314]
12. P. D. Huet, Bishop of Avranches in his Demonstratio Evangelica
ad serenissimum Delphinum claims that mediaeval European numerals
were invented by the Pythagoreans. [Huet (1690)]
13. Dom Calmet (1707) upholds the theory of the Greek origin of
our numerals [Calmet], as does J. F. Weidler, in Spicilegium observa-
tionum ad historiam notarum numeralium pertinentium. [Weidler (1755)]
14. C. Levias (1905), a contributor to the Jewish Encyclopaedia,
states that our numerals were invented by the people of Israel and
were introduced in Islamic countries around 800 CE by the Jewish
scholar Mashallah. [Levias (1905), IX, p. 348]
15. Levi della Vida upholds the theory of the Greek origin of our
numerals. [Levi della Vida (1933)]
INDIAN CIVILISATION
360
16. M. Destombes says that European numerals are derived from
the following letters of the Graeco-Byzantine alphabet I, 0, H, Z
r, B, by reversing the series of letters: B, T, ... Z, H, 0, 1, written in
capitals and graphically adapted to the "shapes of the Visigothic letters
from the third quarter of the tenth century CE”. [Destombes (1962)]
The basis for all these hypotheses is, of course, invalid, because no evidence
has ever been found to support the theory that the Greeks used a similar
system to our own. However, rather than admit defeat in the face of solid
counter-arguments firmly based in reality, the authors of these hypotheses
persevered stubbornly, using all their imagination to come up with some-
thing resembling proof or confirmation of their unlikely theories.
As A. Bouche-Leclerq (1879) remarks, “it is almost tempting to admire
the cunning way in which an unshakeable belief can transform into proof
the very objections which threaten to destroy it, and nothing better demon-
strates the psychological history of humanity than the irresistible prestige
of the preconceived idea.” Bouche-Leclerq is actually denouncing certain
charlatans of Ancient Greece who mastered the art of exploiting trusting
souls through the use of divinatory practices that were based on the inter-
pretation of numerological dreams using the numeral letters of the Greek
alphabet: “Perhaps the most embarrassing case”, he explains, “was one
which involved a dream which promised an elderly man a number of years
that was too high to be added on to his current age and too low to represent
his life-span as a whole. The charlatan, however, found a way to overcome
such a dilemma. If a man of seventy heard someone say, ‘You will live for
fifty years,’ he would live for another thirteen years. He has already lived for
over fifty years and it is impossible that he will live for another fifty years,
being seventy already. So the man will live for another thirteen years
(according to the charlatans) because the letter Nu (N), whilst representing
the number fifty, comes thirteenth in the Greek alphabet!”
It is likely that the same author would have also condemned the meth-
ods of historical scientists, who have been known to be somewhat
economical with the truth. No doubt he would have said something similar
about them if he had heard one particular historian’s rather flimsy “expla-
nation” which was soon adopted by all of his peers. When a shrewd man
asked him why the Greeks had left no written trace of zero or of decimal
place-value numeration, the historian in question, not to be deterred,
replied: “That is because of the level of importance that they placed in oral
tradition and also the great secrecy with which the Neo-Pythagoreans sur-
rounded their knowledge”! If everyone reasoned in this way, history would
amount to little more than a fairy tale.
Bearing in mind the fact that these authors were ardent admirers of
Hellenistic civilisation, it is easy to understand why their theory was sup-
ported solely by claims that were unaccompanied by any shred of evidence,
their main aim being to glorify the famous “Greek miracle”.
The admiration that these authors display for Greek civilisation is, of
course, perfectly justified. The Greeks were responsible for innovations in
such varied fields as art, literature, philosophy, medicine, mathematics,
astronomy, the sciences and engineering; their enormous contribution to
our sciences and culture is undeniable. The paradox lies in the fact that
the very men who wished to add to these achievements that are already
acknowledged by the rest of the world, were unaware of the real story
surrounding the scholars and mathematicians upon whom they wanted to
bestow this undeserved honour. This clearly demonstrates narrow-
mindedness on their part, attributing the development of our place-value
notation solely to the origin of the graphical representation of the nine
numerals in question.
J. F. Montucla (1798) quite rightly points out that “if the characters orig-
inate from Greek letters, they have drastically changed somewhere along
the way. In fact, these letters could only resemble our numerals if they were
shortened and turned about in a very odd fashion.* Moreover, the appear-
ance of these characters is much less important than the ingenious way in
which they are used; using only ten characters, it is possible to express
absolutely any number. The Greeks were a highly intelligent race, and if
this had been their invention, or even if they had simply got wind of it, they
certainly would have made use of it.”
Ancient Greece only had two systems of numerical notation: the first
was the mathematical equivalent of the Roman system and the other was
alphabetical, like the one used by the Hebrews. With a few exceptions
towards the end of the era, neither of these systems were based on the rule
of position, nor did they possess zero. Therefore the systems were not really
of much practical use when it came to mathematical calculations, which
were generally carried out using abacuses, upon which there were different
columns for each decimal order.
Considering that the Greeks had invented such an instrument, the
next logical step would have been their discovery of the place-value
system and zero, through eliminating the columns of the instrument.
* Using such methods, it is always possible to find a way of promoting a theory: it is easy to manipulate the
nine characters in order to “prove” that our nine numerals originate from them. This is precisely how cer-
tain extravagant comtemporary authors, ignoring not only the history of mathematical notation and
writing, but also and above all the laws of palaeography, have come to “demonstrate” that these numerals
derive from the first nine Hebrew letters, or even from the graphical representations for the twelve signs of
the Zodiac. This goes to show that you can put the words of a song to any tune you like; in other words,
appearances can be deceptive.
361
EVIDENCE FROM EUROPE
This would have provided them with the fully operational counting
system that we use today.
However, the Greeks did not bother themselves with such practical
concerns.
INDIA: THE TRUE BIRTHPLACE OF
OUR NUMERALS
The real inventors of this fundamental discovery, which is no less important
than such feats as the mastery of fire, the development of agriculture, or the
invention of the wheel, writing or the steam engine, were the mathematicians
and astronomers of Indian civilisation: scholars who, unlike the Greeks, were
concerned with practical applications and who were motivated by a kind of
passion for both numbers and numerical calculations.
There is a great deal of evidence to support this fact, and even the
Arabo-Muslim scholars themselves have often voiced their agreement.
EVIDENCE FROM EUROPE WHICH SUPPORTS THE CLAIM
THAT MODERN NUMERATION ORIGINATED IN INDIA
The following is a succession of historical accounts in favour of this theory,
given in chronological order, beginning with the most recent.
1. P. S. Laplace (1814): “The ingenious method of expressing every
possible number using a set of ten symbols (each symbol having a
place value and an absolute value) emerged in India. The idea seems so
simple nowadays that its significance and profound importance is no
longer appreciated. Its simplicity lies in the way it facilitated calcula-
tion and placed arithmetic foremost amongst useful inventions. The
importance of this invention is more readily appreciated when one
considers that it was beyond the two greatest men of Antiquity,
Archimedes and Apollonius.” [Dantzig, p. 26]
2. J. F. Montucla (1798): “The ingenious number-system, which
serves as the basis for modern arithmetic, was used by the Arabs long
before it reached Europe. It would be a mistake, however, to believe
that this invention is Arabic. There is a great deal of evidence, much of
it provided by the Arabs themselves, that this arithmetic originated in
India.” [Montucla, I, p. 375]
3. John Wallis (1616-1703) referred to the nine numerals as Indian
figures [Wallis (1695), p. 10]
4. Cataneo (1546) le noue figure de gli Indi, “the nine figures from
India”. [Smith and Karpinski (1911), p. 3]
5. Willichius (1540) talks of Zyphrae Indicae, “Indian figures”.
[Smith and Karpinski (1911) p. 3]
6. The Crafie of Nombrynge (c. 1350), the oldest known English arith-
metical tract: 1 1 fforthermore ye most vndirstonde that in this craft ben vsed
teen figurys, as here bene writen for esampul 0 9 8 a 6 5 4 3 2 1 ... in the
quych we vse teen figurys of Inde. Questio. 1 1 why ten figurys oflnde? Solucio.
For as I have sayd afore thei werefonde first in Inde. [D. E. Smith (1909)]
7. Petrus of Dacia (1291) wrote a commentary on a work entitled
Algorismus by Sacrobosco (John of Halifax, c. 1240), in which he says
the following (which contains a mathematical error): Non enim omnis
numerus per quascumque figuras Indorum repraesentatur . . .: “Not every
number can be represented in Indian figures”. [Curtze (1897), p. 25]
8. Around the year 1252, Byzantine monk Maximus Planudes
(1260-1310) composed a work entitled Logistike Indike ("Indian
Arithmetic”) in Greek, or even Psephophoria kata Indos (“The Indian
way of counting"), where he explains the following: “There are only
nine figures. These are:
123456789
[figures given in their Eastern Arabic form].
“A sign known as tziphra can be added to these, which, according to
the Indians, means ‘nothing’. The nine figures themselves are Indian,
and tziphra is written thus: 0”. [B. N., Paris. Ancien Fonds grec, Ms
2428, f° 186 r°]
9. Around 1240, Alexandre de Ville-Dieu composed a manual in
verse on written calculation (algorism). Its title was Carmen de
Algorismo, and it began with the following two lines: Hacc algorismus
ars praesens dicitur, in qua Talibus Indorum fruimur bis quinque figuris:
“ Algorism is the art by which at present we use those Indian figures,
which number two times five”. [Smith and Karpinski (1911), p. 11]
10. In 1202, Leonard of Pisa (known as Fibonacci), after voyages
that took him to the Near East and Northern Africa, and in particular
to Bejaia (now in Algeria), wrote a tract on arithmetic entitled Liber
Abaci (“a tract about the abacus”), in which he explains the following:
Cum genitor meus a patria publicus scriba in duana bugee pro pisanis mer-
catoribus ad earn confluentibus preesset, me in pueritia mea ad se uenire
faciens, inspecta utilitate et commoditate futura, ibi me studio abaci per
aliquot dies stare uoluit et doceri. Vbi ex mirabili magisterio in arte per
nouem figuras Indorum introductus . . . Novem figurae Indorum hae sunt:
987654321
INDIAN CIVILISATION
362
cum his itaque novem figuris, et cum hoc signo o. Quod arabice zephirum
appellatur, scribitur qui libet numerus: “My father was a public scribe of
Bejaia, where he worked for his country in Customs, defending the
interests of Pisan merchants who made their fortune there. He made
me learn how to use the abacus when I was still a child because he saw
how I would benefit from this in later life. In this way I learned the art
of counting using the nine Indian figures . . .
The nine Indian figures are as follows:
987654321
[figures given in contemporary European cursive form].
“That is why, with these nine numerals, and with this sign 0, called
zephirum in Arab, one writes all the numbers one wishes.”
[Boncompagni (1857), vol.I]
11. C. 1150, Rabbi Abraham Ben Mei'r Ben Ezra (1092-1167), after a
long voyage to the East and a period spent in Italy, wrote a work in
Hebrew entitled: Sefer ha mispar (“Number Book"), where he explains
the basic rules of written calculation.
He uses the first nine letters of the Hebrew alphabet to represent
the nine units. He represents zero by a little circle and gives it the
Hebrew name of galgal (“wheel”), or, more frequently, sifra (“void”)
from the corresponding Arabic word.
However, all he did was adapt the Indian system to the first nine
Hebrew letters (which he naturally had used since his childhood).
In the introduction, he provides some graphic variations of the fig-
ures, making it dear that they are of Indian origin, after having
explained the place-value system: “That is how the learned men of India
were able to represent any number using nine shapes which they fash-
ioned themselves specifically to symbolise the nine units.” [Silberberg
(1895), p. 2; Smith and Ginsburg (1918); Steinschneider (1893)]
12. Around the same time, John of Seville began his Liber algoarismi
de practica arismetrice (“Book of Algoarismi on practical arithmetic”)
with the following:
Numerus est unitatum collectio, quae quia in infinitum progreditur
(multitudo enim crescit in infinitum), ideo a peritissimis Indis sub quibus-
dam regulis et certis limitibus infinita numerositas coarcatur, ut de infinitis
difinita disciplina traderetur et fuga subtilium rerum sub alicuius artis cer-
tissima lege teneretur: “A number is a collection of units, and because the
collection is infinite (for multiplication can continue indefinitely), the
Indians ingeniously enclosed this infinite multiplicity within certain
rules and limits so that infinity could be scientifically defined; these
strict rules enabled them to pin down this subtle concept.
[B. N., Paris, Ms. lat. 16 202, P 51; Boncompagni (1857), vol. I, p. 26]
13. C. 1143, Robert of Chester wrote a work entitled: Algoritmi de
numero Indorum (“Algoritmi: Indian figures”), which is simply a trans-
lation of an Arabic work about Indian arithmetic. [Karpinski (1915);
Wallis (1685), p. 12]
14. C. 1140, Bishop Raimundo of Toledo gave his patronage to a
work written by the converted Jew Juan de Luna and archdeacon
Domingo Gondisalvo: the Liber Algorismi de numero Indorum (“Book
of Algorismi of Indian figures) which is simply a translation into a
Spanish and Latin version of an Arabic tract on Indian arithmetic.
[Boncompagni (1857), vol. I]
15. C. 1130, Adelard of Bath wrote a work entitled: Algoritmi de
numero Indorum (“Algoritmi: of Indian figures”), which is simply a
translation of an Arabic tract about Indian calculation. [Boncompagni
(1857), vol. I]
16. C. 1125, The Benedictine chronicler William of Malmesbury
wrote De gestis regum Anglorum, in which he related that the Arabs
adopted the Indian figures and transported them to the countries
they conquered, particularly Spain. He goes on to explain that the
monk Gerbert of Aurillac, who was to become Pope Sylvester II (who
died in 1003) and who was immortalised for restoring sciences in
Europe, studied in either Seville or Cordoba, where he learned about
Indian figures and their uses and later contributed to their circulation
in the Christian countries of the West. [Malmesbury (1596), P 36 r°;
Woepcke (1857), p. 35]
17. Written in 976 in the convent of Albelda (near the town of
Logrono, in the north of Spain) by a monk named Vigila, the Codex
Vigilanus contains the nine numerals in question, but not zero. The
scribe clearly indicates in the text that the figures are of Indian origin:
Item de figuris aritmetice. Scire debemus Indos subtilissimum ingenium
habere et ceteras gentes eis in arithmetica etgeometrica et ceteris liberalibus
disciplinis concedere. Et hoc manifestum est in novem figuris, quibus quibus
designant unum quenque gradum cuiuslibet gradus. Quorum hec sunt forma:
98765432 1.
“The same applies to arithmetical figures. It should be noted that
the Indians have an extremely subtle intelligence, and when it comes
to arithmetic, geometry and other such advanced disciplines, other
ideas must make way for theirs. The best proof of this is the nine fig-
ures with which they represent each number no matter how high. This
is how the figures look:
98765432 1.”
363
EVIDENCE FROM ARABIC SOURCES
(In the original, the figures are presented in a style very close to the
North African Arabic written form.) [Bibl. San Lorenzo del Escorial,
Ms. lat. d.1.2, P 9v°; Burnam (1912), II, pi. XXIII; Ewald (1883)]
EVIDENCE FROM ARABIC SOURCES WHICH SUGGESTS
THAT MODERN NUMERATION ORIGINATED IN INDIA
The following evidence proves that for over a thousand years, Arabo-Muslim
authors never ceased to proclaim, in a praiseworthy spirit of openness, that
the discovery of the decimal place-value system was made by the Indians.*
1. In Khulasat al hisab (“Essence of Calculation"), written c. 1600,
Beha’ ad din al ‘Amuli, in reference to the figures in question, remarks
that: “It was actually the Indians who invented the nine characters.”
[Marre (1864), p. 266]
2. C. 1470, in a commentary on an arithmetical tract, Abu’l Hasan
al Qalasadi (d. 1486) wrote the following in reference to the nine fig-
ures used in Muslim Spain and Northern Africa: “Their origin is
traditionally attributed to an Indian.” [Woepcke (1863), p. 59]
3. In “Prolegomena” ( Muqqadimah ), written c. 1390, Abd ar Rahman
ibn Khaldun (1332-1406) says that the Arabs first learned about science
from the Indians along with their figures and methods of calculation in
the year 156 of the Hegira (= 776 CE). [Ibn Khaldun, vol. Ill, p. 300]
4. In Talkhisfi a ‘mal al hisab (“Brief guide to mathematical opera-
tions”) written c. 1300, Abu’l ‘abbas ahmad ibn al Banna al Marrakushi
(1256-1321) makes a direct reference to the Indian origin of the figures
and counting techniques. [Marre (1865); Suter (1900), p. 162]
5. C. 1230, Muwaffaq al din Abu Muhammad al Baghdadi wrote a
tract entitled Hisab al hindi (“Indian Arithmetic”). [Suter (1900), p. 138]
6. C. 1194, Persian encyclopaedist Fakhr ad din al Razi (1149-1206)
wrote a work entitled Hada’iq al anwar, which included a chapter
called Hisab al hindi (“Indian Calculation”). [B. N., Anc. Fds pers., Ms.
213, P 173r]
7. C. 1174, mathematician As Samaw’al ibn Yahya ibn ‘abbas al
Maghribi al andalusi, a Jew converted to Islam, wrote a work entitled
Al bahir fi ‘ilm al hisab (“The lucid book of arithmetic”), in which a
direct reference is also made concerning the Indian origin of the fig-
ures and the methods of calculation. [Suter (1900), p. 124; Rashed and
Ahmed (1972)]
* Henceforth, the scientific transcription of Arabic words will not be scrupulously adhered to. “Kh", “gh”
and “sh” will be used in the place of h, g, and s to facilitate the reading of Arabic for those who are
not specialists.
8. In 1172 Mahmud ibn qa’id al ‘Amuni Saraf ad din al Meqi wrote
a tract entitled Fi’l handasa wa’l arqam al hindi (“Indian geometry and
figures”). [Suter (1900), p. 126]
9. C. 1048, ‘Ali ibn Abi’l Rijal abu’l Hasan, alias Abenragel, in a pref-
ace to a treatise on astronomy, wrote that “the invention of arithmetic
using the nine figures belongs to the Indian philosophers”. [Suter
(1900), p. 100]
10. C. 1030, Abu’l Hasan ‘Ali ibn Ahmad an Nisawi wrote a work
entitled al muqni ‘fi'l hisab al hindi (“Complete guide to Indian arith-
metic”). [Suter (1900), p. 96]
11. Between 1020 and 1030, in his autobiography, Al Husayn ibn
Sina (Avicenna) tells of how, when he was very young, he heard con-
versations between his father and his brother which were often about
Indian philosophy, geometry and calculation, and when he was ten (in
the year 990), his father sent him to a merchant who was well-versed in
numerical matters to learn the art of Indian calculation.
In his tract on speculative arithmetic, Ibn Sina writes the following:
“As for the verification of squares using the Indian method ( fi’l tariq al
hindasi ) . . . One of the properties of a cube consists of the way of verify-
ing it using the methods of Indian calculation (al hisab al hindasi ) . . .’’
[Woepcke (1863), pp. 490, 491, 502, 504; Leiden Univ. Lib., Ms. legs
Wamerien, no. 84]
12. C. 1020, Abu’l Hasan Kushiyar ibn Labban al-Gili (971-1029)
wrote a work which carries the Arabic title, Fi usu’l hisab al hind
(“Elements of Indian calculation”), the opening words of which
being: “This [tract] of calculations [written] in Indian [figures] is
formed by . . .” [Library of Aya Sofia. Istanbul. Ms 4,857, fi 267 r;
Mazaheri (1975)]
13. In roughly the same year, mathematician Abu Ali al Hasan
ibn al Hasan ibn al Haytham, from Basra, wrote Maqalat fi
‘ala 7 hisab al hind (“Principles of Indian calculation”). [Woepcke
(1863), p. 489]
14. Astronomer and mathematician Muhammad ibn Ahmad Abu’l
Rayhan al Biruni (973-1048), after living in India for thirty years, and
having been introduced to Indian sciences, wrote a number of works
between 1010 and 1030, including Kitab al arqam (“Book of figures”),
and Tazkirafi'l hisab wa’l mad bi’l arqam alsind wa’l hind (“Arithmetic
and counting using Sind and Indian figures”).
In his work entitled Kitab fi tahqiq i ma li’l hind (which is one of the
most important works about India to be written at that time), in
which he mentions the diversity of the graphical forms of the figures
used in India, and insists that the figures used by the Arabs originated
INDIAN CIVILISATION
364
in India, he makes the following remark: "Like us, the Indians use
these numerical signs in their arithmetic. I have written a tract which
shows, in as much detail as possible, how much more advanced the
Indians are than we are in this field.”
And in Athar wu 7 baqiya (“Vestiges of the past”, or “Chronology of
ancient nations”), he calls the nine figures arqam al hind (“Indian fig-
ures”), and demonstrates both how they differ from the sexagesimal
system (which is Babylonian in origin), and their superiority over the
Arab system of numeral letters. [Al-Biruni (1879) and (1910); Smith
and Karpinski (1911), pp. 6-7; Datta and Singh (1938), pp. 98-9;
Woepcke (1900), pp. 275-6]
15. Curiously, in his “Book of creation and history” (c. 1000),
Mutahar ibn Tahir gives, in the Nagari form of the figures, the decimal
positional expression of a number which the Indians believed repre-
sented the age of the planet. [Smith and Karpinski (1911), p. 7]
16. In 987, historian and biographer Ya ’qub ibn al Nadim of
Baghdad wrote one of the most important works on the history of
Arabic Islamic people and literature: the Al Kitab al Fihrist al 'ulum
(“Book and index of the sciences"), in which he particularly refers to
the work of the great Arabic Muslim astronomers and mathematicians
of his time, and in which he constantly refers to methods of calculation
as hisab al hindi (“Indian calculation”). [Dodge (1970); Suter (1892)
and (1900); Karpinski (1915)]
17. Before 987, Sinan ibn al Fath min ahl al Harran (quoted in
Fihrist by Ibn al Nadim) wrote a work entitled Kitab al takht fi’l hisab al
hindi (“Tract on the wooden tablets used in Indian calculation”). [Suter
(1892), pp. 37-8; Woepcke (1863), p. 490]
18. Also before 987, Ahmad Ben ‘Umar al Karabisi (quoted in Ibn al
Nadim’s Fihrist) wrote Kitab al hisab al hindi (“A tract on Indian calcu-
lation"). [Suter (1900), p. 63; Woepcke (1863), p. 493]
19. Before 987 again, ‘Ali Ben Ahmad Abu’l Qasim al Mujitabi al
Antaki al Mualiwi (who died in 987) wrote a tract entitled Kitab al
takht al kabir fi’l hisab al hindi (“Book of wooden tablets relating to
Indian calculation”). [Suter (1900), p. 63; Woepcke (1863), p. 493]
20. Before 986, Al Sufi (who died in 986) wrote a work entitled
Kitab al hisab al hind “Treatise on Indian calculation”. [Smith and
Karpinski (1911)]
21. C. 982, Abu Nasr Muhammad Ben ‘Abdallah al Kalwadzani
wrote Kitab al takht fi’l hisab al hindi ("Treatise on the tablet relative to
Indian calculation”), quoted in Fihrist by Ibn al Nadim. [Suter (1900),
p. 74; Woepcke (1863), p. 493]
22. C. 952, Abu’l Hasan Ahmad ibn Ibrahim al Uqlidisi wrote a
work entitled: Kitab alfusul fi’l hisab al hind (“Treatise on Indian arith-
metic”). [Saidan (1966)]
23. In 950, Abu Sahl ibn Tamim, a native of Kairwan (now Tunisia),
wrote a commentary on Sefer Yestsirah (a Hebrew work concerning
Cabbala) in which he explains the following: “The Indians invented the
nine signs which denote units. I have already spoken about these at
great length in a book which I wrote on Indian mathematics [he uses
the expression hisab al hindi], known as hisab al ghubar (“calculations
in the dust”). [Reinaud, p. 399; Datta and Singh (1938), p. 98]
24. C. 900, arithmetician Abu Kamil Shuja’ ibn Aslam ibn
Muhammad al Hasib al Misri (his last two names meaning “the
Egyptian arithmetician") wrote an arithmetical work using the rule of
the two false positions, which he attributed to the Indians. This work,
which is only found in Latin translation, is called: “Book of enlarge-
ment and reduction, entitled ‘the calculation of conjecture’, after the
achievements of the wise men of India and the information that
Abraham[?] compiled according to the ‘Indian’ volume”. [Suter, BM3;
Folge, 3 (1902)]
25. Before 873, Abu Yusuf Ya ‘qub ibn Ishaq al Kindi wrote Kitab
risalatfi isti mal 7 hisab al hindi arba maqalatan (“Thesis on the use of
Indian calculation, in four volumes”), quoted in Fihrist by Ibn al
Nadim. [Woepcke (1900), p. 403]
26. C. 850, the Arabic philosopher Al Jahiz (who died in 868) refers
to the figures as arqam al hind (“figures from India”) and remarks that
“high numbers can be represented easily [using the Indian system]”,
even though the author expresses contempt for the Indian system. He
asks the following question: “Who invented Indian figures . . . and cal-
culation using the figures?” [Carra de Vaux (1917); Datta and Singh
(1938), p. 97]
27. C. 820, Sanad Ben ‘Ali, a Jewish mathematician who was con-
verted to Islam, and who was one of Caliph al Ma’mun’s astronomers,
wrote a tract entitled: Kitab al hisab al hindi (“A treatise on Indian cal-
culation”) quoted by Ibn al Nadim in Fihrist. [Smith and Karpinski
(1911), p. 10; Woepcke (1900), p. 490]
28. C. 810, Abu Ja ‘far Muhammad ibn Musa al Khuwarizmi wrote:
Kitab al jam’ wa’l tafriq bi hisab al hind ("Indian technique of addition
and subtraction”), of which there are Latin translations dating from
the twelfth century. The tract begins thus:
"... we have decided to explain Indian calculating techniques using
the nine characters and to show how, because of their simplicity and
conciseness, these characters are capable of expressing any number."
365
HOW RELIABLE IS THIS EVIDENCE?
He goes on to give a detailed explanation of the positional princi-
ple of decimal numeration, with reference to the Indian origin of the
nine numerical symbols and of “the tenth figure in the shape of a
circle” (zero), which he advises be used “so as not to confuse the
positions”. [Allard (1975); Boncompagni (1857)]; Vogel (1963);
Youschkevitch (1976)]
HOW RELIABLE IS THIS EVIDENCE?
All the above evidence points to the same conclusion: the numerical sym-
bols that are used in the modern world were created in India.
However, there still remains the task of judging how reliable this evi-
dence is. According to E. Claparede (1937), “reliable evidence is not the
rule but the exception”. This idea is perhaps best expressed by Charles
Peguy, through the character of Clio, Muse of History ( Oeuvres completes,
VIII, 301-302): “Humankind lies most when giving evidence (because the
testimony becomes part of history), and . . . people lie even more when
giving formal evidence. In everyday life, it is important to be truthful.
When giving evidence, it is necessary to be twice as truthful. It is a well-
known fact, however, that people lie all the time, but people lie less when
not testifying than when they are testifying.”
Etymologically, “testimony” derives from the Latin testis (“witness”),
from which we get the verbs “to attest”, “to contest”, etc. Thus “testi-
mony” means “the written or verbal declaration with which a person
certifies the reality of a fact of which they have had direct knowledge”
(P. Foulquie, 1982).
Often, however, the fact in question is certified by an anterior declara-
tion given by an eye-witness, as if one was testifying to a scene which a
friend had seen and then recounted.
This is precisely the conditions in which nearly all the above declara-
tions were written.
By its very nature, a testimony is never objective:
It is always marred by the subjectivity of its author, the unreliability of
his memory, as well as gaps in perception and the unavoidable distor-
tions of human memory (it is estimated that these errors increase at a
rate of 0.33 per cent per day). Swiss psychologist Edouard Claparede
and Belgian criminologist L. Vervaeck, using their pupils as subjects,
found that correct testimonies were rare (only 5 per cent) and that the
feeling of certainty increased with time ... at the same rate as the
increase in errors! [N. Sillamy (1967)].
It is because of its capital role in courtroom cases that the study of testi-
mony plays such a major part in the applications of judicial psychology (see
H. Pieron, 1979). The courtroom saying, testis unus, testis nullus (one sole
witness is as useful as no witness at all) does not apply here because the
origin of the numerals has been mentioned many times in the space of
more than a thousand years. This case would in fact seem highly plausible.
But are all these accounts really completely independent of one
another? If all these concurring pieces of evidence originate from one single
source, then the proof might as well not exist at all.
The following example, taken from M. Bloch (1949), illustrates this
point very clearly:
Two contemporaries of Marbot - the Count of Segur and General Pelet
- gave accounts of Marbot’s alleged crossing of the Danube which
were analogous to Marbot’s own account. Segur’s evidence came after
Pelet’s: he read the latter’s account and did little more than copy it. It
made no difference if Pelet wrote his account before Marbot; he was
Marbot’s friend and there is no doubt that Pelet had often heard
Marbot recount his fictitious heroic deeds. This leaves Marbot as the
only witness because his would-be guarantors both based their
accounts on what he himself had related about the event.
In this kind of situation there is quite literally no witness at all.
However, Planudes, Fibonacci, Ibn Khaldun, Avicenna, al-Biruni, al-
Khwarizmi and others, of whom many were actual eye-witnesses to the
event, are neither Pelets, nor Segurs, and certainly not Marbots. Their evi-
dence and their accounts, as will be seen later, are firmly rooted in reality.
These men are all in agreement, but this stems from neither a similar state
of mind nor a phenomenon of collective psychology.
Despite the basic unreality of memory and the gaps and distortions
which characterise the evidence given by any member of the human race,
these accounts as a whole might still be an important item to add to the file
for this investigation.
EVIDENCE FROM PRE-ISLAMIC SYRIA
The Arabs and the Europeans were not the first to offer evidence about the
origin of our digits. There were others; people who were around long
before and who lived far beyond the frontiers of Islam. Proof is to be
found in the Middle East, at a time when Muslim religion was only just
beginning to emerge, shortly after the first Ommayad caliph came to
power in Damascus.
INDIAN CIVILISATION
366
At that time there lived a Syrian bishop named Severus Sebokt. He
studied philosophy, mathematics and astronomy at the monastery of
Keneshre on the banks of the Euphrates: a place that was exposed to a
great wealth of knowledge because of its situation at the crossroads of
Greek, Mesopotamian and Indian learning.
Severus Sebokt, then, knew Greek and Babylonian sciences as well as
Indian science. Irritated by the belief that Greek learning was superior to
that of other civilisations, he wrote a short article in the hope of bringing
the Greeks down a peg or two.
Nau, who wrote a commentary on and published this manuscript,
explains the circumstances under which it was written:
In the Greek year 973 (662 in our calendar), Severus Sebokt, clearly
offended by Greek pride, reclaimed the invention of astronomy for
the Syrians. He explained that the Greeks had gleaned their knowl-
edge from the Chaldaeans and the Babylonians, who he claimed were
in fact Syrians. He quite rightly concludes that science belongs to
everyone and that it is accessible to any race or individual who takes
the trouble to understand it; it is not the property of the Greeks
[F. Nau (1910)].
It is in order to reinforce this point that Severus uses the Indians as an
example:
The Hindus, who are not even Syrians, have made subtle discoveries in
the field of astronomy which are even more ingenious than those of
the Greeks [sic] and the Babylonians; as for their skilful methods of
calculation and their computing which belies description, they use
only nine figures. If those who think they are the sole pioneers of sci-
ence, simply because they speak Greek, had known of these
innovations, they would have realised (albeit a little late) that there are
others who speak different languages who are also knowledgeable.
This piece of evidence is indispensable. The “computing that belies descrip-
tion which uses only nine figures” is, to Sebokt’s mind, infinitely superior
to spoken numeration: it is not possible to express all numbers using the
latter method (because, like most oral methods of numeration, it involves a
hybrid principle, using addition and multiplication of the names of the
basic numbers); the Indian system makes it possible to write any number
using only the nine figures.
In other words, the Indian system, as described by Severus Sebokt,
has an unlimited capacity for representation because it has positional
numeration.
This numeration is decimal because it uses nine digits.
It might seem curious that Sebokt does not mention the use of zero, but
this is probably because he only had an abacus upon which to carry out his
mathematical operations. It is likely that his “abacus” was a board sprin-
kled with sand or dust upon which he would write numbers using the nine
Indian symbols within various columns corresponding to the consecutive
decimal denominations. Therefore, zero was not physically represented:
the absence of a unit in a given column was communicated by means of an
empty space.
Sebokt’s evidence proves that the Indian counting system was known
and esteemed outside India by the middle of the seventh century CE.
FROM THE EVIDENCE TO THE ACTUAL EVENT
The above evidence proves that all the preceding accounts are independent
of each other but, however reliable these accounts are, they merely serve as
confirmation of the truth. Alone, they do not constitute what is known as
“historical truth”. As F. de Coulanges said, “History is a science: it is a prod-
uct of observation, not imagination; in order for the observation to be
accurate, authentic documentation is needed.”
A. Cuvillier (1954) explains that history, in the scientific sense of the
word, is
the study of human facts through time. So defined, historical facts are
distinguished from those that are the subject of other sciences by their
unique nature . . . Suspended in time, historical facts are, as a rule, in the
past. Even when dealing with contemporary facts, the historian is still
only personally privy to a very small percentage of the facts. The first
task of a historian is to establish the facts through the use of documents,
in other words the traces of these facts which still remain in the present.
Sociologist F. Simiand said that history is “information gleaned from
left-over traces”. The “traces” which are of interest here are the surviving writ-
ten documents from Indian civilisation or from any culture connected to it.
Of course, it is essential to ensure that these documents are authentic.
The traces in question came from an area of incredible diversity which,
whilst proving the wonderful fertility of Indian civilisation, also shows an
infinite complexity, with an added difficulty (to name but one): the con-
siderable number of fakes produced by members of this same civilisation.
This, then, is the terrain the historian must embark upon; one of unde-
niable cultural wealth, even exuberance, yet it is crucial to remain
367
PROOF OF THE EVENT
extremely cautious when faced with documentation which is often tricky to
date and which has to be closely examined in order to separate the genuine
from the counterfeit, the ancient from the modern, the collective work
from the individual work, a commentary from a copy of the original, etc.*
However, the vital work of historians from India and Southeast Asia
must not be forgotten. For over a century, they have been separating the
authentic from the fake, establishing the source and the date of a great
many documents (even if this chronology is only approximate), restoring
documents which had been damaged by the passage of time to their origi-
nal state, studying the content and the allusions made in each work, and
carrying out many other indispensable tasks.
All these results were collected in random order. To paraphrase H. Poincare
(1902), the science of history is built out of bricks; but an accumulation of his-
torical facts is no more a science than a pile of bricks is a house.
PROOF OF THE EVENT
In the previous chapter we offered a classification of written numbering
systems that are historically attested, and through it we drew out a genuine
chronological logic: the guiding thread, leading through centuries and civil-
isations, taking the human mind from the most rudimentary systems to
the most evolved. It enabled us to identify the foundation stone (and, more
generally, the abstract structure) of the contemporary written numeral
system, the most perfect and efficient of all time. And it is precisely this
chronological logic of the mind which shows us the path to follow in order
to arrive at a historical synthesis. A synthesis intended to show just how the
invention of numerals actually "worked”, and to place it in its overall con-
text, in terms of period, sequence of events, influences, etc.
Using this approach, we will be able to tell the story much more rigor-
ously and to track the invention of the Indian system very closely indeed.
Drawing on all the available evidence to prove that India really was the
cradle of modern numeration, the problem will be divided into the follow-
ing subsections:
* Indian history is a constantly shifting terrain, where “forgeries” or “modern documents presented as
ancient ones" abound in great quantities. It is an area where even documents that are believed to be authen-
tic could quite possibly have been the fruit of several successive corrections or re-workings and the result of
some apparently homogenous fusion of various commentaries, even commentaries on the commentaries
themselves, so that the seemingly authentic document might have absolutely nothing in common with what
the author to whom the work is attributed orginally intended. It is a field where certain specialists, who
have not always been as rigorous as they might have been, have confused the issue by supporting their argu-
ments with documents that have no historical worth whatsoever. This would appear to explain why the
origin of the decimal point system was such an enigma for so long.
To untangle this apparently inextricable knot was no simple task because it involved the elimination of
all unreliable sources (which are still used in a great many scientific publications) in order to include, as far
as possible, nothing but trustworthy sources, from the most ancient documents on Indian civilisation.
1. To show that this civilisation discovered, and put into practice,
the place-value system;
2. To prove that this same civilisation invented the concept of zero,
which the Indian mathematicians knew could represent both the idea of
an “empty space” and that of a “zero number”;
3. To establish that the Indians formed their basic figures in the
absence of any direct visual intuition;
4. To show that the early form of their symbols prefigured not only
all the varieties currently in use in India and in Central and Southeast
Asia, but also the respective shapes of Eastern and Western Arabic fig-
ures as well as the appearance of those figures used today and their
various European predecessors of the same kind;
5. To prove that the learned men of that civilisation perfected the
modern system of numeration for integers;
6. Finally, to establish once and for all that these discoveries took
place in India, independent of any outside influence.
Historical reality, it can be seen, is not as simple as is generally thought:
it is in any case not as simple as what an expression like “the invention of
Arabic numerals”, so cherished by the general public, seems to signify. For
in terms of “invention” there would have to have been not only quite an
exceptional combination of circumstances but also and above all an
improbable conjunction of several great thoughts, created over fifteen cen-
turies ago thanks to the genius of Indian scholars.
This would have taken exceptional powers of reflection, guided over a
long period of time, not by logic or conscience, but by chance and neces-
sity; chance discoveries and the need to remedy the problems engendered.
A. Vandel said, "A new idea is never the result of conscious or logical
work. It emerges one day, fully formed, after a long gestation period which
takes place within the subconscious."
It is true, as J. Duclaux says, that "the essential characteristic of scientific
discoveries is that they cannot be made to order”, because “the mind only
makes discoveries when it is thinking of nothing”.
INDIAN NUMERICAL NOTATION
With the aim of establishing the Indian origin of modern numerical fig-
ures, the following is a review of the numerical notations in common use in
India before and since this colossal event, beginning with the symbols cur-
rently in use in this particular part of the world.*
* Henceforth, the references given relate to the works which write out each of the styles in question. As for
the geographical location of the regions concerned, these are taken mainly from L. Frederic's Dictionnaire de
la civilisation indienne.
INDIAN CIVILISATION
368
It should be made clear straight away that the modern figures 1, 2, 3, 4,
5, 6, 7, 8, 9, 0 acquired their present form in the fifteenth century in the
West, modelled on specific prototypes and adopted permanently when the
printing press was “invented” in Europe. Today they are used all over the
world, thus constituting a kind of universal language which can be under-
stood by East and West alike.
However, this form is not the only one which can express the decimal
positional system. Particular symbols representing the same numbers still
coexist with the figures that we all know in several oriental countries.
From the Near East and the Middle East to Muslim India, Indonesia and
Malaysia, the following symbols are preferred:
1234567890
Ref.
t 0
EIS
Peignot and Adamoff
Pihan
Smith and Karpinski
Geographical area (see Fig. 25. 3):
Used in Libya, Egypt, Jordan, Syria, Saudi Arabia, Yemen, the Lebanon, Syria, Iraq, Iran,
etc., as well as in Afghanistan, Pakistan, Muslim India, Indonesia Malaysia and formerly
in Madagascar.
Fig. 24.2. Current Eastern Arabic numerals (known as “Hindi” numerals)
This is also the case in non-Muslim India, Central and Southeast Asia.
In these countries, symbols are still used that are graphically different
from our own, and whose cursive form varies considerably from one region
to another, according to the local style of writing.
Of course, this diversity dates back to ancient times, as the following
pages will prove.
Nagari figures
In his Kitab fi tahqiq i ma li’l hind, (an account of what he had witnessed in
India, written around 1030) al-Biruni, the Muslim astronomer of Persian
origin, after having lived in India and Sind for nearly thirty years, described
the great diversity of the graphical forms of figures in common use at that
time in different regions of India; his commentary begins thus [see al-
Biruni (1910); Woepcke (1863), pp. 275-6]:
Whilst we use letters for calculation according to their numerical
value, the Indians do not use their letters at all for arithmetic.
And just as the shape of the letters [that they use for writing] is dif-
ferent in [different regions of] their country, so the numerical
symbols [vary].
These are called *anka.
What we [the Arabs] use [for figures] is a selection of the best [and
most regular] figures in India.
Their shapes are not important, however, as long as their meaning is
understood.
The Kashmiris number their pages using figures which resemble ornamen-
tal drawings or letters [= characters used for writing] invented by the
Chinese, which take a long time and a lot of effort to learn, but which are
not used in calculation [which is carried out] in the dust ( hisab ‘ala 't turab).
Amongst the figures which were used long ago and are still used today
most commonly in the various regions of India, the most regular are
Nagari, which are also called Devanagari, from the name of the superb writ-
ing which they belong to (the words literally means “writing of the gods” in
Sanskrit) (Fig. 24.3).
Al-Biruni (who mastered written and spoken Sanskrit), was alluding to
precisely these figures when he said that the Arabs, in adopting the place-
value system from India, had taken, as a means of notation for the nine
units, “the best and most regular figures”.
1234567890
Ref.
? * 3 » h * f -
1 tin (oc p
? a y t, t £ ®
C (U
Desgranges
Frederic, DCI
Pihan
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53) :
Used in the Indian states of Madhya Pradesh (Central Province), Uttar Pradesh (Northern
Province), Rajasthan, Haryana, Himachal Pradesh (the Himalayas) and Delhi.
Fig. 24.3. Modern Nagari (or Devanagari) numerals
369
INDIAN NUMERICAL NOTATION
This point will be confirmed later in a palaeographical study, where it will
be shown how these figures, or at least their ancestors, were over the years
transformed by the hands of Arabic Muslim scribes to provide:
• in the Near East, the forms of the symbols in Fig. 24.2;
• in Northwestern Africa, other graphical representations, which
would gradually be transformed, this time by European scribes, into
the figures that we use today.
Furthermore, a striking resemblance still persists between the first three
and the last of these signs and our own numerals 1,2,3 and 0.
Marathi figures
These figures are used in the west of India, in the state-province of
Maharashtra (capital, Bombay). They are, as a rule, the cursive form of their
corresponding Nagari, except for a slight variation in the shape of the 5 and
the 6 (Fig. 24.3). There is a resemblance between these symbols for 2, 3, and
0 and our own, and the Marathi nine is symmetrical to the European nine.
1234567890
Ref.
U 3 y t £ Oc'e/ 0
Drummond
Frederic, DCI
Pihan
Geographical area (Fig. 24. 27 and 24. 53) :
Used in the area bordered in the west by the coasts of Konkan and Daman, and in the
north by Gujarat and Madhya Pradesh, in the south by Karnataka and in the southeast by
Andhra Pradesh.
Fig. 24.4. Modern Marathi numerals
Punjabijigures
Used in the state of Punjab (capital, Chandigarh), in the northwest of India,
bordering Pakistan. These are the same as the corresponding Nagari fig-
ures, except for the shape of the 7 (Fig. 24.3). There are similarities between
these symbols and our figures 2, 3, 7 and 0:
1
2
3
4
5
6
7
8
9 0
Ref.
1
3
a
M
n
•c;
0
Pihan
Geographical area (Fig. 24. 27 and 24. 53) :
Used in the northwest of India bordering Pakistan where the Indus, the Chenab, the
Jhelam, the Ravi and the Satlej rivers meet; as well as in the states of Himachal Pradesh
and Haryana.
Fig. 24 - 5 - Modem Punjabi numerals
Sindhifigures
These are symbols used in Sind, whose name derives from that of the river
Sindh (the Indus). These signs are more or less identical to their corre-
sponding Nagari, but their shape is generally more cursive than the latter
(Fig. 24.3). The figures 2, 3 and 0 are similar to our own, and the Sindhi 5 is
rather like a symmetrical version of the European 4.
1234567890
Ref.
1 \
Pihan
Stack
Geographical area (Fig. 24. 27 and 24. 53):
Used south of Punjab, on the lower banks of the Indus, in a region bordered in the south
by the Gulf of Oman and in the west by the Thar desert.
Fig. 24.6. Modern Sindhi numerals
Gurumukhi figures
In the city of Hyderabad (on the River Indus, to the east of Karachi, not to
be confused with the other Hyderabad, capital of Andhra Pradesh), the
merchants used to use a slight variant of the preceding figures, known as
Khudawadi.
The traders of Shikarpur and Sukkur, on the other hand, sometimes
used Sindhi or Punjabi figures, sometimes eastern Arabic figures and some-
times Gurumukhi figures, which are a mixture of Sindhi and Punjabi styles
(Fig. 24.5 and 24.6):
12345678 90
Ref.
<1 4 $ « m £ 9 ttf®
Datta and Singh
Stack
Geographical area (Fig. 24. 27):
Used in Sind and Punjab.
Fig. 24.7. Gurumukhi numerals
Gujarati figures
These are used in Gujarat State (capital, Ahmadabad), on the edge of the
Indian Ocean, between Bombay and the border of Pakistan. Again, these
are derived from Nagari figures, but they are more cursive in form, particu-
larly the 6 (Fig. 24.3). There are similarities between the Gujarati figures 2,
3 and 0 and our own numerals, as well as the figure 6.
INDIAN CIVILISATION
370
1234567890
Ref.
Drummond
Forbes
Frederic, DCI
Pihan
Geographical area (Fig. 24. 27 and 24. 53):
Used in the west of India, bordering the Indian Ocean,
between Bombay and the border with Pakistan, on the Gulf of Cambay.
Fig. 24.8. Modem Gujarati numerals
Kaithi figures
Used mainly in Bihar State, in Eastern India, and sometimes in Gujarat
State. They evidently derive from Nagari figures and are similar in form to
Gujarati figures (Fig. 24.3 and 24.8):
1234567 8 90
Ref.
? ^ 3 r ^ s ^ t d ♦
Datta and Singh
Geographical area (Fig. 24. 27):
Used in the east of India, in the region bordered in the east by Bengal, in the north by Nepal,
in the west by Uttar Pradesh and in the south by Orissa. Also sometimes used in Gujarat.
Fig. 24.9. Modern Kaithi numerals
Bengali figures
12345678 90
Ref.
i < C ? ? i 7 V #6 «
K3 5 i 1 i- Z> *
^ $ <C K a) 0
Frederic, DCI
Pihan
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53):
Used in the regions in the northwest of the Indian sub-continent, between Bihar, Nepal, Assam,
Sikkim, Bhutan, and the Bay of Bengal. Also widely used in Assam (along the Brahmaputra).
Fig. 24.10. Modern Bengali numerals
Used in the northeast of the Indian sub-continent in Bangladesh (capital,
Dacca), in the Indian state of West Bengal (capital, Calcutta), and in much
of central Assam (along the Brahmaputra River).
Of all the Bengali figures, there are four which resemble Nagari figures-
2, 4, 7 and 0 (Fig. 24.3). The others, however, are very different from those
used in other parts of India. In one of the following variants, our figures 2
3, 7 and 0 are recognisable; one of the variants of 8 also constitutes a sort of
prefiguration of our 8.
Maithili figures
Used mainly in the north of Bihar State, these derive mainly from Bengali
figures (Fig. 24.10):
1234567 8 90
Ref.
Datta and Singh
Geographical area (Fig. 24. 27):
Used in the region of Mithila, in the north ofBihar, between the Ganges and the southern
frontier of Nepal.
Fig. 24.11. Modem Maithili numerals
Oriyafigures
Used mainly in Orissa State (capital, Bhubaneswar), these are also known
as Orissi figures. Although they derive from the same source as Nagari fig-
ures, they present significant differences (Fig. 24.3):
1234567890
Ref.
\ ; *» v Vi® 5 r °
Frederic, DCI
Pihan
Renou and Filliozat
Sutton
Geographical area (Fig. 24. 27 and 24. 53):
Used in the region to the south of the eastern coast of Deccan, bordered in the north by
Bengal and Bihar, in the west by Madhya Pradesh and in the south by Andhra Pradesh.
Fig. 24.12. Modem Oriya (or Orissi) numerals
Takari figures
In everyday use in Kashmir, alongside eastern Arabic figures. They are also
called Tankri figures, of which a variant, Dogri, is used in the Indian part of
Jammu (in southwestern Kashmir):
371
INDIAN NUMERICAL NOTATION
12345678 90
Ref.
n»atf'<in < is 60
Datta and Singh
Geographical area (Fig. 24. 27):
Used in the region in the extreme northwest of the Indian sub-continent, currently
divided by the Indian-Pakistani border, joining the country of Jammu to the north of
Himachal Pradesh, the plain of Kashmir in the high basin of the Jhelum, the valley of
Zaskar in the north of the Himalayas and that of Ladakh, adjoining Tibet and China.
Fig. 24-13- Modern Takari (or Tankri) numerals
As for the 7 and the 9, they are, respectively, almost identical to the fig-
ures 1 and 7 of Nagari notation (Fig. 24.3).
Nepali figures
Used mainly in the independent state of Nepal (capital, Kathmandu), these
are also called Gurkhali figures.
In one of the following variations, our 1, 2, 3, 4, 7 and 0 can be recog-
nised, as well as our 8 to a certain extent (first set of figures, Fig. 24.15).
Sharada figures
The figures that were used for many centuries in Kashmir and Punjab, from
which, among others, Dogri and Takari figures derived (Fig. 24.13).
12345678 90
Ref
m S’ d •
* *
Pihan
Renou and Filliozat
Smith and Karpinski
Geographical area (Fig. 24. 27 and 24. 53):
Formerly used in Kashmir and Punjab (before the sixteenth century).
Fig. 24.14. Sharada numerals (relatively recent forms)
These figures are connected to Sharada writing, which was used in the
region at least since the ninth century, before it was replaced, relatively
recently, by the Persian Arabic characters that are used for writing.
This notation (even in its most recent form) deserves special attention,
because instead of representing zero with an oval or a small circle, it uses a
dot, the circle being used to denote the number 1 (the shape was slightly
modified according to the base).
The Sharada 2 is like the Nagari 3, except that the lower appendage is
absent from the Sharada figure.
To the untrained eye, it should be pointed out, the figures 2 and 3 are
not sufficiently distinct from one another, although the top of the 3 differs
from that of the 2 because it is long and snaking.
The 6 is symmetrical to the European 6, whilst the 8 is very similar to
the hand-written form of our 3.
12345678 90
Ref
n*iy. J Ecqi-o>o
* *3 * 4 9 «
Datta and Singh
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53):
Formerly used in Kashmir and Punjab (before the sixteenth century).
Fig. 24.15. Current Nepali numerals
There is an obvious similarity between these figures and the Nagari and
Sharada figures, with which they share a common source (Fig. 24.3 and 24.14).
Tibetanfigures
These are the figures used in Tibet. They are similar to Devandgari figures
(Tibetan writing comes from the same source as Nagari, introduced to the
region in the seventh century CE at the same time as Buddhism). The 2, 3,
the 9 (written backwards) and the 0 are alike.
12345678 90
Ref
? 2 J 1) /, f O
Foucaux
Pihan
Renou and Filliozat
Smith and Karpinski
Geographical area (Fig. 24.27 and 24.53):
Used in regions of Tibet, from the border of Pakistan to the border of Burma
and Bhutan.
Fig. 24.16. Tibetan numerals
INDIAN CIVILISATION
372
Tamilfigures
Unlike Northern and Central India, in Southern India, namely Tamil Nadu,
Karnataka, Andhra Pradesh and Kerala, the Dravidian people do not speak
Indo-European languages.
Tamil figures are used in Southeast India, in Tamil Nadu state (capital,
Madras):
123456789 0
Ref.
3 £L ffh ff 1 & 9n GT dm
ffi S P O OT An 06
Frederic, DCI
Pihan
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53):
Used in the region on the eastern coast of the Indian peninsula, from the north of Madras
to the tip of Cape Comorin (Kanya Kumari) and bordered in the east by the Bay of Bengal,
in the west by Kerala, in the northwest by Karnataka and in the north by Andhra Pradesh.
Also used in the north and northwest of Sri Lanka,
Fig. 24.17. Current Tamil numerals (or “Tamoul" numerals, according to an erroneous transcription)
It should be noted, however, that the Tamils do not use zero in this system,
which is only vaguely based on the place-value system.
Along with the signs for nine units, their system actually possesses a
specific sign for 10, 100 and 1,000. To express multiples of 10, or hundreds
or thousands, the sign for 10, 100 or 1,000 is proceeded by that of the corre-
sponding units, which thus play the part of multiplier.
In other words, the Tamil system is based upon a principle which is at
once additive and multiplicative, known as the hybrid principle and which
has been used in many systems since early antiquity (see Fig. 23.20).
Equally, in terms of their appearance, these figures have nothing in
common with the preceding notations.
For these reasons, it was believed that the Tamil figures were an original
creation of the Dravidians, after they came up with the idea of using certain
letters of their alphabet as signs for counting with.
It is true that there is a degree of resemblance between the first ten fig-
ures and what might constitute the corresponding letters of the Tamil
alphabet, although the correspondence is not always very rigorous:
Comparison between the numeral
and the letter
1
9
2
£L
3
ffn
4
&
5
6
7
ffn
7
Gf
8
&\
9
ffn
Fig. 24.18.
9>
ka, ga
e.
u
KJ
na
ff-
sha
6
ra
ffx
cha
GT
e
s\
a
ffeu
ku, gu
Tamil name for the
corresponding number
uru
. 1
irandu
2
munru
3
nalu, nangu
4
a'indu, andju
5
aru
6
erla, ezha
7
ettu
8
onbadu
9
There is one question that cries out to be asked: if the theory is correct, why
were these particular letters used to denote these numerical values? The
obvious answer would be that the initials of the Tamil names for the num-
bers were used, but this is not the case, as the preceding table clearly
demonstrates.
Then why were these letters singled out to represent numbers? Why did
these people not give a numerical value to all the Tamil letters, as the
Greeks and the Jews did with their respective alphabets when they created
their systems of numeral letters?
This theory is rather far-fetched; it is merely a coincidence that these fig-
ures resemble the above Tamil letters. Moreover, the correspondence can
only be established using the modern forms of the letters.
In fact, Tamil letters and figures are connected to all the other systems
used in India: they all derive from the same source. Tamil writing, however,
evolved in an entirely different manner from the others, both in terms of
appearance and linguistic structure, introducing innovations which gave
it its distinctive character. In particular, the characters and numerical sym-
bols are considerably more rounded, with curves and volutes. It is not
impossible that the material on which the characters were written played a
role in this evolution, if it did not actually cause it.
INDIAN NUMERICAL NOTATION
.3 73
In other words, the first nine Tamil figures are from the same family as
the other corresponding Indian numerical symbols, the difference lying in
their style and their adaptation to the unique shape of Tamil writing.
Malaydlam figures
These figures are used by the Dravidian people of Kerala State, on the
ancient coast of Malabar, in the southwest of India. They have the same
name as the form of writing used in the area.
1234567 8 9 0
Ref.
t° a. ca (y @"3 9 nrb
c-S> 1 r*x V ® 9 ^ ^
^ ft cp/Ju6)j?r % /*
Drummond
Frederic, DCI
Peel, J.
Pihan
Renouand Fillozat
Geographical area (Fig. 24. 27 and 24. 53):
Used in the region stretching the length of the southeast coast of India, from Mangalore in
the north to the southernmost point of India, and which is made up of a long coastal strip
stretching from the coast of Malabar and by the Ghats encompassing the peaks of the
Cardamoms.
Fig. 24.19. Current Malaydlam numerals
Like the Tamils, the people of Kerala did not use zero in their notation
system for many centuries: Malaydlam figures are not based on the place-
value system, and there are specific figures for 10, 100 and 1,000. It was
only since the middle of the nineteenth century, under the influence of
Europe, that zero was introduced and combined with the symbols for the
nine units according to the positional principle.
Thus the Tamil and Malaydlam figures were the only ones in India that
did not include zero and were not based on the positional principle until
relatively recently.
However, it should be noted that Tamil figures, a few centuries ago,
before they evolved into their current forms, closely resembled their
Malaydlam cousins which have conserved a style close to the original.
The graphical link with the numerical signs of other regions of India is
more easily seen through examining the original appearance of the Tamil
figures than through looking at their modern form (Fig. 24.17 and 24.19):
The Nagari 1 is easily recognised, whose former shape was almost hori-
zontal (Fig. 24.39) and which evolved in Tibet into a form constituting a
sort of intermediate with the Malaydlam 1 (Fig. 24.16).
The Nagari 2 is also recognisable, although the “head" of the sign is very
neatly rounded at the bottom.
On the other hand, the Malaydlam 3 is much closer to the correspond-
ing Oriya figure (Fig. 24.12), with an extra “tail” which the Nagari 3 also has
(Fig. 24.3).
The 4 is similar to its Sindhi equivalent except for the characteristic
curve on the left (Fig. 24.6).
The 5 is very similar to one of the corresponding Bengali figure
(Fig. 24.10) and is reminiscent of the Malaydlam style.
The 6 resembles its Sindhi counterpart (Fig. 24.6), but it has an extra
loop on the top, the whole figure being in a position which is obtained by
rotating it through 90° anti-clockwise.
The 7 resembles its Marathi, Gujarati and Oriya equivalents (Fig. 24.4,
24.8 and 24.12), whose prototype is found in the ancient Nagari style
(Fig. 24.39).
The 8 is the symmetrical equivalent of the Gujarati 8 (Fig. 24.4).
As for the 9, it particularly resembles the Nagari style of the ninth cen-
tury CE.
There can be no doubt: the Dravidian figures for the nine units have the
same origin as all the others; the similarities found scattered amongst these
diverse figures could not possibly be the product of chance.
The following two varieties of Dravidian figures serve as confirmation of
this fact.
Telugu figures
These are the numerical symbols used by Dravidian people of the former
Telingana, the Indian state of Andhra Pradesh (capital, Hyderabad). They
are also called Telinga figures (Fig. 24.20).
12 3
4
5
6
7 8 9
0
Ref.
O _D 3
£-
Z yj- r*
0
Burnell
V
X
L.
2- c r- f~
0
Campbell
Datta and Singh
0-^3
e_
l Or F~
0
Pihan
Renou and Filliozat
O J 3
V
X
t
1 o' r
0
Smith and Karpinski
Geographical area (Fig. 24. 27 and 24, 53):
Used in the southeast of India, bordered in the southeast by the Bay of Bengal, in the
north by the States of Orissa and Madhya Pradesh, in the northwest by Maharashtra, in
the west by Karnataka and in the south by Tamil Nadu.
Fig. 24.20. Modem Telugu (or Telinga) numerals
INDIAN CIVILISATION
374
Kannara figures
Used by the Dravidian people of central Deccan, including the state of
Karnataka (capital, Bangalore) and part of Andhra Pradesh:
1234567890
Ref.
0-99? v 3L L dir 0
4 (T r 0
Burnell
Datta and Singh
Pihan
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53):
Used mainly in the region stretching from the Mysore mountains to the eastern coast of
the Indian sub-continent, between the Gulf of Oman and the Western Ghats.
Fig. 24.21. Modern Kannara (or Kannada orKarnata) numerals
Sinhalese figures
Used mainly in Sri Lanka and in the Maldives as well as in the islands to the
north of the latter. (In the north and northwest of Sri Lanka, Tamil figures
are also used due to the high number of Tamil people who live in these
areas of the island.)
12345678 90
Ref.
SI Glu £3 SV. 0 £? ®j
SV. GVo eg GW. (0 3<
of m ^
Alwis (de)
Charter
Frederic, DCI
Pihan
Renou and Filliozat
Geographical area (Fig. 24. 27 and 24. 53):
Used in Sri Lanka, in the Maldives, as well as in the islands to the north of the Maldives.
Fig. 24.22. Current Sinhalese (orSinhala) numerals
It should be noted that although Sinhalese writing is linked to Dravidian
forms of writing (even though it is more stylish, striving as it does
towards an ornamental effect), the language of this writing is not
Dravidian. Sinhalese is an Indo-European language: “it is a language that
belongs to Prakrit (dialects) of ‘Middle Indian’, as several inscriptions
written in Brahmi dating from around the second century BCE show.
However, after the fifth century CE, the Sinhalese language, separated
from India’s Indo-European languages by the Tamil area, developed in an
individual style, as did its writing. The two seem to have changed little
since 1250” (L. Frederic).
There are twenty Sinhalese figures. This number of numerical signs is
due to the absence of zero and the fact that the system, which is not based
upon the place-value system, uses a specific figure for every ten units, as
well as special figures that represent 10, 100 and 1,000 (see Fig. 23.18).
Burmese figures
Used in Burma. Formerly used in the kingdom of Magadha, these were
once known as cha lum figures, they are part of Burmese writing, which
itself derives from the former Pali alphabet, introduced to the region by
Buddhists (Fig. 24.23).
12345678 90
Ref.
°J?9D£ c l°(!: 0
o J ? 9D© < \°Cp t>
3 J t $ 3 © ? 0 <5 0
? J ? S 0 Q
; J ? * 3 S ? n 0 0
Carey
Datta and Singh
Latter
Pihan
Geographical area (Fig. 24.27 and 24.53):
Used in the region stretching from Laos to the Bay of Bengal, and from Manipur to Pegu;
also, in a slightly modified form, around Tenasserim and along the coast from Chittagong.
Fig. 24.23. Modern Burmese numerals
In modern Burmese writing, the principal element of the shape of the let-
ters is a little circle, the value of which varies according to the breaks,
juxtapositions or appendages.
The same applies to the figures, or at least to three of them, whose
shapes should not be confused.
These are:
• the 1, formed by a circle, a quarter open on the left;
• the 8, which is a circle that is a quarter open at the bottom;
• and the 0 which is a whole circle.
The 3 is an open circle like the 1, with an appendage which slants
towards the right, and the 4 is formed by the mirror image of the 3.
As for the 9, it is the 6 turned upside-down.
375
INDIAN NUMERICAL NOTATION
However this graphical rationalisation is relatively recent: the Indian
origin (via former Pali figures) of the Burmese figures was still unknown in
the seventeenth century.
Thai- Khmer figures
These are the official numerical symbols of Thailand, Laos and Cambodia.
They also belong to the family of numerical signs that are of Indian origin,
actually belonging to the former Pali style.
1
2
3
4 5
6 7 8 9
0
Ref.
C)
\<D
<n
6 £
V CV {J
O
Pihan
G7
Is
cn
6. &
^ til J f'
0
Rosny
9
a)
&
b rb c/
O
«
O
& *
b ** cfc *•
0
6)
n
gl d
b al
0
9
Is
<n
d d.
'a «) £ <K
0
Geographical area (Fig. 24.53):
Used in Thailand, Laos, Kampuchea, in the State of Chan to the east of Burma, in some
parts of Vietnam, in China in the provinces of Guangxi and Yunnan, as well as in the
Nicobar islands.
Fig. 24.24. Modern Thai -Khmer (known as “Siamese”) numerals
Some of these figures look so alike that they are easily confused. Unlike the
various “true” Indian figures, the Thai-Khmer 2 is more complicated than
the 3. The 5 only differs from the 4 because it has an extra loop at the top.
The 8 is more or less symmetrical to the 6, and the figure 7 is easily con-
fused with the 9.
Balinese figures
These are from Bali, and also developed from the Pali figures.
1234567890
Ref.
V ^ 3 ® ^ J 0
Renou and Filliozat
Geographical area (Fig. 24.53):
Used in Bali, Borneo and the Celebes islands.
Fig. 24.25. Modem Balinese numerals
Javanese figures
The final figures in this list of numerical symbols currently in use in Asia
are those from the island of Java:
1234567 890
Ref.
am tj, o| £ oj aan 43, (urn 0
De Hollander
Pihan
Geographical area (Fig. 24.53):
Used in Java, Sunda, Bali, Madura and Lombok.
Fig. 24.26. Modern Javanese numerals
Apart from the figures 0 and 5 (whose Indian origin is obvious), this nota-
tion actually corresponds to a relatively recent artificial innovation, the
appearance of the figures curiously having been made to resemble the
shape of certain letters of the current Javanese alphabet. Before this, how-
ever, the Javanese people used a notation which belonged to the Pali group
of the family of Indian figures: the notation known as Kawi (attested since
the seventh century CE), which belongs to the writing of the same name
(from which the current Javanese alphabet derives).
Brahmi, “mother" of all Indian writing
Despite the high number of graphical representations of the nine units,
there is no doubt as to their common origin.
Leaving European and Arabic numerals on one side for a moment, each
of the preceding styles were graphically connected to one of the various
styles of writing belonging to either India, Central or Southeast Asia: it is
clear from extensive palaeographical research that they all derive, directly
or indirectly, from the same source.
Therefore, it is worthwhile saying a few words about the history of the
styles of writing of this region.
The oldest known writing of the sub-continent of India appeared on the
stamps and plaques of the civilisation of the Indus (c. 2500 - 1500 BCE),
discovered mainly in the ruins of the ancient cities of Mohenjo-daro and
Harappa. However, as this writing has not yet been deciphered, the corre-
sponding language remains unknown; therefore there is a large gulf
separating these inscriptions of the first known texts in Indian writing and
the language, assuming that a link exists between the two systems.
In fact, the history of Indian writing begins with the inscriptions of
Asoka, third emperor of the dynasty of the Mauryas of the Magadha, who
reigned in India from c. 273 to 235 BCE, whose empire stretched from
INDIAN CIVILISATION
376
Arabian Sea
&
“^Kaveretti
Laccadive o Trivandrum
Islands Gulf
(INDIA) 'V w . of
Male Marinar
Maldives:
INDIAN OCEAN
1
JAMMU and KASHMIR
14
MEGHALAYA
2
HIMACHAL PRADESH
15
NAGALAND
3
PUNJAB
16
MANIPUR
4
HARYANA
17
MIZORAM
5
RAJASTHAN
18
TRIPURA
6
UTTAR PRADESH
19
ORISSA
7
GUJARAT
20
DADRA and NAGAR HAVELI
8
MADHYA PRADESH
21
MAHARASHTRA
9
BIHAR
22
ANDHRA PRADESH
10
BENGAL
23
GOA
11
SIKKIM
24
KARNATAKA
12
ASSAM
25
KERALA
13
ARUNACHAL PRADESH
26
TAMIL-NADU
Fig. 24 . 27 . The states of present-day India
Afghanistan to Bengal and from Nepal to the south of Deccan [see
L. Frederic (1987)]. These inscriptions are mainly edicts carved on rocks or
columns for which diverse styles of writing were used: Greek and
Aramaean in Kandahar and Jalalabad in Afghanistan; the Kharoshthi
system in Manshera and Shahbasgarhi to the north of the Indus; and
Brdhmi writing in all the other regions of the Empire.
Kharoshthi comes directly from the old Aramaean alphabet and is simi-
larly written from right to left. This is why it is also labelled
“Aramaeo-Indian” writing. Probably introduced in the fourth century BCE,
it remained in use in the northwest of India until the end of the fourth cen-
tury CE.
As for the written form of Brdhmi, it was written from left to right and
was used to note the sounds of Sanskrit.
The origin of this writing is still not known. Attempts have been made
to prove that it comes from Kharoshthi writing, but the explanation for this
is far from convincing. Brdhmi certainly derives from the Western Semitic
world, doubtless via some other variety of Aramaean, of which specimens
have not yet been found [see M. Cohen (1958); J. G. Fevrier (1959)].
Since the first millennium BCE, India was already open to outside influ-
ences, due to long-established ties with the Persians and Aramaean
merchants who used the routes which went from Syria and Mesopotamia
to the valley of the Indus.
However, the appearance of Brdhmi probably pre-dates Emperor Asoka,
by whose time it was in widespread use in the different regions of the sub-
continent of India.
This language outlived all the others, becoming the unique source of all
the forms of writing that later emerged in India and her neighbouring
countries. It was given the name Brdhmi, in Hindu religion one of the
names of the seven *mdtrika or "mothers of the world”: one of the feminine
energies (*shakti) supposed to represent the Hindu divinities. Represented
as sitting on a goose, her power was equal to that of Brahma, the
“Immeasurable”, god of the Sky and the horizons, who “endlessly gives
birth to the Creation” and who one day invented Brdhmi writing for the
well-being and diversity of humankind.
According to the edicts of Asoka, Brdhmi appeared, in a slightly modi-
fied form, in contemporary inscriptions of the Shunga Dynasty (185 -
c. 75 BCE on the Magadha, in the present Bihar state, south of the
Ganges, then in those of the Kanva Dynasty (who succeeded the former
from 73 to c. 30 BCE).
377
INDIAN NUMERICAL NOTATION
The following is a more developed exploration of Brahmi, first through
the inscriptions of the Shaka Dynasty (Scythians, who reigned over Kabul
in Afghanistan, Taxila in Punjab and Mathura, from the second century
BCE to the first century CE) and through the coins embossed with the sov-
ereigns of the Shaka Dynasty who reigned from the second to the fourth
century CE in Maharashtra (under the name of Kshatrapa, “Satraps”).
Brahmi evolved a little more in the writing of the Andhra and
Satavahana Dynasties which reigned during the first two centuries CE in
the northwest of Deccan.
Then the system appeared, in an even more developed form, in the
inscriptions of the Kushan emperors (who reigned from the first to the
third century CE, and who, at first based in Gandhara and Transoxiana
attempted to conquer Northwestern India).
Thus through numerous successive and perceptible modifications,
Brahmi gave birth to many highly individual styles of writing; styles which
constitute the main groups currently in use (Fig. 24.28):
1. the group of types of writing in Northern and Central India and in
Central Asia (Tibet and East Turkestan);
2. the writing of Southern India;
3. oriental writing (Southeast Asia).
The apparently considerable differences between the forms of writing
of these various groups is ultimately due either to the specific character of
the language and traditions to which they have been adapted, or to the
techniques of the scribes of each region and the nature of the material
they used.
A parallel evolution: Indian figures
In this context, everything becomes clear: in India and the surrounding
regions, the notation of the nine units evolved in much the same way as the
styles of writing that were born out of Brahmi. In other words, in the same
way as the writing they belong to, the various series of 1 to 9 formerly or
currently in use in India, Central and Southeast Asia all derive more or less
directly from the Brahmi notation for the corresponding numbers.
The numerical symbols of the original Brahmi notation
This notation appeared for the first time in the middle of the third century
BCE in edicts written in both Ardha-Magadhi and Brahmi which the
Fig. 24.28. Indian styles of writing
emperor Asoka had engraved on rocks, polished sandstone columns and
temples hewn out of the rock, in diverse regions of his empire.
But the numerical notation that is found within these edicts is fragmen-
tary, only giving the representations for the numbers 1, 2, 4, and 6:
INDIAN CIVILISATION
378
1234567 890
Ref.
IMF 6
El, III p. 134
A
IA, VI, pp. 155 ff.
\
IA, X, pp. 106 ff.
Indraji, JBRAS XII
Date: third century BCE.
Source: edicts of Asoka written in Brahmi, in various regions of the Empire of the
Mauryas, from the regions ofShahbazgarhi, Manshera, Kalsi, Girnar and Sopara (north
of Bombay) to Tosali and Jaugada in Kallinga (Orissa), Yerragudi in Kannara, Rampurwa
and Lauriya-Araraj in the north ofBihar, Toprah and Mirath north ofDelhi, and
Rummindei and Nigliva in Nepal (Fig. 24.27).
Fig. 24.29. Numerab of the original Brahmi style of writing: our present-day 6 is already recognbable
The numerical symbols ofintemediate notations
The same system appears in the documents of the eras which followed and
this gives a much more precise idea of how Brahmi figures looked.
The following figures appeared at the beginning of the Shunga and
Magadha dynasties in the Buddhist inscriptions which adorn the walls of
the grottoes of Nana Ghat:
1234567890
Ref.
- = f r i ?
* <e 1 ?
Datta and Singh
Indraji, JBRAS XII
Smith and Karpinski
Date: second century BCE.
Source: the caves of Nana Ghat (central India, Maharashtra, c. 150 km from Poona),
Buddhist inscriptions written for a sovereign named Vedishri which mainly concern
various presents offered during religious ceremonies.
Fig. 24.30. Numerab of the intermediary notation of the Shunga: we can already see the prefiguration of
our numerab 4, 6, 7 and 9.
The same series appeared a little later, but in a much more complete form,
in the first or second century CE, in the inscriptions of the Buddhist grot-
toes of Nasik (Fig. 24.31).
Brahmi figures are also found, in more and more varied forms, in
Mathuran inscriptions (Fig. 24.32), Kushana and Andhran inscriptions
(Fig. 24.33 and 24.34), western Satrap coins (Fig. 24.35), the inscriptions of
Jaggayyapeta (Fig. 24.36), and of the Pallava Dynasty (Fig. 24.37) .
As these numerals derive from Brahmi figures and consequently serve as
a go-between with the later forms of the numerals, they shall henceforth be
referred to as the numerical symbols of the intermediate notations.
12345678 90
Ref.
- = = + h^<7s?
’ 1 M f f ^
El, VIII, pp. 59-96
El, VII, pp. 47-74
Biihler
Datta and Singh
Renou and Filliozat
Smith and Karpinski
Date: first or second century CE.
Source: Buddhist caves of Nasik (in Maharashtra, at least 200 km north
of Bombay).
Fig. 24.31. Numerab of the intermediary system of Nasik: we can see the prefiguration of our numer-
als 4, 5, 6, 7, 8 and 9.
12345678 90
Ref.
“ = 5 p m)
- * 2 * b if 7 J ?
* b y n. 9 y
% t 9 7
tl f* ? *
0
Biihler
Datta and Singh
Ojha
Date: first - third century CE.
Source: inscriptions of Mathura (town of Uttar Pradesh, on the banks of the Yamuna 60 km
northwest of Agra), contemporary with a Shaka dynasty.
Fig. 24.32. Numerals of the intermediary system of Mathura
12345678 90
Ref.
' = * H > £ ? *1 ?
= F /» f> 1 *7
F l a \a
l s
5
El, I, p. 381
El, II, p. 201
Biihler
Datta and Singh
Ojha
Smith and Karpinski
Date: first - second century CE.
Source: contemporary inscriptions of the Kushana dynasty.
Fig. 24.33. Numerals of the intermediary system of the Kushana
379
INDIAN NUMERICAL NOTATION
1234567890
Ref.
* > <f 7 S 1
Biihler
Datta and Singh
Ojha
Date: second century CE.
Source: contemporary inscriptions of the Andhra dynasty.
Fig. 24.34. Numerals of the intermediary notation of the Andhra
12345678 90
Ref.
- * = H r V 1 1 1
- - 4 b ? 1 ] 3
- - -- 7 h 0 •> 3
1 ^ J > J
f > ±
* * 5
> >
4
&
JRAS, 1890, p. 639
Biihler
Datta and Singh
Ojha
Smith and Karpinski
Date: second to fourth century CE.
Source: coins of the western Satraps.
Fig. 24.35. Numerals of the intermediary notation of the western Satraps
These intermediate notations spread over the various regions of India and
the neighbouring areas, as did the letters of the corresponding writing,
and, over the centuries, they underwent graphical modifications, finally to
acquire extremely varied cursive forms, each with a regional style.
The origin of the notations of Northern and Central India
One of the first individual notations to appear was Gupta notation, used
during the dynasty of the same name (its sovereigns reigned over the
Ganges and its tributaries from c. 240 to 535 CE) (Fig. 24.38).
1234567 8 90
Ref.
- ~ n/ ^ b ? n h
/ > a/ ^ n ii
; w m J A
^ t/ } } v y
-> v -v * 5 ^
Biihler
Datta and Singh
Ojha
Date: third century CE.
Source: inscriptions of Jaggayyapeta (site of an ancient Buddhist centre established on the
River Krishna, in the present-day state of Andhra Pradesh, in the southeast of the Indian
peninsula, opposite Amaravati, capital of the Andhra kingdom during the Shatavahana
dynasty).
Fig. 24.36. Numerals of the intermediary notation of Jaggayyapeta
1234567 8 90
Ref.
- i V % h f 1 'j ]
m/ 5
Biihler
Datta and Singh
Ojha
Date: fourth century CE.
Source: inscriptions of King Skandravarman (c. 75 CE) of the Pallava dynasty, who
reigned in the southeast of India at the end of the third century CE, after the fall of the
Andhra and Pandya rulers.
Fig. 24.37. Numerals of the first intermediary notation of the Pallava
1234567 8 90
Ref.
■ — - n ^ 1 m
- - 2 . * F P c \
7 2 1 h r ^
? * \> 5
1? J
1
c
Clin, III
Biihler
Datta and Singh
Ojha
Smith and Karpinski
Date: fourth to sixth century CE.
Source: inscriptions of Parivrajaka and Uchchakalpa
Fig. 24.38. Gupta numerals
INDIAN CIVILISATION
380
This notation was the origin of all the series of figures in common use in
Northern India and Central Asia.
The first developments in Nagari notation
As Gupta writing became more refined, it gave birth to Nagari notation (or
“urban” writing, the magnificent regularity of which gave it the name of
Devanagari, or “Nagari of the gods”).
This writing soon acquired great importance, becoming not only the
main writing of the Sanskrit language, but also of Hindi, the great language
of modem Central India.
As numerical notation experienced a parallel evolution, so Nagari fig-
ures were born out of Gupta figures, which later led to the emergence of
modem Nagari figures (see also Fig. 24.3 above):
1
2
3
4
5
6
7
8
9
0
Ref.
n
S3
*>
V
£
0
S
n
O
El, I, p. 122
El, I, P. 162
Si
El, I, p. 186
<1
*
£
<\
r
El, II, p. 19
El, IH, p. 133
7
Sr
S\
T
9
0
El, IV, p. 309
El, IX, p. 1
OL
El, IX, p. 41
1
X
V
1/
Q
3
G
O
El, IX, p. 197
El, IX, p. 198
l
*
A
1
T
<P
El, IX, p. 277
El, XVin,p.87
1
2
e
3
r
V
JA, 1863, p. 392
1
*
\
*
J\
a.
r
9
O
IA, VIII, p. 133
IA, XI, p. 108
IA, XU, p. 155
7
K
n
' 1
ft
V
IA, XII, p.249
IA, XII, p.263
t
a
3
2r
4
1
6
cJ
IA, XIII, p. 250
IA,XTV,p.351
IA, XXV, p. 177
V
V
A
t,
°i
t
i
Biihler
(V
Datta and Singh
\
y
A
4
C
Ojha
7
T
T
V
U
5
3
c
Q?
O
't
Date: seventh to twelfth century CE (Fig. 24.75).
Source: various inscriptions on copper from Northern and Central India.
Fig. 24.39A. Ancient Nagari numerals
12345678 90
Ref.
\ ^ H 'I 0 5 ?
' X ^ * S ?
? * 1 y n
t y Q*
Datta and Singh
Ojha
Smith and Karpinski
Date: eighth to twelfth century CE (Fig. 24.3).
Source: various manuscripts from northern and central India (which use neither zero nor
the place-value system).
Fig. 24.39B. Ancient Nagari numerals
1
2
3
4
5
6
7
8
9
0
Ref.
\
\
V
(1
*
3
T
ft.
O
ASI, Rep. 1903-1904, pi. 72
X
X
i
a
<
ft
0
El, 1/1892, pp. 155-62
Datta and Singh
1
\
\
7S
d
a
r
«l
O
Guitel
Date: 875 to 876 CE (Fig. 24.73).
Source: inscriptions of Gwalior (capital of the ancient princely state of
Madhyabharat, situated between the present-day states of Madhya Pradesh and
Rajasthan, c.120 km from Agra and over 300 km south of Delhi). The two Sanskrit
inscriptions are from the temple of Vaillabhatta-svamim dedicated to Vishnu, and are
from the time of the reign of Bhojadeva, dated 932 and 933 of the Vikrama Samvat
era, or 875 and 876 CE.
Fig. 24.39c. Ancient Nagari numerals
These are the forms that the Arabs used when they adopted Indian numer-
ation: the proof of this will be seen later on; moreover, in the following
tables it can be seen that these figures, if not identical, are very similar to
the numerical symbols that we use today.
Notations which are derived from Nagari
In Maharashtra, via a southern variant, Nagari gave birth to Maharashtri,
which gradually evolved into modem Marathi writing, of which there are
currently two forms: Balbodh (or “academic” writing), used to write
Sanskrit, and Modi, which is more cursive in form, and is only used to write
Marathi. A similar evolution took place for the notation of the nine units
(Fig. 24.4 above).
381
INDIAN NUMERICAL NOTATION
In the state of Rajasthan (bordering Pakistan in the west, Punjab,
Haryana and Uttar Pradesh in the north, Madhya Pradesh in the east and
Gujarat in the south) Nagari evolved into Rajasthani. In the northwest of
India, however, between the Aravalli Range and the Thar Desert, Nagari
diversified into the cursive forms of Marwari and Mahajani, mainly used
for commercial purposes.
After the end of the eleventh century, a notation called Kutila (or “Proto-
Bengali”) was also born out of Nagari, from which, in turn, modem Bengali
evolved, sometime after the beginning of the seventeenth century
(Fig. 24.10), to which Oriya (Fig. 24.12), Gujarati (Fig. 24.8), Kaithi
(Fig. 24.9), Maithili (Fig. 24.11) and Manipur! can be linked.
The development of Sharada notation
After the beginning of the ninth century in Kashmir and Punjab, a north-
ern variant of Gupta led to Sharada notation, which was used in the above
parts of India until the fifteenth century at least (Fig. 24.14).
12345678 90
Ref.
'V? uu
n 3 3- * V \ H
1A, XVII, pp. 34-48
Datta and Singh
Kaye: Bakhshali manuscript
Smith and Karpinski
Date: between the ninth and twelfth century CE. (Fig. 24.14).
Source: Manuscript from Bakshali (a village in Gandhara, near Peshawar, in present-
day Pakistan, where it was discovered in 1881). The manuscript is written entirely in
the Sharada style, in the Sanskrit language, in both verse and prose, by an anonymous
author. It deals with algebraic problems, the numbers being expressed in Sharada
numerals using the place-value system, zero being written as a dot (bindu). This
manuscript could not have been written earlier than the ninth century CE or later
than the twelfth century, but it is possible that it is a copy of- or a commentary on -
an earlier document.
Fig. 24.40A. Ancient Sharada numerals
Notations derived from Sharada
It is from this notation that Takari (Fig. 24.13), Dogri, Chameali, Mandeali,
Kului, Sirmauri, Jaunsari, Kochi, Landa, Multani, Sindhi (Fig. 24.6),
Khudawadi, Gurumukhi (Fig. 24.7), Punjabi (Fig. 24.5), etc., originated.
1234567 8 90
Ref.
KAV
Smith and Karpinski
Date: the fifteenth century CE (approximately).
Source: A Kashmiri document which reproduces the Vedi hymns and texts of the
Atharvaveda in Sharada characters (the document is preserved at Tubingen University).
Fig. 24.40B. Sharada numerals (most recent style)
Nepalese notations
1
2
3
4
5
6
7
8 9 0
Ref.
*\
<v
*
A
?
n
A)
Bendall
T*
t
s
P>
xf A
Datta and Singh
■\
\
%
lo
fl
Ojha
Smith and Karpinski
U
■y
&
di
3)
P
if
1
•a.
3
«i
•S
V*
-
■x
¥
&
<1
-5, T
A
A
d
%
$
*
*5
lx
ZF
Date: eighth to twelfth century CE (Fig. 24.15).
Source: inscriptions from Nepal and various Buddhist manuscripts from Nepal.
Fig. 24.41. Ancient Nepali numerals
Many other systems originated from Gupta. After the fifth century CE, one
variation evolved into Siddhamatrika (or Siddam) writing which was used
mainly in China and Japan for Sanskrit notation. During its development,
some time after the beginning of the ninth century, it gave birth to Limbu
and modern Nepali (also called Gurkhalf), specific notations of Nepal
whose numerical symbols underwent a parallel evolution (Fig. 24.41).
Notations which originated in India and Central Asia
From the time of the Kushana Empire (first to third century CE) until the
Empire of the Guptas, Indian civilisation, along with Buddhism, stretched
to Chinese Turkestan, as well as towards northern Afghanistan and Tibet.
INDIAN CIVILISATION
382
Thus one of the notations to be born out of Gupta reached these regions.
Without any radical change, this notation evolved into the writings of
Chinese Turkestan, which were used to write Agnean, Kutchean and
Khotanese. Each style would have possessed its own figures.
On the other hand, in the various regions of Tibet, the high valleys of the
Himalayas and the neighbouring areas of Burma, Gupta underwent quite
drastic changes to enable spoken languages with very different inflexions to
be written down. This is how the Tibetan alphabet came about, the Guptan
numerical symbols also being adapted to this graphical style (Fig. 24.16).
Mongolian figures
When the great conqueror Genghis Khan died in 1227, the Mongolian
Empire stretched from the Pacific to the Caspian Sea.
J. G. Fevrier (1959) claims that “the Mongolians did not possess any
form of writing and that all their conventions were oral; their ‘contracts’
were alleged to be certain signs carved onto wooden tablets.”
But by conquering nearly all of Asia, these half-savage hordes could no
longer be contented with such rudimentary methods; so they decided to adopt
the writing of the Uighur people of Turfar after they defeated them (the Uighur
alphabet constituting a type of Syriac writing, imported by Nestorian monks).
The Mongolians then decided that they wanted an alphabet that was
more appropriate for writing their language, mainly because of pressure
from the propagators of Buddhism to have their own specific instrument for
translating their texts. Their alphabet was created with the collaboration of
the Uighurs. They wrote in vertical columns which read from left to right.
However, instead of adopting the non-positional system of the afore-
mentioned region, the Mongolians preferred to use Tibetan figures, after the
contact that they had had with the latter. Thus "Mongolian” figures were bom:
12345678 90
Ref.
Pihan
Date: thirteenth to fourteenth century CE.
Fig. 24.42. Mongolian numerals: the numerals 2, 3, 6 and 0 are recognisable, as well as 9 (or rather
its mirror image).
An evolution from the South to the East
Like Gupta, there is another style of writing to come out of Brahmi that is
very different from its origins.
12345678 90
Ref.
« S • % V <4 1 »
Clin, III
^ 1 z f h
Biihler
Datta and Singh
- 7=j ^
Ojha
^
n Sp
Date: fifth to sixth century CE.
Sources: inscriptions from the Paltava dynasty (who reigned in the southeast oflndia in
the region of the lower Krishna on the Coromandel coast, from the end of the third
century CE until the end of the eighth century); Shalankayana inscriptions (a small Hindu
dynasty that reigned from 300 to 450 CE, in Vengi and Pedda Vengi, in the region of the
Krishna river).
Fig. 24.43. Numerals of the intermediary counting system of the Pallava
This is the writing used in inscriptions in Pallava, Shalankayana and
Valabhi (Fig. 24.43 and 24.44) and the more individualised style of
Chalukya and Deccan (Fig. 24.45) and Ganga and Mysore (Fig. 24.46):
1234567 8 90
Ref.
- r r ± j? 9 a if 9
*» = 3 * <T W d P 3
~ : o -h 9 o vS 6)
- ~ & 9 r
C ha 3
= *
Clin, III
Biihler
Datta and Singh
Ojha
Date: fifth to eighth century CE.
Source: inscriptions from Valabhi (a village in Marathi, capital of the Hindu and Buddhist
kingdom which, from 490 to 775, encompassed the present-day States of Gujarat and
Maharashtra).
Fig. 24.44. Numerals of the intermediary counting system of Valabhi
This is the common basis which would lead progressively on the one
hand to the formation of the southern Indian style (attached to Dravidian
383
INDIAN NUMERICAL NOTATION
1234567 8 90
Ref.
' * s i j
Clin, III
Biihler
T & If
Datta and Singh
r
Ojha
Date: fifth to seventh century CE.
Source: inscriptions of the oldest branch of the Chalukya dynasty of Deccan (known as
"de Vatapi”, who lived in Badami, in the present-day district of Bijapur, during the sixth
century CE).
Fig. 24.45. Numerals of the intermediary counting system of the Chalukya of Deccan
1234567 8 90
Ref.
3 rtf Gf T %
**
Clin, III
Biihler
Datta and Singh
Ojha
Date: sixth to eighth century CE.
Source: inscriptions of the Ganga dynasty of Mysore (who ruled over a substantial part of
the present-day State of Karnataka, from the fifth to the sixteenth century).
Fig. 24.46. Numerals of the intermediary counting system of the Ganga of Mysore
styles of writing), and on the other hand to the development of Pali styles,
attached to eastern styles of writing.
Southern (or Dravidian) styles
From one of these systems was derived Bhattiprolu writing.
In Telingana, to the east of Andhra Pradesh and the south of Orissa, this
gradually became Telugu (Fig. 24.20):
In the centre of Deccan, in Karnataka and Andhra Pradesh, it became
Kannara (Fig. 24.21).
1 2 3 4 5
6
7
8 9 0
Ref.
~ 1 2. vi
“I
2
1 J «
Burnell
Renou and Filliozat
Date: eleventh century (Fig. 24.20).
Fig. 24.47. Ancient Telugu numerals
1 2 3
4 5 6 7 8 9 0
Ref.
A ? **)
2 <L l—s'r- O
Burnell
Renou and Filliozat
Date: sixteenth century CE (Fig. 24.21).
Fig. 24.48. Ancient Kannara numerals
To the east of the more southern regions, this became Grantha and Tamil
(Fig. 24.17), as well as Vatteluttu (used primarily on the coast of Malabar
from the eighth to the sixteenth century), whilst in the west this became the
styles known as Tulu and Malayalam (Fig. 24.19).
1234567890
Ref.
r* 2. (n & 6 1 9 V
O. <1 ® ’D 3 erf
Burnell
Pihan
Renou and Filliozat
Date: sixteenth century (Fig. 24.19).
Fig. 24.49. Ancient Tamil numerals
Finally, in the extreme south, primarily in Sri Lanka, Sinhalese was derived
(Fig. 24.22).
The styles of writing of Southeast Asia
At the same time, to the east of India, another variety of intermediate sys-
tems developed to lead to the first forms of Pali. Attached to the ancient
writing Ardha-Magadhi (the ancient language spoken in Magadha), these
diversified, and led to the characteristic forms of writing used (and still
used today) to the east of India and in Southeast Asia.
From this system was derived: Old Khmer (developed some time after
the beginning of the sixth century CE); Cham, used in part of Vietnam,
from the seventh century to some time around the thirteenth century; Kawi
in Java, Bali and Borneo (Fig. 24.50), which dates back to the end of the sev-
enth century, but which has now fallen into disuse; modern Thai writing
(Shan, Siamese, Laotian, etc.), whose first developments date back to the
thirteenth century (Fig. 24.24); Burmese (Fig. 24.24 and 24.51), which
derived from Mon in the eleventh century, used by populations of Pegu
before the Burmese invasion; Old Malay (Fig. 24.51), from which Batak (in
the central region of the island of Sumatra), Redjang and Lampong (in the
southeast of the same island), Tagala and Bisaya in the Philippines, as well
as Macassar and Bugi (from Sulawezi) derived.
INDIAN CIVILISATION
384
1
2
3
4
5
6
7
8
9
0
Ref.
S'
3
1
T
T
c.
&
V-
zz
O
Burnell
Damais
0
3
s
5
'b
V
tL
0
Renou and Filliozat
$
J
S
s
K
V
t
0
G\
3
X
K
r
t
3
5
h
V fc
0
Date: seventh to tenth century CE.
Fig. 24.50. Kawi (ancient Javanese and Balinese) numerals
1234567890
Ref.
? j { 5 9 © ? 0 e 0
j 1 % <Z
Latter
Smith and Karpinski
Date: seventeenth century CE (approx.).
Fig. 24.51. Ancient Burmese numerals
Types of numerals that derive from Brahmi
These fall into three categories, like the types of writing of the same names
(Fig. 24.52 and 24.53):
I. The family of writing styles from Northern and Central India and
Central Asia, which developed from Gupta writing:
1. Forms of writing derived from Nagari:
a. Maharashtri numerals;
b. Marathi, Modi, Rajasthani, Marwari and Mahajani (derived
from Maharashtri) numerals;
c. Kutila numerals;
d. Bengali, Oriya, Gujarati, Kaithi, Maithili and Manipuri (derived
from Kutila) numerals.
2. Forms of writing derived from Sharada:
a. Takari and Dogri numerals;
b. Chameali, Mandeali, Kului, Sirmauri and Jaunsari numerals;
c. Sindhi, Khudawadi, Gurumukhi, Punjabi (etc.) numerals;
d. Kochi, Landa, Multani (etc.) numerals.
3. Types of Nepalese writing:
a. Siddham numerals (influenced by the Nagari style);
b. Modern Nepali numerals (derived from Siddham numerals).
Fig. 24.52. Numerals which derived from Brahmi numerals
385
INDIAN NUMERICAL NOTATION
4. Types of Tibetan writing:
a. Tibetan numerals (derived from Siddham numerals);
b. Mongolian numerals (derived from Tibetan numerals).
5. Types of writing from Chinese Turkestan (derived from Siddham
numerals).
Fig. 24.53. Geographical areas where writing styles of Indian origin are used
II. The family of writing styles from southern India, which developed from
Bhattiprolu, a distant cousin of Gupta:
1. Telugu and Kannara numerals;
2. Grantha, Tamil and Vatteluttu numerals;
3. Tulu and Malayalam numerals;
4. Sinhalese (Singhalese) numerals.
III. The family of “oriental” writing styles, which developed from Pali writ-
ing, which itself derives from the same source as Bhattiprolu:
1. Old Khmer numerals;
2. Cham numerals;
3. Old Malay numerals;
4. Kawi (old Javanese and Balinese) numerals;
5. Modern Thai-Khmer (Shan, Lao and Siamese) numerals;
6. Burmese numerals;
7. Balinese, Buginese, Tagalog, Bisaya and Batak numerals.
As we will see later, Arabic numerals. East and North African alike, derive
from the Indian Nagari numerals, and the numerals that we use today come
from the Ghubar numerals of the Maghreb. Thus these diverse signs can be
placed in the first category of group I.
The mystery of the origin of Brahmi numerals
Having demonstrated how the above types of numerals all derived from
Brahmi, it is now time to explain the origin of the Brahmi numerals themselves.
For some time now, this writing has conserved an ideographical repre-
sentation for the first three units: the corresponding number of horizontal
lines. However, since their emergence, the numerals 4 to 9 have offered no
visual key to the numbers that they represent (for example, the 9 was not
formed by nine lines, nine dots or nine identical signs; rather, it was repre-
sented by a conventional graphic). This is a significant characteristic which
has yet to be satisfactorily explained. To try and understand this enigma,
let us now examine the principle hypotheses that have been put forward on
this subject over the last century.
First hypothesis: The numerals originated in the Indus Valley
S. Langdon (1925) believed that Indian styles of writing and numerals
derived from the Indus Valley culture (2500 - 1800 BCE).
The first objection to this theory concerns the claim that there is a link
between Indian letters and Proto-Indian pictographic writing.
We have just seen that Brahmi writing actually developed from the
ancient alphabets of the western Semitic world through the intermediate of
a variety of Aramaean: this link has been satisfactorily established, even if
samples of this intermediate writing have not yet been found (Fig. 24.28).
Documents from this civilisation are separated from the first texts written
in Brahmi and in a purely Indian language by the space of two thousand
years. However, the fact that the writing of Indus civilisation has not yet
been deciphered does not concern us here.
It is not known whether the ancient civilisations of Mohenjo-daro and
Harappa still existed when India was invaded by the Aryans, or if their writ-
ing had developed during this interval.
Moreover, no mention is made of this link in Indian literature, and with
good cause: the invaders probably found writing repugnant because, like
all Indian European peoples, they attached great importance to oral tradi-
tion [see T. V. Gamkrelidze and V. V. Ivanov (1987); A. Martinet (1986)].
INDIAN CIVILISATION
386
It is almost certain that when the Aryans arrived in India they brought no
form of writing with them, as happened in Greece and the rest of Europe,
whilst various Indo-European peoples came in successive waves to conquer
the continent. Their intellectual and spiritual leaders would certainly have
had “a knowledge of the great religious poems ; but it seems that their litera-
ture was written at a later date, and then the literate men would doubtless
have preferred to keep the oral tradition going as long as possible to perpet-
uate their prestige and their privileges” (M. Cohen (1958)].
Therefore, Langdon’s hypothesis has no foundation, because we do not
know if any link exists between numerals used by the Indus civilisation and
“official” Indian numerals. The theory becomes even more unlikely when
one considers that the documentation which survived from the Indus
Valley is very fragmentary and does not provide enough information for us
to reconstruct the system as a whole.
Second hypothesis: Brahmi numerals derive from “Aramaean-Indian" numerals
Since Indian letters derive from the Aramaean alphabet, would it not be
natural to presume that Brahmi numerals were similarly the offshoot of
one of the ancient systems of numerical notation of the western Semitic
world? At first glance, this hypothesis seems plausible, since a numerical
notation which derives directly from Aramaean, Palmyrenean,
Nabataean, etc., can be found in many inscriptions from Punjab and
Gandhara. This style is known as “Aramaean-Indian”, and is related to
Karoshthi writing (see Fig. 24.54 and Fig. 18.1 to 12).
However, once we have looked at Brahmi numerals for numbers higher
than the first nine units, we will see that this system is too different from
Aramaean-Indian for this hypothesis to be taken into consideration.
On the one hand, Aramaean-Indian reads from right to left, whilst
Brahmi (and nearly all the styles of writing related to it) reads from left
to right.
On the other hand, in the Karoshthi system, the numbers 4 to 9 are gen-
erally represented by the appropriate number of vertical lines, whilst the
Brahmi system gives them independent signs which give no direct visual
indication as to their meaning.
Moreover, the original Brahmi system possesses specific numerals for
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, etc., whilst the Karoshthi system
only has specific figures for 1, 10, 20 and 100.
Finally, the initial Indian system is essentially based on the principle of
addition, whilst Aramaean-Indian is based on a hybrid principle combin-
ing addition and multiplication.
Thus this hypothesis must be discarded.
Edicts
of Asoka
Karoshthi inscriptions from the Shaka
and Kushana dynasties
1
I
/
30
2
H
11
40
33
3
m
50
?33
X
4
im
60
5
urn
IX
70
6
IIX
80
333?
7
MIX
100
V
8
122
itffl
9
200
u
10
?
274
*1 3?yf»
20
3
300
/">
Date: third century BCE to fourth century CE.
Sources: inscriptions written in Karoshthi from the edicts of Asoka (3rd c. BCE), where
the numerals are partially attested; and Karoshthi inscriptions (2nd c. BCE - 4th c. CE)
from the north of Punjab and the former province of Gandhara (region in the north-west
of India, the extreme north of Pakistan and the northeast of Afghanistan, which was part
of the Persian Empire, before it was conquered in 326 BCE by Alexander the Great), where
these numerals are more fully attested.
Fig. 24.54. "Aramaean-Indian” numerical notations
Third hypothesis: Brahmi numerals derive from the Karoshthi alphabet
Another hypothesis (suggested by Cunningham, and later shared by
Bayley and Taylor), proposes that Brahmi numerals derived from the
letters of the Brahmi alphabet, used as the initials of the Sanskrit names
of the corresponding numbers. The following table demonstrates
this theory:
387
INDIAN NUMERICAL NOTATION
Forms given to
Karoshthi letters
by supporters
of this theory
Brahmi numerals:
forms found in
Asoka's edicts,
and the
inscriptions
of Nana Ghat
and Nasik
Names of
numbers
in Sanskrit
Karoshthi letters:
forms found in
Asoka’s edicts
cha
if *
4
+ * ¥
4
chatur
*
cha
pa
h
5
H h
5
pancha
h
pa
sha
f
6
4 > f
6
shat
T
sha
sa
7 1
7
7 /? J
7
sapta
n
sa
na
1 ^
9
i 'I
9
nava
1
na
Fig. 24.55.
The link that has been established here, however, is too tenuous, for at least
three reasons.
The first is that the forms given by the supporters of this theory actually
come from inscriptions from different eras, most often from later eras,
therefore holding little significance for the problem in question, which con-
cerns a graphical system. This is how such evolved forms like those of
Kushana inscriptions in the northwest of Punjab (second to fourth century
CE) came to be confused with more ancient styles such as inscriptions from
the Shaka era (second century BCE to first century CE) or those from
Asoka’s time (third century BCE).
The second reason is that the signs which are given for the presumed
phonetic values are very similar (if not identical) to letters which are known
to represent other numerical values.
Thirdly, the supporters of this hypothesis allowed themselves to get so
carried away that they themselves actually added the final touch to the
Aramaean-Indian letters which was needed in order to prove their theory.
There is another even more fundamental reason, however, why the
above two theories must be rejected: they presume that Karoshthi pre-dates
Brahmi, whilst today’s specialists believe precisely the opposite.
It is certain that Karoshthi writing derives from the Aramaean alpha-
bet, because several of its characters are identical (in form and
structure) to their Aramaean equivalents; and, like Semitic writing, it
reads from right to left. Karoshthi remains very different from the latter
style of writing, however, because it was adapted to the sounds and
inflexions of Indian-European languages. It was probably introduced to
the northwest of India in Alexander the Great’s time (c. 326 BCE), and
was used there until the fourth century CE, and until a slightly later date
in Central Asia.
Nevertheless, Brahmi does not derive from this writing. Brahmi stems
from another variant of Aramaean, whose characters were adapted to
Indian languages whilst the direction of the writing was changed so that it
read from left to right.
It is highly probable that Brahmi had been around long before Asoka’s
time, because by then it was not only fully established, but also and above
all it was in use in all of the Indian sub-continent. Therefore, it is very likely
that Karoshthi was not used in other parts of India except for the regions of
Gandhara and Punjab because, as J. G. Fevrier (1959) has already pointed
out, it emerged when an Indian style of writing already existed, namely
Brahmi, which was in use since roughly the fifth century BCE.
Thus it would seem unlikely that Karoshthi could have influenced the
formation of Brahmi letters and numerals.
Fourth hypothesis: Brahmi numerals derive from the Brahmi alphabet
This hypothesis would initially seem quite feasible when one considers that
many kinds of numerals have developed in this way.
The Greeks and the southern Arabic people, for example, gave, as a
numerical symbol, the initial letter from their respective alphabets of the
name of the number.
We also know that the Assyro-Babylonians, who had no numeral for 100
in their Sumerian cuneiform system, decided to use acrophonics; thus, the
syllable me was used to denote this number, the initial of the word meat,
which means “hundred” in Akkadian.
Ethiopian numerals, which now appear to be completely independent
from Ethiopian writing, actually derive from the first nineteen letters of the
Greek numeral alphabet; this dates back to the fourth century CE, when the
town of Aksum (not far from the modern town of Adoua) was the capital of
the ancient kingdom of Abyssinia.
Thus the theory that the numerals of a given civilisation derive from its
own alphabet is quite feasible.
In other words, the Indians could quite possibly have used a certain
number of the letters of the Brahmi alphabet to create a corresponding
numbering system. This is the substance of J. Prinsep’s hypothesis
(1838); he believed that the prototypes of the Indian numerals consti-
tuted the initial, in Brahmi characters, of the Sanskrit names for the
corresponding numbers.
INDIAN CIVILISATION
388
However, as this hypothesis has never been confirmed, it remains in the
realm of conjecture. Moreover, the author also mixed archaic styles with
later ones, and “customised" the characters in question to make his theory
appear to hold water.
Fifth hypothesis: Brdhmi numerals derive from a previous numeral alphabet
B. Indraji (1876) put forward the theory that Brdhmi numerals derived
from an alphabetical numeral system that was in use in India before
Asoka’s time.*
If we compare the shapes of the numerals with the letters that appear in
Asoka’s Brdhmi inscriptions of Nana Ghat and Nasik, we can see that there
are quite obvious similarities. The numeral for the number 4, a kind of “+” in
Asoka’s edicts, is identical to the sign used to write the syllable ka. Likewise,
the 6 is very similar to the syllable (Fig. 24.29). The 7 resembles the syllable
kha, whilst the 5 has the same appearance as the ia, ha, etc. (Fig. 24.30).
However, this link which has been established between the original
Indian numerals and the letters of the Brdhmi alphabet is not convincing.
First, the Brdhmi numerals for 1, 2 and 3 do not resemble any letter:
they are formed respectively by one, two and three horizontal lines (Fig.
24.29 to 35). Moreover, no phonetic value was assigned to the ancient
forms of the Brdhmi numeral which represented multiples of 10 (see Fig.
24.70). Even where this relationship has been established, there is too
much variation in the attribution of phonetic values to the signs in ques-
tion. Thus, whilst the numeral 4 has been connected to the syllable ka, in
its diverse forms it can equally be said to resemble the letter pka or the syl-
lables pna, Ika, tka or pkr. Similarly, the shape of the numeral representing
the number 5 resembles the syllable tr as well as the following: ta, ta, pu, hu,
ru, tr, tra, na, hr, hra or ha [see B. Datta and A. N. Singh (1938), p. 34].
In other words, if such a system did exist in Asoka’s time, it is impossi-
ble to discover the principle by which it might have functioned.
* Along with the various styles of numerals, the Indians have long known and used a system for represent-
ing numbers which involves vocalised consonants of the Indian alphabet which, in regular order, each have
numerical value. These are known as *Vamasankhyd in Sanskrit, the system of “letter-numbers”. The
system varies according to the era and region but always follows the method of attributing numerical values
to Indian letters, sometimes even following the principle used in numerical representations (the place-value
system or the principle of addition). Included in the numerous systems of this kind is the one which the
famous astronomer Aryabhata (c. 510 CE) used to record his astronomical data; there is also the system
called Katapayadi used (amongst others) by the ninth- century astronomers Haridatta and
Shankaranarayana, as well as Aksharapalli frequently used in *Jaina manuscripts. These systems are still in
common use today in various regions of India, from Maharashtra, Bengal, Nepal and Orissa to Tamil Nadu,
Kerala and Karnataka. They are also used by the Sinhalese, the Burmese, the Khmer, the Siamese and the
Japanese, as well as the Tibetans, who often use their letters as numerical signs, mainly to number their reg-
isters and manuscripts. Details can be found under the entries *Vamasankhyd, * Aksharapalli, *Numeral
Alphabet, * Aryabhata and * Katapayadi numeration of the Dictionary.
Despite the shakiness of their explanations, the supporters of Indraji’s
theory conjectured that the idea of assigning numerical values to letters of
the alphabet dated back to the most ancient of times, their reasoning being
that “Indian, Hindu, Jaina and Buddhist traditions attribute the invention
of Brdhmi writing and its corresponding numeral system to Brahma, the
god of creation.”
(Of course, such an argument cannot be taken seriously, especially in the
case of Indian civilisation, where such traditions were actually only developed
relatively recently and are due to two basic traits of the Indian mentality.
First, there is the desire of some of these theorists to make such concepts hold
more water in the eyes of their readers, disciples or speakers, in attributing
their invention to Brahma. There are also those who, convinced of the innate
character of the Indian letters and numerals, do not even consider it neces-
sary to give a historical explanation for their existence. In the first instance,
the motive was to make these concepts sacred, and in the second, to make
them timeless. The latter conveys India’s fundamental psychological charac-
ter; an obsession with the past which always involves wiping out historical
events and replacing fact with religious history, which takes no account of
archaeology, palaeography or, most importantly, history.)
The pioneer of the above theory even went so far as to claim that the first
Indian numeral alphabet dates back to the eighth century BCE. According to
Indraji, it was Panini (c. 700 BCE) who first had the idea of using the conso-
nants and vowels of the Indian alphabet to represent numbers.
JT
ka
kha
g a
gha
ha
FT
3T
cha
chha
j a
jha
ha
7
7
I
5*
m
ta
tha
da
dha
na
FT
Q
K
tT
ta
tha
da
dha
na
cr
HT
*T
pa
pha
ba
bha
ma
<
FT
ya
ra
la
va
q-
sha
sha
sa
ha
Fig. 24.56. Consonants of the Devanagari (orNagari) alphabet
389
INDIAN NUMERICAL NOTATION
Panini is the famous grammarian of the Sanskrit alphabet: born in
Shalatula (near to Attack on the Indus, in the present-day Pakistan), he is con-
sidered to be the founder of Sanskrit language and Literature; his work, the
Ashtadhyayi (also known as Paniniyatri), remains the most famous work on
Sanskrit grammar [see L. Frederic (1987)]. We have no exact dates in Panini's
life, and there is much doubt surrounding the work which is attributed to him.
In other words, the date suggested by Indraji for the invention of the
first Indian alphabetical numeral system has no foundation at all, espe-
cially when one considers that there is no known written document, nor
specimen of true Indian writing, which dates so far back in Indian history.
It goes without saying, then, that this hypothesis must also be rejected.
The origin of Indian alphabetical numerals
So where does the idea of writing numbers using the Indian alphabet
come from?
It must be made clear straight away that the idea did not come from
Aramaean merchants, who brought their own writing system to India (Fig.
24.28). With a few later exceptions, the northwestern Semites never used
their letters for counting; their numerals were of the same kind as the
Karoshthi (or Aramaean-Indian) system (see Fig. 24.54 and Fig. 18. 1 to 12).
One could attribute the idea to a Greek influence, in the light of Alexander
the Great’s conquest of the Indus in 326 BCE, and moreover because this kind
of system was in use in Greece since the fourth century BCE. However, this
hypothesis is not plausible, because no Indian inscriptions, during or after
Asoka’s reign, show any evidence of alphabetical numeration.
In fact, the first numeral system of this kind was invented in the Indian
sub-continent by the famous mathematician and astronomer, ‘Aryabhata.
This system was undeniably unique compared to all the other previous and
contemporary systems; not only has it been quoted in numerous works and
commentaries over the centuries, but it has also inspired a considerable
diversity of authors, commentators and transcribers, in various eras, to
draw comparisons with it and both similar and very different systems (see
* Aryabhata and *Katapayadi in the Dictionary).
Sixth Hypothesis: Brahmi numerals came from Egypt
Here are some other hypotheses put forward as to the origin of Brahmi
numerals.
Biihler (1896) and A. C. Burnell (1878) believed that the Indians owed
their Brahmi writing to Pharaonic Egypt. Biihler claimed that it derived
from hieratic writing (see Fig. 23.10), and Burnell believed that Brahmi
writing derived from demotic writing.
Biihler’s theory is not totally unfounded, because there is a much
stronger similarity between Brahmi writing and Egyptian hieratic writing
than there is between the former and the demotic writing of the same civili-
sation. However, is this partial resemblance significant enough to suggest
that Egypt had such an influence on India’s distant past?
Arabia, the legendary land of “Pount”, was a staging post for Egyptian
trade. Ships sailed to the Red Sea along the eastern Nile delta and along a
canal first to the Bitter Lakes, then to the Gulf of Suez. It is possible that
these same merchant ships, in their quest for eastern goods, travelled fur-
ther than Arabia, not only to the areas around the Persian Gulf, but also as
far as the mouth of the Indus [see A. Aymard and J. Auboyer (1961)].
Conversely, during Alexander the Great’s time, India communicated
with the Caspian and the Black Sea by river navigation, notably along the
Amu Darya; overland routes also led from Europe to India through Bactria,
Gandhara and the Punjab, giving access to ports on the western coast of
India. Commercial relations became more and more firmly established
between Egypt and India, and ships even sailed as far as the coast of
Malabar, in particular to the port of Muziris (now the town of Canganore)
[see Aymard and Auboyer (1961)].
These relations, however, occurred at a relatively late time and do not
really prove anything in terms of the transmission of Egyptian hieratic
numerals: the amount of time which separates these numerals from their
Brahmi counterparts is too great to allow this hypothesis to be taken into
consideration.
(It must be remembered that hieratic numerals were almost obsolete in
Egypt by the eighth century BCE; therefore, if this system was transmitted
to India, the transmission cannot have taken place any later than this time.
As we possess no information about India at this time, this hypothesis
cannot be proved.)
Moreover, the above comparison only concerns units; there is a clear
difference between the other symbols (numerals representing 10 or above,
which have not yet been discussed: see Fig. 24.70). Therefore the compari-
son only concerns a small percentage of the numerals.
The origin of the first nine Indian numerals
There is another hypothesis which seems much more plausible, even in the
absence of any documentation.
Basically, we have already proved this hypothesis: different civilisations
have had the same needs, living under the same social, psychological, intel-
lectual and material conditions. Independently of one another, they have
followed identical paths to arrive at similar, if not identical results.
INDIAN CIVILISATION
390
This explains the existence of certain numerals of other civilisations
which resemble and often represent the same numbers as Brahmi numer-
als, and which generally date back several centuries before Asoka’s time.
On consulting Figs. 24.57 and 24.29 to 35, we can see signs which are
not Indian, yet which are very similar to the various ways of writing the
numbers 1, 2 and 3 in Indian civilisations. We can also see the evident simi-
larity between the Nabataean or Palmyrenean “5” and that of ancient India,
as well as the physical resemblance between the Egyptian hieratic or
demotic “7” and “9” and their Indian counterparts.
What these analogies actually prove is not the unlikely theory that the
first nine Indian numerals came from one of the other civilisations, but
rather that there are universal constants caused by the fundamental rules
of history and palaeography. These similarities occur because other civilisa-
tions used similar writing materials to those of ancient India, for example
the calamus (a type of reed whose blunted end was dipped in a coloured
substance), and which was used by Egyptian and western Semitic scribes
(Aramaeans, Nabataeans, Palmyreneans, etc.) to write on papyrus or
parchment, which was long used on tree bark or palm leaf in Bengal, Nepal,
the Himalayas and in all of the north and northwest of India.
We know to what extent the nature of the instrument influenced, on the
one hand, Egyptian manuscript writing, and on the other hand, the writ-
ings of the ancient Semitic world.
In the first case, the use of the calamus turned the hieroglyphics of mon-
umental Egyptian writing into cursive hieratic signs, changing the detailed,
pictorial symbols into a shortened, more simplified form, perfectly adapted
to the needs of manuscript writing and rapid notation.
In the latter case, the same writing apparatus was used to transform the
rigid and angular shape of Phoenician writing into the rounder, more cur-
sive and fluid forms, like that of Elephantine Aramaean scribes.
Thus the superposition of two or three horizontal lines, first trans-
formed into one complete sign by a ligature, gave birth to the same forms
as the Indian numerals for 2 and 3, whose palaeographical styles vary
considerably according to the era, the region and the habits of the scribe
(Fig. 24.58).
This explanation relies on the assumption that horizontal lines formed
the first three of the ancient ideographical Indian numerals, and this is
what Brahmi inscriptions written after the third century BCE (Shunga,
Shaka, Kushana, Mathura, Kshatrapa, etc.) would suggest (Fig. 24.30 to
38). This figurative representation was still in use during the time of the
Gupta Dynasty (third to sixth century CE), and even persisted in some
areas until the eighth century.
5
6
7
8
9
x a
v
*3 k
T
i d
^ 1
< q
4*
^ m
V
yy
r
— X i
n
V
*1 J
V
- Egyptian numerals: a (HP 1, 618, Abusir); b (HP I, 618, Elephantine); c (HP II, 619,
Louvre 3226); d (HP II, Louvre 3226); e (HP II, 619, Gurob); f (HP 1, 620, IUahun); g (HP I,
620, Bulaq 18); h (HP II, 620, Louvre 3226); i (HP II, 620, P. Rollin); j (HP III, 620,
Takelothis); k (HP I, 621, Elephantine); 1 (HP 1, 621, Illahun); m (HP I, 621, Math); n (HP
I, 621, Ebers); o (HP III, 621, Takelothis); p (HP I, 622, Abusir); q (HP 1, 622, Illahun);
r (HP 1, 622, Illahun); s (HP 1, 622, Bulaq, Harris); t (HP II, 622, P. Rollin); u (HP II, 622,
Gurob); v (HP II, 622, Harris); w (DG, 697, Ptol.); Nabataean numerals: x (CIS, III, 212);
Palmyrenean numerals: y (CIS, 113, 3913).
Fig. 24.57. Numerals which have the same appearance and numerical value as their Brahmi equiva-
lents. [Ref. Moller (1911-12); Cantineau; Lidzbarski (1962)1
Fig. 24.58, Evolution of Indian numerals 2 and 3
However, if we examine Asoka’s edicts (c. 260 BCE), we can see that
throughout the Mauryan empire, the numbers 1, 2 and 3 were not repre-
sented by superposed horizontal lines, but by one, two or three vertical
lines (Fig. 24.29).
Why did this change of direction occur? And why did it happen
between the third and second century BCE, when the representations had
391
INDIAN NUMERICAL NOTATION
been horizontal since the time of the Buddhist inscriptions of Nana Ghat
(Fig. 24.30)?
The second question is difficult to answer, as no documentation has
been found from that time on this subject (if indeed anyone took the trou-
ble to write about something which must have seemed so insignificant).
However, this is of little importance; we are only interested here in how
such a change came about.
Could it have occurred due to aesthetic reasons? This is as unlikely as
the possibility that the new notation evolved for practical reasons. To draw
a line one, two or three times, whether vertically or horizontally, has no
aesthetic value, and involves practically the same amount of exertion,
unless it goes against what one is accustomed to writing.
On the other hand, this modification could have been due to the realisa-
tion that a vertical representation of the first three units was likely to be
confused with the danda. This is a punctuation mark in the form of a small
vertical line ( I ), which the Indians have long used in their Sanskrit texts to
mark the end of a line or of part of a sentence, which they double ( 1 1 ) to
indicate the end of a sentence, couplet or strophe.
The invention of the danda in the second century BCE could be respon-
sible for the change in direction of the lines representing the numbers 1 to
3. This is mere conjecture which for the moment remains without proof or
confirmation.
Here is another question: why did the Indians conserve these represen-
tations of the first three units for so long, when the numerals for 4 to 9 had
already graphically evolved into independent numerals, which offered no
visual clue as to the numbers they represented (Fig. 24.29 to 38)? This is
not only true of the Indians: many other civilisations have offered us simi-
lar puzzles over the ages, notably those of China and Egypt.
The explanation lies in a basic human psychological trait, which was dis-
cussed in Chapter 1. Whilst it was necessary to have other signs than four
or five to nine lines for the numbers 4 or 5 to 9, it was not judged necessary
to change the signs for units which were lower or equal to 4; this was not
only because these symbols could be drawn quickly and easily, but also and
above all because without needing to count, the eye can easily distinguish
between a number of lines when they number four or less. Four is the limit,
beyond which the human mind has to begin to count in order to determine
the exact quantity of a given number of elements.
So what was the reasoning behind the formation of the other six Brahmi
numerals? Are they purely conventional signs, created artificially to supply
a need? Probably not. Taking the universal laws of palaeography into
account, and the evidence surrounding the formation of similar numerals
in other cultures, it is more likely that the numerals were born out of proto-
types formed by the primitive grouping of a number of lines representing
the value of the unit.
In other words, to all appearances, the Brahmi numerals of Asoka’s
edicts were to their ideographical prototypes as Egyptian hieratic numerals
were to their corresponding hieroglyphic numerals.
As the lines representing the numbers 1 to 3 were vertical before they
were horizontal, one could reasonably presume that the first nine Brahmi
numerals constituted the vestiges of an old indigenous numerical notation,
where the nine numerals were represented by the corresponding number of
vertical lines; a notation, doubtless older than Brahmi itself’, where, like
the Egyptian hieroglyphic system, the Cretan or the Hittite system for
example, the vertical lines were set out as in Figure 24.59.
I II
1 2
II III
II II
III till
III III
8 9
Fig. 24 . 59 . A plausible reconstruction of the original Indian ideographical notation : the starting
point of the evolution which led to the Brahmi numerals for 4 to 9 (those for 1 to 3 retaining their
ideographical form for many centuries, although represented horizontally rather than vertically)
To enable the numerals to be written rapidly, in order to save time, these
groups of lines evolved in much the same manner as those of old Egyptian
Pharaonic numerals. Taking into account the kind of material that was
written on in India over the centuries (tree bark or palm leaves) and the
limitations of the tools used for writing (calamus or brush), the shape of
the numerals became more and more complicated with the numerous liga-
tures (Fig. 24.60), until the numerals no longer bore any resemblance to the
original prototypes. Thus a primitive numbering system became one of
numerals of distinct forms which gave no visual indication as to their
numerical value: the Brahmi numerals of the first three centuries BCE.
Taking into consideration the universal constants of both psychology and
palaeography, this is currently the most plausible explanation of the origin
of the nine Indian numerals. The summary at the end of this chapter
demonstrates the likely stages of the development of Brahmi numerals.
Therefore, it appears that Brahmi numerals were autochthonous, that is to
say, their formation was not due to any outside influence. In all probability,
they were created in India, and were the product of Indian civilisation alone.
In other words, one could say that the problem of the origin of our present-
day numerals has been satisfactorily solved. This is also demonstrated in
‘This is not at all surprising if one considers that, on the one hand, the ancient civilisation of the Indus
which preceded the Aryans used exactly this type of notation (Fig. 1.14), and that on the other hand,
Sumerian civilisation developed a numeral system even before creating its own writing system.
INDIAN CIVILISATION
the palaeographical tables of Fig. 24.61 to 69, which constitute the complete
historical synthesis of the question, and which have been set out taking into
account all the details proved both previously and subsequendy.
Fig. 24.60. Results of the graphical evolution of the signs which were originally formed by the juxta-
position or superposition of several vertical lines, these lines being drawn on a smooth surface and
written with a calamus with a blunt tip dipped into a coloured substance. This evolution took place
among the Egyptians.
392
r Cb Kannara
" 0 — --o — p —
Gupta, Pallava \
_ , Sharada \i .
Telugu o—*~o
Nepali, Bengali
Fig. 24.61. Origin and evolution of the numeral 1. (For Arabic and European numerals, see
Chapters 25 and 26.)
393
Gu P ta Pallava
■ — > <i/
.cvjoq:
, Cham
"(ffl Thai
Western Arabic
^ (Ghubar)
Bengali
"'r*T*5r
European
(apices)
Oriya
^ ^ ^ -7—7 —1 Telugu
Gupta, Pallava, Valabhi, Nepali \
_ 5> -► — * -*• — * Telugu, Kanr
/Nagari, Gujarati, Sindhi Sharada, Nepali,
\ / JaV3neSe
* - ^ ^Tibetan, Nepali^ « 03 **V3 Khmer
Gupta '
Nagari
\\
\ ?f-f ?n in
\ Western Arabic Cursive European
\ (Ghubar) (algorisms)
Western Arabic 2 - ^
(Ghubar) Cursive European
(algorisms)
Fig. 24.62. Origin and evolution of the numeral 2. (For Arabic and European numerals, see
Chapters 25 and 26.)
INDIAN NUMERICAL NOTATION
.m Charters on copper (various Indian styles)
^3-3-, J-oS-G 3
t 1 \ Cham, Khmer Kawi
' Valabhi N epali
Sharada Pali
Bengali
Tibetan Pali
?? ■? “
Arabic (Ghubar) European cursive (algorisms)
3 -cs'r 1 r— 0
' Eastern Arabic Eastern
( Arabic
Western Arabic ^ 2^ 2 •)!
Nagari \ (Ghubar)
\ European
\ (apices)
5 ? } — -0
Western Arabic (Ghubar) Modern
and European cursive
(algorisms)
Fig. 24.63. Origin and evolution of the numeral 3. (For Arabic and European numerals, see
Chapters 25 and 26.)
INDIAN CIVILISATION
\S-«
t Valabhi Khmer Khmer
\/ Gujarati,
, 0 Marathi
Sindhi
/
j( 5>* jf)
Qk„U»
Brahmi Shunga Shaka, . ,
v ,« Kshatrapa,\
Kushana, r
j ,, Pallava
Andhra,
c.
Eastern
Eastern Arabic
Arabic /,
r 6 /
Western Arabic
(Ghubar)
. f 3|/R
fcfio***
European
(apices)
/*-,**? 4-0
Cursive European modern
(algorisms)
Fig. 24.64. Origin and evolution of the numeral 4. (For Arabic and European numerals, see
Chapters 25 and 26.)
394
3 X5 *13 & & 5 5 ^5 %
Khmer Khmer Khmer
t
C" Valabhi
Gupta^ IF ■* Tr>-~Tj^h — *2 ~*“J\ ~*\>\ ~*~Sl
j various Indian l 1 1 ' v
’ manuscripts Ne P Sli various Indian Pah
^ g _ p manuscripts
_ ^^*Cham Cham \ Cham
? < t' \ ^ ^ J* Balinese
Nepali 'sKhmer Sindhi Marathi /
x e. u
Nagari j 1 V (/
£j H (} Zl — *• 6 & *9 $ — 0 £
^ u , various Indian manuscripts, jft 4 4) o S— H
Y Sharada
Nagari, Punjabi,
Gujarati, Sharada
charters on copper,
inscriptions from Gwalior
> H f
H t 1 t
Western Arabic
(Ghubar)
•0 b
European (apices)
3 V <j1
ft h h 3
European (apices)
(algorisms)
Fig. 24.65. Origin and evolution of the numeral 5 . (For Arabic and European numerals, see
Chapters 25 and 26)
395
INDIAN NUMERICAL NOTATION
Nepali, Valabhi
/j) ^ Khmer Javanese, Balinese
^Charn^ -~ q ) -q)
various Indian manuscripts
Western Arabic
(Ghubar)
Cham
Sharada
Khmer
^ ^ Nagari, Punjabi
Marathi, Sindhi
£ Tibetan, Mongolian
Gujarati
9 M t
^ ^ ^ Eastem
Eastern Arabic Arabic
b Jb lb P European
Jr ef la If P> 1> (apices)
— -e 5 trtrir-H
Cursive European
(algorisms)
modern
Fig. 24.66. Origin and evolution of the numeral 6. (For Arabic and European numerals, see
Chapters 25 and 26.)
Shunga,
Shaka,
Kushana,
Andhra,
Gupta
T
q
7
i
f\
Addition of a sign above the numeral
to avoid any confusion with the
corresponding numeral "1”
* 4 i
A *0 ^
Cham Cham Khmer
various Indian manuscripts
Nepali
various Indian manuscripts
and charters on copper
A
1
1
Pali
Pallava
Bengali
1
3
3
3 ^ \f *)
Kshatrapa,
Tibetan,
Pali, Nagari, Gujarati, Marathi, Oriya
Valabhi
? ? ? 0 J
A 7 7 ^ c*
7/ 4 7 l*
Western Arabic (Ghubar)
1
*iyi^ a/^in^
European (apices)
U \j <3
\ 1 V V
v y y
Eastern Arabic
[Vj
Eastern
Arabic
i
''-'- 1777 - 0
Cursive European modern
(algorisms)
Fig. 24.67. Origin and evolution of the numeral 7. (For Arabic and European numerals, see
Chapters 25 and 26.)
INDIAN CIVILISATION
-6— 3S-3
Nepali Sharada
inscriptions
jr J J from Gwalior
Valabhi / /
/ r ~?\
/ Pali Balinese '
y — - T v 0ri y a
Kshatrapa
I"
Nagari, Pali
Cham
Khmer Kawi
V— v-^
Khmer
V Khmer
v-
t; t; cr
Gupta
T-
Nagari Pali Punjabi
AT
Telegu,
Kannara
Kshatrapa various Indian Telegu,
manuscripts Kannara
5 * ^ % C 9^
Western Arabic T
(Ghubar) g 8 $ 5 S
European
(apices)
-7 A. ^
TO< A“»
Eastern
Arabic
Cursive European
(algorisms)
Fig. 24.68. Origin and evolution of the numeral 8. (For Arabic and European numerals, see
Chapters 25 and 26.)
396
" ^ -*■ ^ -* *■ ^ “*■ j* Mon 8 olian
Kushana, Kshatrapa ^ ^ ^ ^ ^
Nagari, Marathi p ^
Ni “"
- — y — ^ — *- |V( Gujarati Punjabi
Kushana \ Oriya
Gupta, Khmer fQ fiU
^ *^hmer
/ Gupta, \ Cham ^ rf) O'
Nagari C/e)
Andhra, /
Gupta ( 3 ) ^^Nagari
Nagari O / »■ (jj ♦ ^ ^
various Indian manuscripts, charters^"]
on copper and inscriptions from Gwalior Nagari Sharada
93 tlj
Western Arabic (Ghubar)
3 VTf 1 v|T
Eastern Arabic Eastern
1 9 ? f 2 ‘S 3 i_3 C* 9
’6£G<fb*£>Ti> £&* Cursive £uropean modern
European (apices) (algorisms)
Fig. 24.69. Origin and evolution of the numeral 9. (For Arabic and European numerals, see
Chapters 25 and 26.)
397
OLD INDIAN NUMERALS! A VERY BASIC SYSTEM
OLD INDIAN NUMERALS: A VERY BASIC SYSTEM
As the preceding diagrams have shown, Indian numerals, even in their
earliest forms, were the forerunners of the nine basic numerals of our
present-day system. In other words, it was from these signs that, some
centuries later, the numerals that we wrongly call “Arabic” were derived.
As we shall see later, modern numerals are the descendants of North
African numerals, which themselves are cousins of the eastern Arabic
numerals, which in turn are linked to Nagari numerals, which belong to
the family of decimal numeral systems currently in use in India and South-
east and central Asia.
From a graphical point of view, then, the first nine Brahmi numerals
shared one of the fundamental characteristics of our present-day numerals.
This, however, was the only aspect which they originally had in common.
If we examine Brahmi inscriptions or intermediate Indian inscriptions,
from Asoka’s edicts to those of the Shungas, Shakas, Kushanas, Andhras,
Kshatrapas, Guptas, Pallavas or even the Chalukyas, that is to say from the
third century BCE to the sixth and seventh centuries CE, we can see that the
corresponding principle of numerical notation is very rudimentary.
For a decimal base, this system relied largely upon the principle of addi-
tion, attributing a specific sign to each of the following numbers (Fig. 24.70):
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
200
300
400
500
600
700
800
900
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
This written numeral system had special numerals, not only for each basic
unit, but also for every ten, hundred, thousand and ten thousand units. To
represent a number such as 24,400, one needed only to juxtapose, in this
order, the numerals 20,000, 4,000 and 400 (Fig. 24.71):
Vf m
20,000 4,000 400
Of course, if these numerals had belonged to a place-value system, the
number in question would have been written in the following way, using
the style of numerals in use at that time, the zero being represented by a
little circle, as it appears in later Indian inscriptions:
2 4 4 0 0
UNITS
1
2
3
4
5
6
7
8
9
Third century BCE:
Brahmi of Asoka
1
II
+
Second century BCE:
Inscriptions ofNana
Ghat
—
—
**
?
First or second
century CE:
Inscriptions ofNasik
-
—
=
f *
b
7
First to second
century CE: Kushana
inscriptions
—
=
**
P" Y
/> fn
Ip £
t
n n
?
*7 ‘I
? T
T
First to third century
CE: Andhra, Mathura
and Kshatrapa
inscriptions
—
=
r:
w>
bhb
rr
\ i
V
*73
? $
I'f
<5
Fourth to sixth cen-
tury CE: Gupta
inscriptions
a 2
y*
F h
£ ft
9
h
t'T;
3
Sixth to ninth century
CE: Inscriptions of
Nepal
-
=\
'b
Fifth to sixth century
CE: Pallava
inscriptions
n 0
n
9 n
on
m
m
m
0
*
V
<=>
1
Sixth to seventh cen-
tury CE: Valabhi
inscriptions
d
$
G)
Various Indian manu-
scripts
?
1
‘i
&
V
1 b,
3
s <■
T3
0)9
Fig. 24.70A Numerical notation linked to Brahmi writing and its immediate derivatives. There is evi-
dence of the signs formed by straight lines ; the others are reconstructions based on a comparative
study of forms. (For references, see Fig. 24. 29 to 38 and 24.41 to 46.)
INDIAN CIVILISATION
398
TENS
20
30
40
50
60
70
80
90
Third century BCE:
Brahmi of Asoka
G 0
■
Second century BCE:
Inscriptions of Nana
Ghat
B
1
■
1
-)
1
<P
■
First or second century
CE: Inscriptions of
Nasik
cX
oc
0
%
■
First to second century
CE: Kushana
inscriptions
OC 0 L
OC
64$
•0 u
u
w?
X X
eec
vyy
B
0)<D
OO
First to third century
CE: Andhra, Mathura
and Kshatrapa
inscriptions
X <*
oc
O C
80 4
J u
**
K "H
'J 1
T >
» *
£
©
£
Fourth to sixth
century CE: Gupta
inscriptions
oC
CC
to
0
V
f
03
e
90
Sixth to ninth
century CE:
Inscriptions of Nepal
T
ct
W
6 G
Fifth to sixth century
CE: Pallava
inscriptions
1
Sixth to seventh
century CE: Valabhi
inscriptions
£
&
•J
F*
If
1/
w
OQ
Various Indian
manuscripts
e&
BQL
e
If
9/
V
Ed
O
/©i
*69"
Fig. 24.70B.
HUNDREDS, THOUSANDS AND TENS OF THOUSANDS
4,000
6,000
8,000
10,000
20,000
70.000
1
Second century BCE:
Inscriptions of Nana
Ghat
fTH
W=
Mf
First or second century
CE: Inscriptions of
Nasik
B
B
B
7f
B
B
B
fl
First to second century
CE: Kushana inscriptions
11
1
11
y
Tv
>
l*
First to third century CE:
Andhra, Mathura
and Kshatrapa
inscriptions
V
1
1
1
1
1
1
Fourth to sixth century
CE: Gupta inscriptions
1
T
H
Sixth to ninth century
CE: Inscriptions of
Nepal
5IH
si
B
1
1
1
1
Sixth to seventh century
CE: Valabhi
inscriptions
13
”3?
■vi
•>
9
1
fl
Various Indian
manuscripts
i
W
*
l
100+1x100
a votical line is added to"100”
T
§
a
1
s
8-1
*1
1 i
X
§
T
*
I
1,000+1x1,000
a vertical line is added to “1,000"
1,000+2x1,000
two vertical lines are added to “1,000"
T
■XT
X
t
CD
X
t
OO
X
t
2
X
1,000 x 20
t
B
X
Fig. 24.70c.
399
THE INDIAN PLACE-VALUE SYSTEM
Like certain other systems of the ancient world, this numeration was very
limited. Arithmetical operations, even simple addition, were virtually
impossible. Moreover, the highest numeral represented 90,000: therefore
such a system could not be used to record very high numbers.
This comment is significant because it constitutes a numerical “palin-
drome”: the number reads the same from left to right or from right to left,
which is only possible if we are dealing with the place-value system:
12345654321
srrFTTZ <;*
d a; gij,oc ; >\pi - *
>VIl» rtg F?tv V
j£MpiwdW©?-fc4'xvG /-
fT* t *» tr X ft f nl Tbc T
f Cruf Ty
- rd
Fig. 24.71. Detail of a Buddhist inscription in Brahmi characters adorning one of the walls of the
cave at Nana Ghat (second century BCE). The shaded section shows the Brahmi notation for the
number 24,400. [Ref Smith and Karpinski (1911), p. 24]
< >
It should be noted that these types of numbers possess unusual properties;
take the following, for example:
1 2 = 1
ll 2 = 121
111 2 = 12321
llll 2 = 1234321
lllll 2 = 123454321
llllll 2 = 12345654321
mill l 2 = 1234567654321
nilllll 2 = 123456787654321
lnmill 2 = 12345678987654321
THE PROBLEM OF THE DISCOVERY OF THE INDIAN
PLACE-VALUE SYSTEM
Thus the ancestors of our numerals remained static for a long time before
acquiring the dynamic and manageable character that they have today
thanks to the place-value system.
This leads us to ask two fundamental questions, which we will tackle
through an archaeological, epigraphic and philological examination of the
mathematical, astrological and astronomical texts of India: When and how
did the first nine numerals of this rudimentary system come to be governed by this
essential rule? And when was zero first used?
The first significant clues
Before we look at archaeology and epigraphy, it is worthwhile investigating
whether some clues about zero and the place-value system can be discov-
ered in Indian Sanskrit mathematical literature.
Here, for example, is an extract from the Ganitasarasamgraha (Chapter 1,
line 27) by the mathematician Mahaviracharya who, giving 12345654321 as
the result of a previous calculation, defines this number in the following way
[see B. Datta and A. N. Singh (1938)]:
ekadishadantani kramena hinani
which means the quantity “beginning with one [which then grows] until it
reaches six, then decreases in reverse order”.
These are properties that could not have been worked out using a non-positional
system, due to its inconsistencies and the rules that would have governed it.
In other words these types of numbers could only have been discov-
ered after the place-value system was invented. As we know that the
Ganitasarasamgraha is dated c. 850 CE, we can infer that the place-value
system was discovered before the middle of the ninth century.
Here is another piece of evidence which places the discovery of the place-value
system at an earlier date: the arithmetician Jinabhadra Gani, who lived at the end
of the sixth century, gave to the number 224,400,000,000 the following Sanskrit
expression, in his Brihatkshetrasamdsa 1, 69 (see Datta and Singh, p. 79):
dvi vimshati cha chatur chatvarimshati cha ashta shunyani
“twenty-two and forty-four and eight zeros” (=224400000000).
This proves that the Indians knew of zero and the place-value system in the
sixth century.
The preceding examples do not constitute “proof” in the strictest sense
of the word, but they show that the place-value system must have been in
use for some time if its subtleties were understood and appreciated by the
contemporary public.
Evidence found in Indian epigraphy
The first known Indian lapidary documents to bear witness to the use of
zero and the decimal place-value system actually only date back to the
second half of the ninth century CE.
INDIAN CIVILISATION
400
\
X
5
u,
<
4
T
Si
V
1
2
3
4
5
6
7
8
9
10
w
\t
n
\ir
K
vr
\<3\
11
12
13
14
15
16
17
18
19
20
1\
21
11
22
H
23
24
25
26 '
Ref.: AS1, Rep. 1903-1904, pi. 72; El, 1/1892. p. 155-162; Datta and Singh (1938); Guitel;
Smith and Karpinski (1911).
Fig. 24.72. Numerals from the first inscription of Gwalior
These are two stone inscriptions from Bhojadeva’s reign, discovered in
the nineteenth century in the temple of Vaillabhattasvamin, dedicated to
Vishnu, near the town of Gwalior (capital of the ancient princely state of
Madhyabharat, situated approximately 120 kilometres from Agra and a
little over 300 kilometres south of Delhi).
The first inscription is quite clearly dated 932 in the Vikrama calendar
(932 - 57 = 875 CE, see *Vikrama, Dictionary). It is in Sanskrit, and con-
sists of twenty-six stanzas, which are numbered in the following manner
using Ndgari numerals (the signs for the numbers 1, 2, 3, 7, 9 and 0 already
strongly resemble their modern equivalents) (Fig. 24.72).
The second inscription is dated (in numerals) the year 933 in the
Vikrama calendar ( = 876 CE). Written in Sanskrit prose, it gives an account
of the offerings the inhabitants of Gwalior made to Vishnu. It tells mainly
of the offering of a piece of land 270 x 187 hasta, which was to be turned
into a flower garden, and of fifty garlands of seasonal flowers which the gar-
deners of Gwalior were to bring to the temple as a daily offering. The
number denoting the date (933), as well as the three other numbers men-
tioned, are represented by Ndgari numerals as they appear in Fig. 24.74.
There is no question as to the authenticity of these two inscriptions, and
they clearly demonstrate the extent to which the inhabitants of the region
were familiar with zero and the place-value system during the second half
of the ninth century.
The inscriptions from Gwalior are not the oldest documents to contain
evidence of the use of this system. Of the numerous other examples, of
which there follows a list in ascending chronological order, there are docu-
ments engraved on copper which come from diverse regions of central and
western India and date back to the era between the end of the sixth century
and the tenth century CE.
Fig. 24.73. Detail from the second inscription of Gwalior (876 CE). The shaded section shows the
representation of the numbers 933 and 270. 1 Ref. : El, I, p. 1601
w
°
933
270
187
50
Ref.: El, I, p. 160, lines 1, 4, 5 and 20.
Fig. 24.74. Numerals from the second inscription of Gwalior
These documents are legal charters written in Sanskrit and engraved in
ancient Indian characters. They record donations given by kings or wealthy
personages to the Brahmans. Each one contains details of the religious
occasion when the donation(s) was (or were) offered and gives the name of
the donor, the number of gifts plus a description of them, as well as a date
which corresponds to one of the Indian calendars ( *Chhedi , *Shaka,
* Vikrama, etc.; see Dictionary).
These dates are usually expressed in both letters and numerals, with the
basic numerals, written in various Indian styles, varying in value according
to their decimal position (Fig. 24.75 and 76).
The preceding evidence led historians, in the nineteenth century, to con-
clude that our present-day numerals were of Indian origin, and that they
had been in use at least since the end of the sixth century CE (Fig. 24.75).
The foundations of this evidence seemed to crumble, however, at the
beginning of the twentieth century, when three science historians, G. R.
Kaye, N. Bubnov and B. Carra de Vaux, who were among those the most
opposed to the idea that our numerals originated in India, questioned the
authenticity of the copper inscriptions. They claimed that these documents
had been re-written, altered or falsified at a much later date than the years
given in the lists. It was concluded that, of all the texts which had been
thought to be of Indian origin, only the inscriptions carved in stone could
be regarded as proof of the existence of the system in question.
401
THE INDIAN PLACE-VALUE SYSTEM
DOCUMENTS AND SOURCES
972
Donation charter of The number 894 is expressed:
Amoghavarsha of the
Rashtrakutas, dated 894 in the C* ) O
*Shaka calendar ' CJ
(= 894 + 78 = 972 CE).
IA, XII,
p. 263
933
Donation charter of Govinda IV The number 855 is expressed:
of the Rashtrakutas, dated 855
in the *Shaka Samvat calendar \
(= 855 + 78 = 933 CE). V J V.
IA, XII,
p. 249
917
Donation charter of Mahipala, The numbers 974 and 500 are
dated 974 in the *Vikrama expressed:
Samvat calendar e\ r ft
(= 974 - 57 = 917 CE). (O'! ^ t| O 0
IA, XVI,
p. 174
837
Bauka inscription. Dated 894 in The number 894 is expressed:
the *Vikrama Samvat calendar ^ ,
(= 894 - 57 = 837 CE). 3^0
El, XVIII,
p. 87
815
Donation charter of Nagbhata The number 872 is expressed:
ofBuchkala. Dated 872 in the
*Vikrama Samvat calendar I *2
(= 872 - 57 = 815 CE). 1 '
El, IX,
p. 198
793
Donation charter of The number 715 is expressed:
Shankaragana of Daulatabad.
Dated 715 in the *Shaka ^
calendar (= 715 + 78 = 793 CE). ^
El, IX,
p. 197
753
Donation charter of Dantidurga The number 675 is expressed:
of the Rashtrakutas. Dated 675
in the *Shaka Samvat calendar A
(= 675 + 78 = 753 CE). \ ' X
IA, XI,
p. 108
753
Inscription of Devendravarman. The number 20 is expressed:
Dated 675 in the *Shaka calen-
dar (= 675 + 78 = 753 CE). £? °
El, III,
p. 133
737
Donation charter of Dhiniki. The number 794 is expressed:
Dated 794 in the *Vikrama
Samvat calendar SJ Si Q
(=794 -57 = 737 CE). "
IA, XII,
p. 155
594
Donation charter of Dadda III, The number 346 is expressed:
of Sankheda in Gujarat (see Fig. 24.76)
(Bharukachcha region). Dated
346 in the *Chhedi calendar I H ^
(= 346 + 248 = 594 CE). (
El, II,
p. 19
Fig. 24.75-
Since the inscriptions of Gwalior (875/876 CE) constituted the first evi-
dence of this kind, these authors surmised that in India, zero and the
place-value system could not have been used much before the second half
of the ninth century CE.
It is true that amongst the charters recorded copper found in India, the
authenticity of a certain number of them has been questioned, and quite
rightly so, by Indianists (including Torkhede, Kanheri and Belhari, dated
respectively 813, 674 and 646 CE [El, III, p. 53; IA, XXV, p. 345; JA, 1863,
p. 392], Therefore, we have eliminated them from our investigation. As for
the other documents of this kind, their authenticity has never been ques-
tioned by anyone except for Kaye and others who shared his motives.
The evidence was questioned in the hope of proving that Greek mathe-
maticians were the “real” inventors of our numeral system, and that
historians had been mistaken in attributing the invention to the Indians.
However, as we have already seen, this hypothesis had no historical foun-
dation, it was simply concocted in order to extend the tradition of the
“Greek miracle”.
The questioning of the authenticity of the Indian charters has never
been satisfactorily justified.
The authors of the controversy would have it that these documents were
“fabricated” at a later date, when the opportunity presented itself to a
group of dishonest people who wished to take possession of the wealth
which had long belonged to religious institutions and which the local
authorities had confiscated or requisitioned some time before.
This explanation sounds feasible; however, there is no evidence to prove
it, and the event was given a totally arbitrary date (some time during the
eleventh century).
It was alleged that on the oldest known dated charter (Fig. 24.76), the
numerals 3, 4 and 6, which come at the end of the inscription and which
denote the *Chhedi year 346 (594 CE), were added at a later date.
If this were true, then why is the numeral 3 written as three horizontal
lines? At the end of the sixth century (which corresponds with the date on
the document), this way of writing the number was still used, although it
was already becoming obsolete. It had disappeared completely by the next
epoch, to be replaced by the non-ideographical sign belonging to the same
style as the 4 or 6 which appear in the same document.
Of course, it could be argued that the forger (if there had been a forger)
could have studied the palaeography of Indian numerals before imitating the
style in question. The date on the legal document which tells of offerings
made by someone’s ancestor (authentic or not) would have been important
to the descendant or person claiming to be so in order for them to prove that
they were the rightful owners of the property mentioned on the charter.
INDIAN CIVILISATION
402
But why would someone go to so much trouble, when the date is already
given in the text in the form of the names of the numbers in Sanskrit? At
that time, this indication was quite acceptable on its own; it was even more
reliable than the numerals, whose appearance was susceptible to so many
alterations in the hands of scribes and engravers.
What would have been the point of such an addition? And why would
the date have been written in keeping with the place-value system when
non-positional notation derived from the Brahmi system was still fre-
quently used (at least by the lay person) to write this type of legal document
(Fig. 24.70)?
In other words, if the document was forged, why was the place-value
system favoured rather than the old non-positional system?
No acceptable answers to these questions have been provided by those
who put forward the theory of a forgery. On the other hand, to support his
theory, Kaye did not hesitate to cite the charters inscribed on copper con-
taining dates written using the old system and dating back to the era
between the end of the sixth century and the ninth century (source: IA, VI,
p. 19 ; El, III, p. 133, etc.).
The most amusing part of this story is that the above dispute only cen-
tred on the oldest copper charters containing examples of the use of the
place-value system, and not on the numerous other documents of the same
nature which were written after or at the same time as the Gwalior inscrip-
tions (876 CE). As for those containing examples of numbers written in the
old non-positional Indian numerals whose date oscillates between the sixth
and eighth century, their arithmetic was never questioned by Kaye. Thus
we can see that these authors had worked out their conclusions far better
than their arguments.
Fig. 2476. Donation charter of Dadda III, from Sankheda in Gujarat (region of Bharukachcha).
Dated 346 in the *Chhedi calendar (= 346 + 248 = 594 CE), this document is the oldest known
formal evidence of the use of the place-value system in India (in the shaded section, the number 346 is
expressed according to this system). [Ref: El, II, p. 19]
We must be careful, however, because there is no way of ascertaining
whether or not any of these copper charters are authentic; it is easy to make
forgeries with copper, and we are dealing with a region of the world where
counterfeiters, since time immemorial, have been masters at their craft.
The preceding counter-arguments would seem to suggest that these
charters could well be authentic.
For the benefit of the doubt, however, we will not use these documents
as evidence in our investigation, even though, from a purely graphical
point of view, the letters and numerals they contain are indisputably
authentic, unless the “forgers” pushed their talents to the limit to make
exact copies of the contemporary and regional styles for each of the char-
ters in question.
The fact remains that the history of the Indian decimal place-value
system owes much to men such as Kaye. They proved that the subject was a
lot more complicated than it seemed at first, and that all the documenta-
tion must be scrutinised very closely in establishing the facts. The
controversy obliged scholars to go back to square one and apply stricter
rules to their analysis of the facts and documents in this very rich and fer-
tile field where they had not always exercised the correct degree of caution.
On the other hand, men such as Kaye displayed a certain narrow-
mindedness in limiting themselves to the literal frontiers of this civilisation
which spread across a geographical area of truly continental dimensions,
and which influenced and was witness to the flourishing of many other cul-
tures which were situated beyond the limits of its own territory.
The following demonstrates that there are a great many other (unques-
tionably authentic) documents, which prove that zero and the place-value
system are truly and exclusively Indian inventions, and that their discovery
dates back even further than the oldest known inscription on copper.
Proof found in epigraphy from Southeast Asia
The texts that we will consider now are of considerable value to this investiga-
tion, for at least two reasons: first, they are all carved in stone, which means
that there can be no doubt as to their authenticity; secondly, they are extracts
from dated inscriptions, the oldest of which date back into the distant past.
These inscriptions are written either in Sanskrit or in vernacular lan-
guage, that is to say in the regional language, be it Old Khmer, in Old
Malay, in Cham, in Old Javanese, etc. Many of them record offerings to
temples, their interest being an indication of the date (the year in which the
inscription was written) and a detailed description of the offerings.
The way in which the corresponding numbers are expressed gives us the
most significant indication of the use of the Indian place-value system.
403
THE INDIAN PLACE-VALUE SYSTEM
If we only look at the indigenous inscriptions for the moment (those
which are unique to each of the civilisations in question), we can see a very
interesting particularity: the commonly-used numbers are not expressed in
the same manner as the dates.
For the common numbers (expressing units of length, surface areas or
capacities; the number of slaves, objects or animals; the quantity of gifts
offered to the divinities and temples, etc.), the engravers usually simply
expressed them in the letters of their vernacular language.
However, Cambodia is an exception to this rule; the Khmer engravers
often preferred to use their local numeral system, which is immediately iden-
tifiable due to its undeniably primitive character (Fig. 24.77). This system
uses one, two, three or four vertical lines to represent the first four units,
although the fourth is often represented by a sign which gives no ideographi-
cal clue to the number it represents. As for the units 5 to 9, these are also
represented by independent signs. This system also has a particular sign for
10, 20 and 100. As the system relies on the additive principle to represent
numbers below 100, the multiples of 10, from 30 to 90, are expressed by com-
binations of the numerals for 20 and 10, according to the following rule:
30 = 20 + 10 Juxtaposition of the signs 20 and 10
40 = 20 + 1 x 20 A vertical line is added to the sign for 20
50 = 40 + 10 Juxtaposition of the signs 40 and 10
60 = 20 + 2 x 20 Two vertical lines are added to 20
70 = 60 + 10 Juxtaposition of the signs 60 and 10
80 = 20 + 3 x 20 Three vertical lines are added to 20
90 = 80 + 10 Juxtaposition of the signs 80 and 10
The multiples of 100 are expressed in much the same manner, the numeral
100 being accompanied by the corresponding unit:
200 = 100 + 1 x 100 A vertical line is added to the sign 100
300 = 100 + 2 X 100 Two vertical lines are added to 100
The system seems to be limited to numbers below 400: there is no example
of a higher number than this; above this quantity the Khmers wrote the
names of the numbers in the letters of their language.
Thus, in terms of graphical representation, the ancient Khmer vernacu-
lar numeral system derived from the old Brahmi system, as can be seen in
the above table.
On the other hand, the structure of the system stems from the counting
system of the Old Khmer language, for which we know the base was 20
11
10
X or S or S
2 II
20
9 or -»?
= 20 + 10
3111
30
1*
= 20 + 1 X 20
4 llll or
40
a line is added to “20”
J.
= 40 + 10
5 5 or
5" or ^5 50
%*
= 20 + 2x20
6 <5 or
^ or^ 60
t or^
two lines are added to “20”
7 1 or
^ or \ 70
P
= 60 + 10
f or«f
= 20 + 3x20
8 ^ or
V or ? 80
three lines are added to “20”
907"
or f 90
t'
= 80 + 10
100
•7 T
= 100 + 1 x 100
-L.
200
n
one line is added to “100”
JL.
rTi
= 100 + 2x100
300
e L
two lines are added to “100"
Examples
taken from two Khmer inscriptions of Lolei (in the region of Siem Reap in
Cambodia), dated 815 in the Shaka calendar (= 893 CE).
m- —in
ssi iS“
10 + 2
10 + 3 20 + 10 + 5
80 + 7 100 + 80 + 2 200 + 10 + 6 300 + 80 + 10 + 8
>
> >
> >
> >
12
13 35
87 182
216 396
Ref. Aymonier (1883); Guitel (1975).
Fig. 24 . 77 - The written numeration of the ancient Khmers: a system which uses the additive principle
and which contains a curious trace of base 20. Used until the thirteenth century CE in vernacular
inscriptions of Cambodia to express everyday numbers.
(which explains the presence of a special sign for 20 and its multiples). As
Coedes observes:
The numeral system was not decimal, and today, despite the fact that
Siamese numerals are used to represent multiples of ten above thirty,
and likewise for 100, 1,000, etc., it still is not completely decimal: the
names of the numbers from six to nine are expressed as five-one, five-
INDIAN CIVILISATION
404
two, five-three, five-four, and special names for four and many of the
multiples of twenty are still in common use. In ancient times, the
Khmer people used no more than the names for one, two, three, four,
five, ten, twenty and some multiples of twenty to express numbers, no
matter how high the numbers were, and they used the Sanskrit word
*shata for “hundred”, to which the term slika was added, which they
also used to express the number 400 (= 20 2 ).
In other words, the spoken Khmer numeral system constituted a
kind of compromise between Indian decimal numeration and a very old
and far more primitive indigenous system, based both on 4 and 5 [see
BEFEO, XXIV (1924) 3-4, pp. 347-8; JA, CCLXII (1974) 1-2,
pp. 176-91].
On the other hand, in order to express dates, the stone-carvers of the
diverse civilisations of Southeast Asia never used their vernacular numeral
system or wrote the numbers in word form in their own language; as we
will see, this fact is of great significance.
They only recorded dates using one of the two following methods: either
the names of the numbers in Sanskrit, or, more frequently after a certain
date, a decimal numeral system using nine numerals and a zero in the form
of a dot or a little circle, strictly adhering to the place-value system (Fig.
24.50 and 24.78 to 80).
There is evidence of this use of the place-value system until the thir-
teenth century, from the ninth, eighth and even the seventh century CE,
depending on the region.
In Champa, it was used consistently, at least since the Shaka year 735
(813 CE), which is the date of the oldest known Cham inscription of Po
Nagar (Fig. 24.80).
In the Indian islands, however, the system appeared much earlier:
• at the end of the eighth century in Java; the oldest vernacular
inscription (in Kawi writing) to bear witness to the use of the place-
value system on this island is from Dinaya, dated the * Shaka year 682
(760 CE) (Fig. 24.80);
• at the end of the seventh century at Banka; the most ancient vernac-
ular inscription (in Old Malay) which attests its use in this island is
that of Kota Kapur, dated 608 Shaka (686 CE) (Fig. 24.80);
• at the end of the seventh century in Sumatra; the oldest vernacular
inscriptions (in Old Malay) to bear witness to its use in the region
come from Talang Tuwo and Kedukan Bukit in Palembang, dated the
respective Shaka years 606 and 605 (684 and 683 CE) (Fig. 24.80);
• and also at the end of the seventh century in Cambodia; the oldest
vernacular inscription (in Old Khmer) to bear witness to its use in this
7th century 8th century 9th century 10th century Uth century 12th— 13lh century
1
90G\<3\ 9 ej Q Q
KKKKKKK K
315 314 325 330 848 215 324 31
2
\ 33\i\3 13 *3 3
KKKKKKK
291 125 292 292 158 216 247
3
3363 “d O J
K K K K K K K
291 253 682 125 292 933 850
4
S,*13 (5 3 2 , iScg
KKKKKKK
253 682 245 253 31 206 207
5
£ 575 15 5
K K K K K K
127 713 328 325 156 254
6
^ ^e) &
K K K K K K
127 215 660 206 246 850
7
n ^ &
K K K
215 216 410
8
KKKKKK K K
314 713 328 327 682 231 239 247
9
fU
KKKKKKKKK
292 848 933 158 216 31 410 207 241
0
• O • •
K K K K
127 315 214 254
Fig. 24.78. A selection of palaeographical variants (dated) of numerals of the place-value system
which, in vernacular inscriptions of Cambodia written in Old Khmer, were exclusively used to express
dates of the Shaka calendar. (For the K references, see IMCC)
country is from Trapeang Prei, in the province of Sambor, the respec-
tive Shaka year 605 (683 CE) (Fig. 24.80).
In Cambodia, however, this is not the oldest existing dated vernacular
inscription. There is one which dates back even further, the earliest possi-
ble inscription to contain a date; it is from Prah Kuha Luhon, dated the
Shaka year 596 (674 CE), this date being written in letters using the
405
THE INDIAN PLACE-VALUE SYSTEM
9th century 10th century 11th century 12th century 13th— 14th century
1
i q n ^
C23 C 30 C 17 C4C4C4C3
2
c? ^ q ^ ^
C 120 C 119 C 17 C 4 C 4 C3
3
3 fj
C 37 C 17 C3C5C4C4
1
C 4 C 4 C 5 C 5
f 2. \ U
C 37 C 23 C3C4C5
1
| » J* %
C 30 C 17 C 4 C 4
1
? * iu
C 37 C 23 C 119 C 126 C 122 C5
8
i l
C4 C 5
9
&&& a
C 119 C 120 C 126 C 122 C3
0
© o
C 30 C 4
Fig. 24 . 79 . A selection of palaeographical variants (dated) of numerals of the place-value system
which, in (Cham) vernacular inscriptions of Champa, were exclusively used to express dates of the
Shaka calendar. (For the C references, see IMCC)
Sanskrit names for the numbers (see IMCC, K 44, 1. 6; CIC, IV):
shannavatyuttarapahchashataShakaparigraha
“the Shaka [year] numbering five hundred and ninety-six”
Thus in the vernacular inscriptions of Southeast Asia, the everyday
numbers were always expressed through the names of the numbers in the
DOCUMENTS AND SOURCES
HOW THE DATE IS RECORDED
IN THE SHAKA CALENDAR
1084
Cham inscription of Po Nagar,
northern tower. Dated 1006 in
the Shaka calendar
(= 1006 + 78 = 1084 CE).
ri o o ^
10 0 6
IMCC,
C 30
BEFEO,
XV, 2,
p. 48
1055
Cham inscription of Lai Cham,
region of Hanoi. Dated 977 in
the Shaka calendar
(= 977 + 78 = 1055 CE).
*
9 7 7
IMCC,
C 126
BEFEO,
XV, 2,
pp. 42-3
1055
Cham inscription of Phu-qui,
province of Phanrang.
Dated 977 in the Shaka calendar
(= 977 + 78 = 1055 CE).
9 7 7
IMCC,
C 122
BEFEO,
XV, 2,
p. 41
BEFEO,
xn, 8,
P-17
1050
Cham inscription of Po Klaun
Garai (first inscription).
Dated 972 in the Shaka calendar
(= 972 + 78 = 1050 CE).
w
9 7 2
IMCC,
C 120
BEFEO,
XV, 2,
p. 40
1050
Cham inscription ofPo Klaun
Garai (second inscription).
Dated 972 in the Shaka calendar
(= 972 + 78 = 1050 CE).
£4?
9 7 2
IMCC,
C 120
BEFEO,
XV, 2,
p. 40
1007
Khmer inscription of Phnom
Prah Net Prah (foot of
southern tower). Dated 929
in the Shaka calendar
(= 929 + 78 = 1007 CE).
g3
9 2 9
IMCC,
K 216
BEFEO,
XXXIV,
p. 423
1005
Khmer inscription of Phnom
Prah Net Prah (foot of
southern tower). Dated 927
in the Shaka calendar
(= 927 + 78 = 1005 CE).
g\3 <3 ^
9 2 7
IMCC,
K 216
BEFEO,
XXXV,
p. 201
880
Balinese inscription of Taragal.
Dated 802 in the Shaka calendar
(= 802 + 78 = 880 CE).
V“3
8 0 2
Damais,
P- 148, g.
Fig. 24.80A.
INDIAN CIVILISATION
406
DOCUMENTS AND SOURCES
HOW THE DATE IS RECORDED
IN THE SHAKA CALENDAR
878
Balinese inscription of Mamali.
Dated 800 in the Shaka calendar
(= 800 + 78 = 878 CE).
Y 0 0
8 0 0
Damais,
p. 148, f.
877
Balinese inscription of
Haliwanghang. Dated 799 in
the Shaka calendar
(= 799 + 78 = 877 CE).
tefcjfcr
7 9 9
Damais,
p. 148, f.
829
Cham inscription of Baku!
Dated 751 in the Shaka calendar
(=751 + 78 = 829).
7 5 1
IMCC,
C 23
ISCC,
p. 238
BEFEO,
XV, 2,
p. 47
813
Cham inscription of Po Nagar,
northwest tower. Dated 735
in the Shaka calendar
(= 735 + 78 = 813 CE).
This is the first Cham inscription
to contain the date written in
numerals.
7 3 5
IMCC,
C 37
JA 1891,
i, p. 24
BEFEO,
XV, 2,
p. 47
686
Malaysian inscription of Kota
Kapur (isle of Banka). Dated
608 in the Shaka calendar
(= 608 + 78 = 686 CE).
&ofr
6 0 8
BEFEO,
XXX,
pp. 29ff.
Kern VIII,
p. 207
684
Malaysian inscription of Talang
Tuwo, Palembang (Sumatra).
Dated 606 in
the Shaka calendar
(= 606 + 78 = 684 CE).
/g) O c)
6 0 6
BEFEO,
XXX,
pp. 29ff.
ACOR, II,
p. 19
683
Malaysian inscription of
Kedukan Bukit, Palembang
(Sumatra). Dated 605 in
the Shaka calendar
(= 605 + 78 = 683).
This is the oldest inscription in
Old Malay to be dated in
numerals.
6°!>
6 0 5
BEFEO,
XXX,
pp. 29ff.
ACOR, II,
p. 13
Fig. 24.80B.
HOW THE DATE IS RECORDED
DOCUMENTS AND SOURCES IN THE SAKA CALENDAR
683
Khmer inscription of Trapeang
Prei, province of Sambor. Dated /J) ^ Q
605 in the Shaka calendar (= V V.
605 + 78 = 683 CE). '
6 0 5
This is the first Khmer inscription
dated in numerals.
IMCC,
K 127
CIC,
XLVII
Note: In Java, the oldest Kawi inscription (written in Old Javanese) to be dated in
numerals is of Dinaya, which bears the date 682 in the Shaka calendar = 682 + 78 = 760 CE.
IRefi Tijdschrift, LVII, (1976), p. 411; LXIV, (1924), p. 227],
Fig. 24.80c.
indigenous language or its very rudimentary numerals. For the dates of the
Shaka calendar, however, either the Sanskrit word, or, more commonly, the
decimal place-value system was used, from no later than the end of the sev-
enth century CE.
As we have seen, the different numerals used throughout Southeast Asia
were actually nothing more than palaeographical variations of Indian
numerals, which themselves derived from the Brahmi form of the first nine
units (Fig. 24.52, 24.53, and 24.61 to 69). The only difference between
these diverse systems is their complete transformation into local cursive
forms (Khmer, Javanese, Cham, Malay, Balinese, etc.), according to the
habits of the scribes and engravers of the region.*
On the other hand, as well as the use of Sanskrit (the learned language
of Indian civilisation) to record the dates, all the vernacular inscriptions
reveal that the dates were written exclusively, for many centuries, according
to a system whose Indian origin is indisputable: the Shaka era of the Indian
astronomers [see R. Billard (1971)].
* It should be noted that the interpretation of the Cham numerals posed considerable difficulties because of
mistakes made as to their values when they were first deciphered. 7 was mistaken for 1, the (more recent
form of) 1 was mistaken for the very ancient form of 5, 7 for 9. etc. This caused even graver errors in the
nineteenth century when it came to dating the inscriptions, which in turn led to mistakes in the interpreta-
tion of historical events and chronology. Thus it appeared that inscriptions of the same person, referring to
the same event, were written at completely different points in time. The inscriptions of King
Parameshvaravarman, for example, gave the Shaka date 972 (1050) in Sanskrit chronograms, whilst other
inscriptions, or the same one, gave the date as the Shaka year 788 (866). Another example is an inscription
of Mi-s’on (BEFEO, IV, 970, 24) which lists a series of religious foundations set up by King Jaya Indravarman
of Gramapura; logically, these dates should have been listed in chronological order; however, when the
values that were believed to be correct were applied, the dates emerged in the following incoherent manner:
1095, 1096, 1098, 1097, 1070 and 1072. There were many similar enigmas, which seemed to have no solu-
tion and were distorting all the acquired data, until L. Finot (BEFEO, XV, 2, 1915, pp. 39-52) discovered the
true origin of the Cham numerals and with it the solution which had eluded his predecessors. The inaccu-
rate interpretations of the Cham numerals were due to the very unusual variations that they had undergone
over the centuries because of the whims and aesthetical preoccupations of the corresponding engravers.
407
OUTSIDE INFLUENCE OR INDIAN INVENTION?
These facts are even more significant because they concern the ancient
civilisations of Indo-China and Indonesia (Cambodia, Champa, Java, Bali,
Malaysia), which were strongly influenced by India in the early centuries
CE, partly because of the widespread nature of Shivaism and Buddhism,
and also because of the important intermediate role they played in the
trading of spices, silk and ivory between India and China [see G. Coedes
(1931), (1964)].
Champa is the ancient kingdom that stretched along the southeast coast
of what is now Vietnam with the region of Hue at its centre. The native
inhabitants of Champa had become Hindu by religion due to their frequent
contact with Indian traders. Champa first became a powerful nation at the
beginning of the fifth century CE, under the rule of Bhadravarman, who
dedicated the shrine of Mi-so’n (which would remain the religious centre
of the kingdom) to Shiva, one of the greatest divinities of the Indian
Brahmanic pantheon.
Not far from Champa is Cambodia, which previously belonged to the
Hindu kingdom of Fu Nan from the first to the sixth century CE, before
becoming the centre of Khmer civilisation, which, conserving the founda-
tions of an Indian culture, flourished until the fourteenth century.
In Java, too, which entered into relations with India at the beginning of the
second century CE, and all of ancient Indonesia, early developments owe much
to Indian civilisation through the influence of Buddhism and Brahmanism.
Thus we can see the great influence that Indian astronomers and mathe-
maticians once exercised over the various cultures of Southeast Asia. All the
preceding facts are highly significant for they show how the Khmer, the
Cham, the Malaysian, the Javanese, the Balinese, and other races, were pro-
foundly influenced by Indian culture, and borrowed elements of Indian
astronomy, in particular the Shaka calendar, and conformed to the corre-
sponding arithmetical rules [see F. G. Faraut (1910)].
In these regions, the appearance of zero and the place-value system coin-
cides directly with the appearance of the dates of the Shaka calendar:
The place-value system was used in Indo-China and Indonesia from
the seventh century CE, in other words at least two centuries before
Kaye claimed to find evidence of its use in India itself. However, unless
zero and the Arabic numerals came from the Far East [which is pre-
cisely the opposite of what did happen] evidence of their use in the
Indian colonies would suggest that they were in use in India at an even
earlier date [G. Coedes (1931)].
Thus we have confirmed the words of the Syrian Severus Sebokt, who
wrote, in the seventh century, that the Indian place-value system was
already known and held in high esteem beyond the borders of India.
THE INDIAN PLACE-VALUE SYSTEM: OUTSIDE
INFLUENCE OR INDIAN INVENTION?
The place-value system is unquestionably of Indian origin, and its discov-
ery doubtless dates back much further than the seventh century CE. The
question we must now ask is whether this concept was inspired by an out-
side influence or whether it was a purely Indian discovery.
We know that during the course of history the Babylonians, the Chinese,
the Maya and, of course, the Indians, succeeded in inventing a place-value
system. If the Indians were influenced by any other civilisation, it would
have to have been one of the other three we have just mentioned, either
directly or through an intermediary.
Putting to one side the Maya civilisation, which apparently had no con-
tact with the ancient world, this leaves us with the Chinese and Babylonian.
The possibility of Babylonian influence
If the Indian place-value system was derived from that of the Babylonians,
it might have been through the intermediary of Greek civilisation.
In 326 BCE, Alexander the Great took possession of the land of the
Indus and of the ancient province of Gandhara, from the northeast of
Afghanistan and the extreme north of present-day Pakistan to the north-
west of India, before these regions were governed by the “Indian-Greek”
Satraps c. 30 BCE. We know that many elements and methods of
Babylonian astronomy were introduced into India shortly before the begin-
ning of the first millennium CE, probably through the northwest of the
Indian sub-continent; no doubt this took place in the eastern part of the
present-day state of Gujarat, probably in the region of the port of
Bharukachcha, which saw the development of both cultural and maritime
activities and trade with the West during the first centuries CE. Thus, as
R. Billard (1971) stresses, the period between the third century BCE and the
first century CE is characterised by the appearance of the tithi, a unit of
time used in the Babylonian tablets and corresponding to the thirtieth of a
synodic revolution of the Moon, more or less the equivalent of a day or
nychthemer: elements which are known to have been transmitted to the
Greek astronomers by their Mesopotamian colleagues no later than the
Hellenistic era.
We know that Babylonian scholars had invented and used a place-value
system with 60 as a base since the nineteenth century BCE, and they had
used zero since the fourth or third century BCE. As the sexagesimal system
was “one of the elements that the Greeks had acquired from Babylonian
astronomy, the mother and wet-nurse of their own astronomy” [F. Thureau-
Dangin (1929)], it is possible to suppose that the idea of the place-value
system arrived in India at the same time as Babylonian astronomy.
INDIAN CIVILISATION
408
Although this hypothesis cannot be ruled out, we can nevertheless raise
one serious objection to it. The Greek astronomers only used the
Babylonian sexagesimal system to write the negative values of 60, in other
words the sexagesimal fractions of the unit, whilst the system was origi-
nally developed in order to express whole numbers as well as fractions.
Thus, if a similar influence was exercised over the Indians by the Greeks
(for if the Indians were influenced by the Babylonian system it could only
have been via the Greeks), how could an incomplete system, only used to
record fractions, moreover with a base of 60, have influenced the invention
of a decimal place-value system which was originally invented to record
whole numbers? This is an obvious flaw, which makes this hypothesis
appear rather paradoxical.
The possibility of a Chinese influence
Therefore, at first glance, a Chinese influence would seem more plausi-
ble. We know that since the time of the Han (206 BCE to 220 CE),
Chinese scholars used a decimal place-value system known as suan zi
(“calculation using rods”). A regular system which combined horizontal
and vertical lines was used to represent the nine basic units, constituting
a written transcription of a concrete counting system which used reeds,
standing on one end or placed horizontally on a counting board like aba-
cuses in columns.
Thus one could be forgiven for assuming that following the links estab-
lished between China and India at the beginning of the first millennium
CE, Indian scholars were influenced by Chinese mathematicians to create
their own system in an imitation of the Chinese counting method.
However, this hypothesis is contradicted by the fact that zero only
appeared in the suan zi system relatively late. The Chinese scholars over-
came the difficulties this caused by expressing a number such as 1,270,000
either in the characters of their ordinary counting system (a non-positional
system which did not require the use of zero) or by placing their rod
numerals in a series of squares, the missing units being represented by
empty squares:
1
—
IT
1 2 7 0 0 0 0
It was only after the eighth century CE, and doubtless due to the influence of
the Indian Buddhist missionaries, that Chinese mathematicians introduced
the use of zero in the form of a little circle or dot (signs that originated in
India), thus representing the preceding number in the following manner:
I I = IT O O O O I
1 2 7 0 0 0 0
A symbol for zero is mentioned in the Kaiyun zhanjing, a major work
on astronomy and astrology published by ‘Qutan Xida between 718
and 729 CE. The chapter of this work devoted to the jui zhi calendar of
718 CE contains a section on Indian calculating techniques [J.
Needham (1959)].
After saying that the (Indian) figures are all written in the cursive form in
just one stroke, Qutan Xida continues:
When one or another of the nine numerals has to be used to express the
number ten [literally: "when it reaches ten”], it is then written in a preced-
ing column [before the numerals for the units] ( qian wei). And each time
an empty space appears in a column, a dot is always written [to convey the
empty space] ( meigong wei qu henganyi dian).
The author of this book on astronomy was not Chinese: ‘Qutan Xida was
actually an adaptation of the Indian name *Gautama Siddhanta, the
famous Indian Buddhist mathematician and astronomer living in China
and the head of a school of astronomy at Chang’an since approximately 708
CE. According to L. Frederic (1987), he was the one to introduce the notion
of zero in China as well as the division of a circle into sixty sections.
This remarkable account confirms the influence, which has already been
proved, of the rapid expansion of the Buddhist movement which accompa-
nied the propagation of Indian science in the Far East. It also adds an
important piece of evidence to our investigation of the origin of our
modem numeral system:
Living in China, doubtless knowing all the subtleties of the Chinese
language, Qutan Xida insists not only on the fact that Indian numerals
were written in a cursive form, but also that each one was written in
just one stroke [G. Guitel, (1975)].
In the Chinese place-value system (the "learned” system), the units were
written by juxtaposing or superposing one or more vertical or horizon-
tal lines:
I II III III! mil T IT TIT HIT
123456 789
409
THE NUMERICAL SYMBOLS OF THE INDIAN ASTRONOMERS
The Chinese numerals in common usage are formed by lines, in vari-
ous positions, and written in a strict order, lifting the writing tool several
times, the symbol for 2 being formed by two lines, 4 by six lines, 6 by
four lines, and so on (Figure 21. 1 above). This will become clearer later,
when we see the succession of lines that forms the Chinese numeral for
100 (see Fig. 21. 8 above):
This could only have surprised the learned men amongst which
Gautama Siddhanta (alias Qutan Xida) lived, because Chinese words
are grouped according to the number of lines their drawing requires;
for each character, one is taught the order in which the successive lines
must be drawn [G. Guitel (1975)].
The nine numerals (of Indian origin) that we use today, on the other hand,
are drawn in just one stroke of a pen or pencil. This is one of the character-
istics of our numeral system, whose remarkable simplicity we forget
because we have been using it all our lives.
This evidence proves that at the beginning of the eighth century, zero
and the place-value system had spread as far as China. At the same time, it
almost completely rules out any possibility of a Chinese influence over the
development of our present-day numerals.
THE AUTONOMY OF THE INDIAN DISCOVERIES
Thus it would seem highly probable under the circumstances that the dis-
covery of zero and the place-value system were inventions unique to Indian
civilisation. As the Brahmi notation of the first nine whole numbers (incon-
testably the graphical origin of our present-day numerals and of all the
decimal numeral systems in use in India, Southeast and Central Asia and
the Near East) was autochthonous and free of any outside influence, there
can be no doubt that our decimal place-value system was born in India and
was the product of Indian civilisation alone.
THE NUMERICAL SYMBOLS OF THE INDIAN
ASTRONOMERS
We are now going to look at a truly remarkable method of expressing num-
bers which is frequently found on mathematical and astronomical texts
written in Sanskrit; there is no doubt that these texts are of Indian origin.
It is to curious to note that historians of science have not always accorded
it the importance it deserves. It constitutes the main piece of evidence of our
investigation: added to all the other evidence, it allows us not only to prove
beyond doubt that our present-day numeration is of Indian origin, and
Indian alone, but also and above all to date the discovery even earlier than
the seventh century CE. Moreover, it is even more significant when we con-
sider that the nature of this system is unique in the history of numerals.
By way of introduction, here is a passage from the first modern Indian
historian, the Persian astronomer al-Biruni, who wrote the following
c. 1010, in his famous work on India [see al-Biruni (1910); F. Woepcke
(1863), pp. 283-90]:
When [the Indian scholars] needed to express a number composed
of many orders of units in their astronomical tables, they used cer-
tain words for each number composed of one or two orders. For each
number, however, they used a certain number of words, so that, if it
was difficult to place one word in a certain place, they could choose
another from “amongst its sisters” [amongst those which denoted
the same number], Brahmagupta said: If you want to write one,
express it through a word which denotes something unique, like the
Earth or the Moon; likewise, you can express two with any words
which come in pairs, like black and white [this is probably an allu-
sion to the “half black” and “half white” of the month, which
corresponds to a division used by the Indians], three by things that
come in threes, zero with the names for the sky, and twelve by the
names of the sun . . . [Such is the way the system works] as I have
understood it. It is an important element in the analysis of their [the
Indians’] astronomical tables . . .
Instead of the word *eka, which means “one”, the Indian astronomers used
names such as *adi (the “beginning”), *tanu (“the body”), or *pitamaha
("the Ancestor”, which alludes to *Brahma, considered to be the creator of
the universe).
Instead of *dvi, which means "two”, they used all the words which
express ideas, things or people which come in pairs: *Ashvin (“the twin
gods”), *Yama ("the Primordial Couple”), *netra ("the eyes”), *bahu (“the
arms”), *paksha (“the wings”), etc.
In other words, rather than using the ordinary Sanskrit names for the
numbers 1 to 9 (*eka, *dvi, *tri, *chatur, *pancha, *shat, *sapta, *ashta,
*nava), the Indian scholars expressed them by names which had symboli-
cal value. For each number, there was a wide choice of words, whose
literal translation evoked the numerical value they denoted in the
reader’s mind.
INDIAN CIVILISATION
410
It is difficult to give an exhaustive list of these diverse symbolic words,
there being an abundant, if not infinite, number of synonyms. However,
the reader will get some idea of the variety of these words from the follow-
ing examples:
ONE
eka:
Ordinary name for the number 1
pitamaha :
First father
adi :
Beginning
tanu\
Body
kshiti, go...:
Words meaning “Earth”
abja, indu, soma ...:
Words meaning “Moon”
TWO
dvi :
Ordinary name for the number 2
Ashvin:
Horsemen
Yama :
Primordial Couple
yamala, yugala . ..:
Words meaning twins or couples
netra:
Eyes
bahu:
Arms
gulphau :
Ankles
paksha:
Wings
THREE
tri:
Ordinary name for the number 3
guna:
Primordial properties
loka:
[Three] worlds
kala:
Time
agni, vahni . .
Fire
Haranetra:
“Eyes of Hara”
FOUR
chatur.
Ordinary name for the number 4
dish:
The [four] cardinal points
abdhi, sindhu . .
The [four] oceans
yuga:
The [four] cosmic cycles
irya :
The positions [of the human body]
Haribahu:
The arms of Vishnu
brahmasya:
The faces of Brahma
FIVE
pahcha:
Ordinary name for the number 5
bana, ishu . . .:
Arrows
indriya:
The [five] senses
rudrasya:
The [five] faces of Rudra
bhuta:
The elements
mahayajha:
The sacrifices
SIX
shat:
Ordinary name for the number 6
rasa :
The senses
anga:
The [six] limbs [of the human body]
shanmukha:
The [six] faces of Kumara
SEVEN
sapta:
Ordinary name for the number 7
ashva:
Horses
naga:
Mountains
rishi:
The [seven] sages
svara:
The vowels
sagara:
The [seven] oceans
dvipa:
The island-continents
EIGHT
ashta: Ordinary name for the number 8
gaja: The [eight] elephants
naga: Word meaning “serpent”
murti: Forms
NINE
nava :
Ordinary name for the number 9
anka:
Numerals
graha:
Planets
chhidra:
The orifices [of the human body]
ZERO
shunya:
Ordinary name for 0
bindu:
The point or dot
kha.gagana . . .:
Words meaning “sky”
akasha:
Ether
ambara, vyoman ...:
Atmosphere
411
The Sanskrit language, which is very learned and rich, lends itself admirably
to this system, as it does to poetry and the Indian way of thinking.
These symbols are all taken from nature, human morphology, animal or
plant representations, everyday life, legends, traditions, religions, attrib-
utes of the divinities of the Vedic, Brahman, Hindu, Jaina or Buddhist
pantheons, as well as from the associations of traditional or mythological
ideas or from diverse social conventions of Indian civilisation.
With this unique system of numerical notation, we have now entered
into the world of symbols of Indian civilisation.
To give the reader a better understanding of the characteristic way of
thinking of Indian philosophers, astrologers, cosmographers, astronomers
and mathematicians, (the true “inventors” of our present-day counting
system), we have included the “Dictionary of Numerical Symbols of Indian
Civilisation” at the end of this chapter, the necessity and usefulness of
which will become clear in the course of the following pages. *
THE PLACE-VALUE SYSTEM OF THE INDIAN NUMERICAL
SYMBOLS
To give us some idea of the principle this system was based on, here is a lit-
eral translation of a Sanskrit verse taken from a work on astronomy
entitled Surya Siddhanta (or “Astronomical canon of the Sun”; [see Anon.
(1955), 1, 33; Burgess and Whitney (I860)]:
Chandrochchasydgnishunyashvivasusarparnavdyuge
Vamam pdtasya vasvagniyamashvishikhidasrakdh
“The apsids of the moon in a yuga
Fire. Vacuum. Horsemen. Vasu. Serpent. Ocean,
and of its waning node
Vasu. Fire. Primordial Couple. Horsemen. Fire. Twins”
This verse is incomprehensible to a reader who does not know that the
words “Fire. Void. Horseman. Vasu. Serpent. Ocean” (dgnishunyash viva -
susarparnava ) and “Vasu. Fire. Primordial Couple. Horseman. Fire.
Horseman” ( vasvagniyamashvishikhidasra ), in the minds of the Indian
astronomers, represented the numbers 488,203 and 232,238 respectively.
Here is a comprehensible translation of the verse:
“[The number of revolutions] of the apsids of the moon in a yuga [is]:
488,203, and [of] its waning node: 232,238.”
* For each of the word-symbols in question ( *Ashvin, *Graha, *Kha, etc.), the reader might find it interest-
ing to consult the corresponding rubric, where the symbol is denoted by |S], then defined in terms of its
numerical value and its literal meaning in Sanskrit, before, as far as possible, its implied symbolism is
explained. To find the list of Sanskrit word-symbols used (in their abundant synonymy) for a given number,
one only need consult the corresponding English word in the Dictionary ( *One, *Two, *Zero, etc.).
THE PLACE-VALUE SYSTEM OF THE INDIAN NUMERICAL SYMBOLS
Thus the author of this text expressed, in his own way, two pieces of
astronomical numerical data, concerning a *yuga or “cosmic cycle” (in this
case a cosmic cycle named *Mahayuga and corresponding to a period of
time of 4,320,000 years).
The key to the system lies in knowing that, in a number-system which has
10 as its base, the first nine whole numbers, 10 and each multiple of 10 have a
specific name; thus one expresses a given number by placing the name for
“ten” between that of the units of the first order and that of the units of the
second order, then the name for “hundred” between those of the second and
third orders, and so on, respecting a previously agreed method of reading.
The number 8,237, for example, might be expressed in the following
manner: “eight thousand, two hundred, three times ten and seven”, accord-
ing to this mathematical breakdown of the components:
8 X 10 3 + 2 X 10 2 + 3 X 10 + 7 = 8,237.
As well as writing the number in terms of decreasing powers of ten, it
can also be written in the opposite order, in increasing powers of ten, start-
ing with the smallest unit, for example:
“Seven, three times ten, two hundred, eight thousand”.
This is exactly how the Indian astronomers expressed numbers when
they used the Sanskrit names of the numbers. Thus the preceding number
can be mathematically broken down in the following way:
7 + 3 X 10 + 2 X 10 2 + 8 x 10 3 = 8,237.
The method of expressing numbers that we are interested in here is the
“oral” method, because it uses Sanskrit words, the difference being that it
simply gives a succession of the corresponding names of the units, in keep-
ing with the method of representation that we have just seen. In other
words, there is no mention of the names which indicate the base and its
various powers (“ten”, “hundred”, “thousand”, etc.) Thus the preceding
number would be expressed in the following manner:
Seven. Three. Two. Eight.
In the same way, two.eight.nine.three corresponds to the value:
2 + 8 x 10 + 9 x 10 2 + 3 x 10 3 = 3,982.
In other words, the Sanskrit names for the numbers 1 to 10 had a vary-
ing value according to their position in the description of numbers of
several orders of units. In saying one, three, nine for 931 for example, the
word one is given the simple value of one unit, three is given the power of
ten and nine the value of a multiple of one hundred.
INDIAN CIVILISATION
412
Thus there can be no doubt that we are dealing with a decimal place-
value system. This seems even more remarkable when we consider that the
Indian scholars were the only ones to invent a system of this kind.
The above example, however, poses a fundamental question. We have
just seen that in this system, a number such as 931 can be expressed rela-
tively easily, by writing one, three, nine. On the other hand, it is difficult to
express a number such as 901, where there is an empty space, if you like, in
the decimal order (the “ten” column). To write this number, one could
obviously not simply write one, nine, because this would convey the
number 91 (= 1 + 9 x 10), and not 901. How, then, do we communicate that
there is nothing in the decimal order?
In other words, when one rigorously applies the place-value system to
the nine simple units, the use of a special terminology is indispensable to
indicate the absence of units in a certain order.
The Indian astronomers overcame this obstacle by using the Sanskrit word
*shunya meaning “void" and by extension “zero”. Thus they were able to
express the number 901 in words which can be translated in the following way:
One. Zero. Nine (= 1 + 0 x 10 + 9 x 10 2 = 901).
The word shunya (“zero”) actually became the concept it signified; it
played the role of zero in the place-value system, and thus prevented any
confusion as to the value of the number expressed.
If we return to the verse quoted above, the Sanskrit numerical expres-
sion agnishunyashvivasusarpdrnava (which represents the number 488,203)
can be broken down as follows:
agni.shunya.ashvi.vasu.sarpa.arnava
The words which act as components of this expression, however, are not
the ordinary Sanskrit names of numbers. They are word-symbols, the lit-
eral translation of which, due to the association of ideas which
characterises the Indian way of thinking, evoked a numerical value, rather
like the way that the words pair and triad evoke the numbers two and three
in our minds, except that the Sanskrit language had a greater choice of syn-
onyms. Indian astronomers nearly always chose to express their numerical
data using this almost infinite synonymy.
In order to represent the above number, the word-symbols appeared
with the value indicated below:
*agni = “fire” = 3
*shunya = “void” = 0
ashvi ( = *Ashvin) = “horsemen” = 2
*vasu = 8
*sarpa = “serpent” = 8
*arnava = “ocean” = 4
Thus one can translate the above expression in the following manner:
Fire.Void. Horsemen. Vitiii. Serpent. Ocean.
3 0 2 8 8 4
Remembering the earlier explanation of the system, we can see that the
number represented is:
3 + 0x 10 + 2 xl0 2 + 8xl0 3 + 8 xl0 4 + 4xl0 5 = 488,203.
The second numerical expression that appears in the verse is vas-
vagniyamashvishikhidasra, which can also be broken down in the following way:
vasv. agni.yama. ashvi. shikhi. dasra
These are also word-symbols possessing the following numerical values:
Vasv ( = *Vasu) =8
*agni = “fire” = 3
*yama = “Primordial Couple” =2
ashvi (= *Ashvin) = “Horsemen” =2
shiki ( = *Shikhin) = “fire” = 3
* dasra - “(one of the) Twins” = 2
Which is interpreted as:
Vfeu. Fire. Primordial Couple.Horsemen.Fire.Twins.
8 3 2 2 3 2
This corresponds to the number:
8 + 3 x 10 + 2 x 10 2 + 2 x 10 3 + 3 x 10 4 + 2 x 10 5 = 232,238.
This method of expressing numbers shows a perfect understanding of zero
and the place-value system using 10 as a base.
It is a type of symbolic representation subject to many variations, yet the
numerical symbols were always perfectly comprehensible to the Indian
astronomers. Even if the value of certain words could vary according to the
author, region or the time when they were written, the context always con-
firmed the intended numerical value.
Dating the Indian numerical word-symbols
When were these word-symbols first used? The answer is highly significant
because the concept of zero and the place-value system in India are at least
as old as this method of expressing numbers.
Dates on Sanskrit inscriptions from Southeast Asia
In India itself, as well as outside India, many documents exist which prove
that this method of counting was, for a great many years, the privileged
413
system of the Indian scholars, from the end of the sixth century at least
until a relatively recent date.
The dated Sanskrit inscriptions of Southeast Asia figure very promi-
nently amongst these documents.
It is important to make a clear distinction between the vernacular
inscriptions and those written in Sanskrit. Both, however, date back to
the Shaka era of the Indian astronomers. Primarily, in both types of
inscriptions, the dates were recorded in words using the Sanskrit names
for the numbers.
In the vernacular inscriptions (according to the region, written in Old
Khmer, Old Javanese, Cham, etc.), these dates were then expressed using
the nine numerals and zero of the Indian place-value system (Fig. 24.80).
In the Sanskrit inscriptions, however, the dates were recorded exclu-
sively in the Indian word-symbols observing the place-value system and
using 10 as the base. Here are some examples, taken from the oldest docu-
ments found in each of the regions in question.
The oldest dated Sanskrit inscription from Java is the Stela of Changal,
the Shaka date of which is expressed in the following way [see H. Kern,
VII, 118]:
shrutindriyarasair
This can be broken down into separate words:
shruti. indriya. rasair
On consulting the Dictionary, under the headings *shruti, *indriya and
*rasa ( = rasair), the following meanings are obtained:
* shruti - Veda = 4
*indriya = properties = 5
*rasair = senses = 6
Bearing in mind that the numbers are always written according to the
decimal place-value system, beginning with the smallest unit, in ascending
powers of ten (which the Indian astronomers called *ankanam vamato
gatih, or the principle of the “movement of the numbers [the numerical
symbols] from the right to the left”), we can see that the date in question
can be interpreted as:
Veda.Properties.Senses.
4 5 6
This corresponds to the number:
4 + 5x10 + 6x10 2 = 654.
THE PLACE-VALUE SYSTEM OF THE INDIAN NUMERICAL SYMBOLS
Thus the inscription in question dates back to the Shaka year 654 + 78
(732 CE).
The oldest dated inscription from Champa is the Stela of Mi-so’n, the
Shaka date of which is written in the following numerical symbols [see
G. Coedes and H. Parmentier (1923), C 74 B; BEFEO, XI, p. 266):
anandamvarashatshata
which can be translated as follows (bearing in mind that ananda means
the “(nine) Nanda”; amvara = *ambara = “space” = 0; and shatshata = “six
hundreds"):
Space. Six.Hundred.
9 0 6 x 100
which corresponds to the date: 9 + 0xl0 + 6x 100 = 609 + 78 Shaka (687 CE).
The use of the term shatshata to denote six hundred shows a certain
inexperience in the writing of numerical symbols, because the number 609
can be written anandamvarashat, which places the symbols for nine
{ananda), zero ( amvara ) and six (shat) in order.
The oldest dated Sanskrit inscription from Cambodia is that of Prasat
Roban Romas, in the province of Kompon Thom. This is also the oldest dated
Sanskrit inscription in the whole of Southeast Asia. It contains the following
Shaka date [see Coedes and Parmentier (1923), K 151; BEFEO, XLIII, 5, p. 6]:
khadvishara
Here is the literal translation (where *kha = “space” = 0; *dvi = “two” = 2;
and *shara = arrows = 5):
Space.Two.Arrows.
0 2 5
which corresponds to the date: 0 + 2 x 10 + 5 x 10 2 = 520 + 78 Shaka
(598 CE).
This proves that the use of Sanskrit word-symbols to express numbers
was already widespread in Indo-China and Indonesia at the end of the sixth
century CE.
As the civilisations were greatly influenced by Indian astronomers and
astrologers, we can quite rightly presume that Indian scholars were using
this technique at an even earlier date.
Evidence from the astronomers and mathematicians of India
There is a great deal of evidence pointing to the fact that the system was
used by Indian scholars from the sixth century CE until a relatively recent
INDIAN CIVILISATION
414
date, as the following (non-exhaustive) list of Indian texts (containing
many examples of the word-symbols) shows. The list is written in reverse
chronological order (after R. Billard, 1971):
1. Trishatika by Shridharacharya (date unknown) [B. Datta and
A. N. Singh (1938) p. 59]
2. Karanapaddhati by Putumanasomayajin (eighteenth century CE)
[K. S. Sastri (1937)]
3. Siddhantatattvaviveka by Kamalakara (seventeenth century CE)
[S. Dvivedi (1935)]
4. Siddhantadarpana by Nilakanthaso-mayajin (1500 CE) [K. V. Sarma
(undated)]
5. Drigganita by Parameshvara (1431 CE) [ Sarma (1963)]
6. Vakyapahchadhyayi (Anon., fourteenth century CE) [Sarma and
Sastri (1962)]
7. Siddhantashiromani by Bhaskaracharya (1150 CE) [B. D. Sastri (1929)]
8. Rajamriganka by Bhoja (1042 CE) [Billard (1971), p. 10]
9. Siddhantashekhara by Shripati (1039 CE) [Billard (1971), p. 10]
10. Shishyadhivrddhidatantra by Lalla (tenth century CE) [Billard
(1971), p. 10]
11. Laghubhdskariyavivarana by Shankaranarayana (869 CE) [Billard
(1971), p. 8]
12. Ganitasarasamgraha by Mahaviracharya (850 CE) [M. Rangacarya
(1912)]
13. Grahacharanibandhana by Haridatta (c. 850 CE) [Sarma (1954)]
14. Bhaskariyahhasya by Govindasvamin (c. 830 CE) [Billard (1971),
p. 8.]
15. Commentary on the Aryabhatiya by Bhaskara (629 CE)
[K. S. Shukla and K. V. Sarma (1976)]
16. Brahmasphutasiddhanta by Brahmagupta (628 CE) [S. Dvivedi
(1902)]
17. Pahchasiddhantika by Varahamihira (575 CE) [O. Neugebauer and
D. Pingree (1970)]
Examples taken from the work of Bhaskara I
We will now look at some examples in their original form, taken from some
of the oldest texts, which give a clearer indication than the above table of
the earliest uses of this system in India.
The first concerns an example of how the number of years (4,320,000)
that make up a *chaturyuga (see also *yuga) was expressed in word-
symbols. It is an extract from the commentary which Bhaskara I wrote in
629 CE on the * Aryabhatiya [see Shukla and Sarma (1976) p. 197]:
viyadambarakashashunyayamaramaveda
This can be broken down in the following manner:
viyad. ambara. akasha. shunya.yama. rama. veda
On consulting the Dictionary, the following meanings are obtained:
*viyat (here written viyad )
- “sky”
= 0
*ambara
= “atmosphere”
= 0
*akasha
= “ether”
= 0
*shunya
= "void”
= 0
*yama
= “(the) Primordial Couple”
= 2
*rama
= “(the) Rama”
= 3
*veda
= “(the) Veda”
= 4
This gives the following translation, with the corresponding mathematical
breakdown:
Sky.Atmosphere.Ether.Void.Primordial Couple.Rama.Veda.
0 0 0 0 2 3 4
= 0 + 0 x 10 + 0 x 10 2 + 0 x 10 3 + 2 x 10 4 + 3 x 10 5 + 4 x 10 6 = 4,320,000.
Here are three lines from the same work by Bhaskara (Commentary on the
Aryabhatiya, manuscript R 14850 of the Government Oriental Manuscript
Library, Madras, Dashagitika, [see R. Billard (1971), pp.105-6], in Sanskrit,
with the corresponding translation (the numerical word-symbols are
underlined to distinguish them from the rest of the text):
tadanayanam idanim kalpader adyanirodhdd ay am abdarashir itiritah
khagnyadriramarkarasavasurandhrendavah
te chankair api 1986123730.
Before we look at the translation, it should be noted that the above word-
symbols can be broken down in the following way:
kha.agny.adri. rama.arka. rasa. vasu. randhra. indavah
The Dictionary gives the following meanings for these words:
*kha
u n
= space
= 0
agny (= *agni)
= “fire”
II
w
*adri
= “mountains”
= 7
*rama
= "(the) Rama”
= 3
*arka
= “sun”
= 12
*rasa
«< *»
= senses
CO
II
*vasu
= 8
*randhra
= “orifices”
= 9
indavah (= *indu)
= “moon”
= 1
415
Thus the following translation is obtained for the preceding extract from
the Sanskrit text:
“In order to carry out the translation, here are the number of years
which have transpired since the beginning of the [current] *kalpa until the
present day:
Space. Fire. Mountain. Rdma. Sun. Sense, Orifice. Moon.
“In figures this reads (te chankair api): 1986123730”.
As with the above example, here is the meaning of the word-symbols:
Space. Fire. Mountain. Kdtfw. Sun. Sense. Vfrvu. Orifice. Moon.
03 7 3 12 689 1
This corresponds to the following number:
0 + 3 x 10 + 7 x 10 2 + 3 x 10 3 + 12 x 10 4
+ 6 X 10 6 + 8 X 10 7 + 9 x 10 8 + 1 x 10 9 = 1,986,123,730.
One might be surprised to find, in a place-value system, word-symbols
denoting values higher than or equal to ten, such as the word *arka
(= “sun" = 12) which is used here to express a number which contains two
orders of units. Later, however, we will see why this symbol is used here,
which does not constitute an exception to the rule of position where 10 is
the base. If in this example, the word arka, on its own, expresses the
number 12, it only acquires the value of 120,000 (= 12 x 10 4 ) because of the
place it occupies in the above expression.
Moreover, the value (1,986,123,730) of the preceding word-symbols is
clearly indicated “in figures” according to the place-value system, in the
third line of the Sanskrit text, accompanied by the words “in figures this
reads . . .”, evidently in order to prevent any ambiguity as to the intended
value. Thus we have a bilingual text of sorts which reinforces the above
explanations.
This is not the only instance where Bhaskara felt the need to give the
number in its corresponding numerals (using the place-value system of
nine units and zero) as well as in astronomical word-symbols. Here is
another example, this time involving a much higher number than the pre-
vious one [see Shukla and Sarma (1976), pp. 155]:
shunydmbarodadhiviyadagniyamdkdshasharasharddri-
shunyendurasdmbardngdnkddrishvarendu
ankair api 1779606107550230400.
As in the previous example, this compound word can be literally trans-
lated in the following way (given that: *shunya = 0, *ambara = 0, *udadhi
[= dadhi] = 4, viyad (= *vyant) = 0, *agni = 3, *yama = 2, *akasha = 0, *shara
THE PLACE-VALUE SYSTEM OF THE INDIAN NUMERICAL SYMBOLS
= 5, *shara = 5, *adri = 7, *shunya = 0, *indu = 1, *rasa = 6, *ambara = 0,
*anga = 6, *anka = 9, *adri = 7, *Ashva = 7 and *indu = 1), where the follow-
ing two consecutive expressions constitute two ways of writing the same
number according to the same principle:
Void.Sky.Ocean.Sky.Fire.Couple.Space.Arrow.Arrow.Mountain.
004032 0 5 5 7
Void.Moon.Sense.Atmosphere.Limb.Numeral.Mount.Horse.Moon
016 0 69 771
“In figures this reads: 1,779,606,107,550,230,400.”
The number expressed in word symbols is the one expressed “in
figures”; according to the text itself:
1,779,606,107,550,230,400.
Here Bhaskara uses the Sanskrit word anka, the “numerals”, not only to
indicate the equivalent of the number concerned in the place-value system
using nine numerals ( ankair api, “in figures this reads . . .”), but also to des-
ignate the number 9. This is of great importance, because the basic meaning
of anka is “a mark” or “a sign”, which by extension can mean “numeral”,
although there is no connection between its other meanings and the
number 9. Therefore, Bhaskara ’s use of anka to represent the number 9
proves that nine numerals and the place-value system were already being
used to write numbers in India when the commentary was written.
Bhaskara gives the number “in figures” as well as in word-symbols, and
this leaves no doubt that he was alluding to the nine basic numerals of the
decimal place-value system which was invented in India: which, along with
zero, enabled the Indian astronomers not only to represent any number,
however high it might have been, but also and above all to carry out any
mathematical operation with the minimum of complication.
Thus, in 629, the methods of expressing numbers either in numerals or
in word-symbols were widely recognised by the learned men of India.
Examples found in the work ofVarahamihira
Here are some more examples from the Pahchasiddhantika, the astronomical
work ofVarahamihira (VIII, lines 2,4 and 5). [See S. Dvivedi and G. Thibaut;
O. Neugebauer and D. Pingree] [Personal communication of Billard]:
1) How the number 110 is expressed:
shunyaikaika = *shunya*eka.eka
= void, one.one
Oil
= o+ixio + ixio 2 = no.
INDIAN CIVILISATION
416
2) How the number 150 is expressed:
khatithi = *kha.*tithi
= space.day
0 15
= 0 + 15x10 = 150.
3) How the number 38,100 is expressed:
khakharupashtaguna = *kha. *kha.*rupa.*ashta.*guna
= space.space.shape.eight.quality
= 00 18 3
= 0 + 0x10 + lx 10 2 + 8xl0 3 + 3xl0 4 =
38,100.
This astronomical text was written c. 575 CE. This proves that zero and the
place-value system were already in use in India in the second half of the
sixth century CE.
THE EARLIEST KNOWN EVIDENCE OF THE INDIAN
PLACE-VALUE SYSTEM
We will now look at the most important source of evidence relative to the
history of the place-value system: the *Lokavibhaga (or The Parts of the
Universe), a work on *Jaina cosmology which constitutes the oldest known
use of word-symbols.
Besides the fact that the “minus one” is expressed by ruponaka (literally:
"diminished form”, rupo = *rupa = “shape” or “form” = 1) and that the con-
cept of zero is expressed by *shunya (void) or by words such as *kha,
*gagana or *ambara (“sky”, “atmosphere”, “space”, etc.), we find the follow-
ing expression used for the number 14,236,713 [source: Anon. (1962),
Chapter III, line 69, p. 70] [Personal communication of Billard]:
triny ekam sapta shat trini dve chatvary ekakam
As the words used here are all names of numbers, they can be translated as fol-
lows (given that eka - 1 [= ekaka, the suffix ka here being a device used to
regulate the metre of the line]; dve = 2; trini = 3; chatvary = 4; shat = 6; sapta = 7):
Three.One.Seven.Six.Three.Two.Four.One
3 1 7 6 3 2 4 1
(= 3 + 1 x 10 + 7 x 10 2 + 6 x 10 3 + 3 x 10 4 + 2 x 10 5 + 4 x 10 6 + 1 x 10 7 =
14,236,713).
The author of this text seems generally to have avoided the abundant syn-
onyms for the numerals and chosen to almost exclusively use the ordinary
Sanskrit names of the numbers (eka, dvi, tri, chatur, pahcha, etc.).
The reason for this is, perhaps, that the word-symbols were not suffi-
ciently well-known outside “learned” circles. However, there is another
probable reason: the author wanted to make his work accessible in order to
promote the merits of the philosophy of his religion and the superiority of
Jaina science to the public at large, and therefore avoided technical terms.
Nevertheless, at times the author does use certain word-symbols, as in
this expression of the number 13,107,200,000 [see Anon. (1962), Chapter
IV, line 56, p. 79]:
pahchabhyah khalu shunyebhyah param dve sapta
chambaram ekam trini cha rupam cha...
five voids, then two and seven, the sky, one and three and the form
00000 2 7 0 1 3 1
(= 0 + 0 x 10 + 0 x 10 2 + 0 x 10 3 + 0 x 10 4 + 2 x 10 5 + 7 x 10 6 + 0 x 10 7 + 1 x
10 8 + 3 x 10 9 + 1 x 10“ = 13,107,200,000).
However, each time the author uses one of these expressions, careful not to
confuse his readers, he feels obliged to:
• either be more precise by adding:
*kramat, “in order”,
or *sthanakramad, “in positional order ( *sthana )”
• or, which is even more remarkable, to add the following explanation:
*ankakramena, in the order of the numerals ( *anka )”.
In other words, the concept of zero and the place-value system was wide-
spread in India in the fifth century CE and had probably already been
known for some time in “learned” circles.
In fact, the *Lokavibhaga is the oldest known authentic Indian docu-
ment to contain the use of zero and decimal numeration. As we shall see, it
dates back to the middle of the fifth century CE.
We even know the exact year of the document thanks to the following
verses [see Anon. (1962), Chapter XI, lines 50-54, pp. 224ff.] [Personal
communication of Billard]:
vaishve sthite ravisute vrshabhe cha jive
rajottareshu sitapaksham upetya chandre
grame cha patalikanamani panarashtre
shastram pura likhitavan munisarvanandi (verse 52)
samvatsare tu dvavimshe kahchishah simhavarmanah
ashityagre shakabdanam siddham etach chhatatraye (verse 53).
Here is the translation:
Verse 52: “This work was written long ago by the Muni Sarvanandin, in
the town called Patalika, in the kingdom of Pana, when Saturn was in
417
Vaishva, Jupiter in Taurus, the Moon in Rajottara, on the first day of the
light fortnight.”
Verse 53: “Year twenty-two [of the reign] of Simhavarman, king of
Kanchi, three hundred and eighty Shaka years.”
In verse 52, we are told that when the text (or the copy of it) was written,
the Moon was in Rajottara. This word means the nakshatra' called
Uttaraphalguni : one of the twenty-seven constellations of the sidereal
sphere, divided according to the sidereal revolution of the Moon. As it is
the tenth constellation which is referred to here, this position corresponds
(according to reliable mathematical calculations) to the interval between
146° 40' and 160° of sidereal longitude. We are also told that the Moon was
in its phase corresponding to the first day of the “light fortnight”: the first
half of the month. We can determine that the work was written in the
Shaka year 380, the corresponding date being written “entirely in letters”
using the ordinary Sanskrit names for the numbers.
Looking at the information given in the verses, which has been inter-
preted according to the elements of Indian history and astronomy, we have:
• the year, namely the Shaka year 380;
• the day of the month, in other words, the Moon is in the first day of
the first fortnight of the month;
• and the position of the Moon: 146° 40' / 160° of sidereal longitude,
which allows us to determine the month.
Without going into too much detail about the methodology used to deter-
mine the dates and to study the astronomical data, suffice to say that the
information given leaves us in no doubt as to the date expressed here; the
date, in the Julian calendar, corresponds exactly to:
Monday, 25 August, 458 CE.
This is the precise date of the Jaina cosmological text, * Lokavibhaga (or The
Parts of the Universe”).*
We can now add the other two pieces of information given in verse 52:
the planet Jupiter was in Taurus, in the second sign of the zodiac, thus
occupying a position of 30° to 60° of sidereal longitude; at the same time,
* Here, this word is used to explain the “lunar mansions" in equal divisions. See * Nakshatra.
+ We also see in verse 53 that this text is dated the 22nd year of the rule of Simhavarman, king of Kanchi
(the “Golden Town”, sacred place of the Hindus, in Tamil Nadu, approximately 60 km southwest of
Madras). According to Frederic, DO (1987), pp. 819-20, this king, the son of Skandavarman II, issued from
ones of the lines of the Pallava Dynasty, reigned from 436. As this was the 22nd year of his reign, this date
corresponds to 436 + 22 = 458 CE. We do not know, however, if the chronology of this sovereign was estab-
lished by specialists using the text of the * Lokavibhaga. If this is the case, then this information is of no
interest to us. On the other hand, if this is not the case, if the dates of the reign of Simhavarman were estab-
lished from another inscription, then we have real confirmation of the date we have just determined using
the astronomical data in the text in question.
THE EARLIEST KNOWN EVIDENCE OF THE INDIAN PLACE-VALUE SYSTEM
Saturn was in Vaishva, the *nakshatra called Uttarashadha (the nineteenth
constellation of the sidereal revolution), therefore between 266° 40' and
280° of sidereal longitude. As this data agrees with the preceding date, the
date is astronomically confirmed.
Whilst this information allows us to date the Lokavibhaga with preci-
sion, it also irrefutably proves the authenticity of the document, due to the
very nature of one of the preceding pieces of astronomical data.
Because Jupiter is situated in the text according to its position in a zodia-
cal sign, we can also find, for astrological reasons, the position of Saturn in
the * nakshatra system.
This is an irrefutable archaism characterised by the very history of Indian
astrology. After this time, there are no more examples where the positions of
the planets (with the exception of the Moon) are described in *nakshatra,
they are only expressed in relation to the position of the twelve signs of the
zodiac (previously unpublished information given by Billard).
The very existence of this archaism and its almost total disappearance
from later Indian texts prove the complete authenticity of its usage, of the
document, and all the information it gives us. Moreover, the Lokavibhaga as
a whole, from an astronomical and cosmological point of view, is undeni-
ably archaic in character in comparison with later texts of the same genre.
Let us now look even more closely at the problem in hand. This text was
“written” long before by a Muni named Sarvanandin, but the word “writ-
ten” is ambiguous because in Sanskrit it can mean “copied” as well as
“written”. The Lokavibhaga appears to be the Sanskrit translation of an ear-
lier work written in Prakrit (probably in a Jaina dialect), judging from the
translation of verse 51:
The *Rishi Simhasura translated into the Language [= Sanskrit] that
which the uninterrupted line of doctors had transmitted [in dialect],
which the revered arhant Vardhamana [= the *Jina] delivered to the
saints during the grand assembly of the gods and men, namely all that
[the disciples of Jina such as] the Sudharma know about the creation of
the universe. Let him be praised by all ascetics.
This could and very probably does mean that the current version of the
Lokavibhaga is an exact reproduction of an original which was written
before 458 CE.
Of course we must be wary of relatively recent Indian texts which are
frequently attributed to the *Rishi, the “Sages” of the Vedic era (twelfth to
eighth century BCE) who are said to have received the great “Revelation”
from the divinities.
The Lokavibhaga, however, is much more modest, as it attributes its
writing to a Muni. This Muni could well have lived one or two generations
before the above date.
INDIAN CIVILISATION
418
This seems even more likely when we consider that, on the one hand,
the numbers which appear in this text conform totally to the rules of the
decimal place-value system, and on the other hand, the care that the author
took to popularise the text. As we have already seen, when this text was
written, Indian scholars were already familiar with the place-value system.
Who, then, is a Muni ? The answer to this question is in the text itself, in
verse 50:
Muni is he who achieves perfection, and, displaying [the] strength of a
lion, escapes the terrible [cycle of renaissance], through obeying the
decree of respect to all animal life, the exercises of piety such as the
vow of honesty, the holiness which conquers all false doctrine and all
futility, dominates the empire of the senses, and even defeats the eter-
nal Karma through the fire of fervent austerities.
That, in a nutshell, is the doctrine of Jaina, as well as what became of the
Muni Sarvanandin to whom the writing of the Lokavibhaga is attributed.
When did this Muni live? A hundred or two hundred years previously?
We will never know. What we do know for certain is that the discovery of
our present-day numeral system was made well before that famous
Monday 25 August, 458 CE.
HIGHLY CONSISTENT EVIDENCE
Considering the quantity and extreme diversity of the information con-
tained in this chapter, it would seem appropriate to present a summary of
all the historical facts which have been established concerning the discov-
ery of zero and the place-value system. The following is a list in reverse
chronological order, with references to the Dictionary for those wishing to
know more details.
Summary of the historical facts relating to the place-value system
1150 CE. The Indian mathematician ‘Bhaskaracharya (known as Bhaskara
II) mentions a tradition, according to which zero and the place-value
system were invented by the god Brahma. In other words, these notions
were so well established in Indian thought and tradition that at this time
they were considered to have always been used by humans, and thus to
have constituted a "revelation” of the divinities. See *Place-value system.
1010-1030 CE. Date of evidence given by the Muslim scholar of Persian
origin, *al-Biruni, about India and in particular her place-value system and
methods of calculation; a highly documented piece of evidence to add to
the others from the Arabic-Muslim world and the Christian West.
End of the ninth century CE. The philosopher ‘Shankaracharya makes a
direct reference to the Indian place-value system.
875-876 CE. The dates of the inscriptions of Gwalior: the oldest known
“real” Indian inscriptions in stone to use zero (in the form of a little circle)
and the nine numerals (in Nagart) according to the place-value system. See
*Nagari numerals, and Figs. 24.72 to 74.
869 CE. The Indian astronomer ‘Shankaranarayana frequently uses the
place-value system with word-symbols.
c. 850 CE. The Indian astronomer *Haridatta invents a system of
numerical notation using letters of the Indian alphabet and based on the
place-value system using zero (randomly represented by two different let-
ters): this is the first example of a place-value system which uses letters of
the alphabet. See *Katapayadi numeration.
850 CE. The Indian mathematician ‘Mahaviracharya frequently uses
the place-value system with the nine numerals or with Sanskrit numerical
symbols [M. Rangacarya (1912)].
c. 830 CE. The Indian astronomer ‘Govindasvamin frequently uses the
place-value system [R. Billard (1971), p. 8].
813 CE. This is the date of the oldest known vernacular inscription of
Champa (Indianised civilisation of Southeast Asia), the Shaka date of which
is indicated using the nine Indian numerals and zero. See *Cham numerals,
and Fig. 24.80.
760 CE. The date of the oldest known vernacular inscription of Java, the
Shaka date of which is expressed using the nine numerals and zero from
India. See *Kawi numerals, and Fig. 24.80.
732 CE. Date of the oldest known Sanskrit inscription from Java, the
Shaka date of which is expressed using the place-value system and word-
symbols of the Indian astronomers [H. Kern (1913-1929)].
718-729 CE. Date of the Kai yuan zhan jing, a work on astronomy and
astrology by the Chinese Buddhist *Qutan Xida, who was in fact of Indian
origin, real name ‘Gautama Siddhanta, who lived in China from c. 708 CE,
and who, in his work, describes zero, the place-value system of the nine
numerals and the Indian methods of calculation.
Seventh century CE. The poet ‘Subandhu makes direct references to the
Indian zero (in the form of a dot) as a mathematical processing device. Thus
zero and the place-value system were so well-established in India that the poet
could use such subtleties with his metaphors. See ‘Zero and Sanskrit poetry.
687 CE. Date of the oldest known Sanskrit inscription of Champa,
the Shaka date of which is expressed using the place-value system and
the word-symbols of the Indian astronomers [G. Coedes and
H. Parmentier, C 74 B;BEFEO, XI, p. 266].
683 CE. The date of the oldest known vernacular inscription from
Malaysia, the Shaka date of which is written in the Indian numerals (includ-
ing zero). See Fig. 24.80.
419
HIGHLY CONSISTENT EVIDENCE
683 CE. Date of the oldest known vernacular inscription of Cambodia,
the Shaka date of which is written in Indian numerals (including zero). See
*01d Khmer numerals and Fig. 24.80.
662 CE. Syrian bishop Severus Sebokt writes of the nine numerals and
Indian methods of calculation.
629 CE. Indian mathematician and astronomer ‘Bhaskara I frequently uses
the place-value system with the word-symbols, often also expressing the number
using the nine numerals and zero [K. S. Shukla and K. V. Sarma (1976)].
628 CE. Indian astronomer and mathematician Brahmagupta fre-
quently uses the place-value system with the nine numerals as well as with
the word-symbols. He also describes methods of calculation using the nine
numerals and zero (very similar to the methods we still use today). He also
provides fundamental rules of algebra, where zero is present as a mathe-
matical concept (the number nought), and talks of infinity, defining it as
the opposite of zero. See *Zero. ‘Infinity. *Khachheda.
598 CE. Date of the oldest known Sanskrit inscription of Cambodia, the
Shaka date of which is written in word-symbols according to the place-
value system [Coedes and Parmentier, K 151; BEFEO, XLIII, 5, p. 6],
594 CE. Date of the donation charter engraved on copper of Dadda III of
Sankheda, in Gujarat. This is the oldest known Indian text to bear witness
to the use of the nine numerals according to the place-value system (see
Fig. 24.75). As we saw earlier, there can be no doubt as to the authenticity
of this document.
End of the sixth century CE. The arithmetician ‘Jinabhadra Gani gives
several numerical expressions which prove that he was well acquainted
with zero and the place-value system [Datta and Singh (1938)].
c. 575 CE. Indian astronomer and astrologer ‘Varahamihira makes fre-
quent use of the place-value system with Sanskrit numerals. See ‘Indian
astrology.
c. 510 CE. ‘Aryabhata invented a unique method of recording numbers
which required perfect understanding of zero and the place-value system.
Moreover, he used a remarkable process of calculating square and cube
roots, which would have been impossible without the place-value system,
using nine different numerals and a tenth sign which performed the func-
tions of zero. See ‘Aryabhata (Numerical notations of), ‘Aryabhata’s
numeration, ‘Square roots (How Aryabhata calculated his).
(Monday 25 August) 458 CE. The exact date of the *Lokavibhaga, ( The
Parts of the Universe ), the Jaina cosmological text: the oldest known Indian
text to use zero and the place-value system with word-symbols.
Thus one can see the impressive amount of evidence proving that our
modern number-system is of Indian origin, and that it was invented long
before the sixth century CE. All the evidence points to the fact that this
invention is entirely Indian, and born out of a very specific context.
Moreover, we are not dealing with one isolated piece of evidence, or
even a limited number of documents, but a huge collection of proofs from
all the disciplines, dating from the most significant eras, which have been
situated through the study of the palaeography, epigraphy and philology of
Indian civilisations both within and outside India.
THE MOST LIKELY TIME OF THESE DISCOVERIES
It is most likely that the place-value system and zero were discovered in the
middle of the reign of the Gupta Dynasty, whose empire stretched the
whole length of the Ganges Valley and its tributaries from 240 to approxi-
mately 535, known as the “classic” period.
This period saw the highest forms of Indian art (sculpture, painting, in
the caves of Ajanta for example, etc.) reach maturity. It was also a classic
period because, as Coomaraswamy said, "almost everything that belongs to
the Asian spiritual conscience is of Indian origin and dates back to the
Gupta Dynasty.”
This era coincides with a kind of rebirth of Brahmanism, before it
evolved in the wider sense of Hinduism in the following centuries.
Trade was also flourishing at this time, with the Near East, via Persia,
and across the sea with the Roman Empire, particularly through Lata or the
eastern area of the present-day state of Gujarat.
Medicine was also developing at this time, particularly dissection.
In the field of literature, Sanskrit, previously the official language of the
court and of Brahmanism, was adopted by the Jainas and Buddhists, who
did much for the development of the language. And it was probably in this
period too that Sanskrit grew to be a much richer language than it had
been in the time of the Vedas. This time also saw the beginnings of the
*Mahdbharata, one of the greatest Indian epic poems, and of the
Dharmashastra, collections of texts, mainly about customs, laws and castes.
It was during this time that the *Darshana - the six systems of Indian
philosophy - were developed.
The stories and fables, such as those of the *Pahchatantra, (the main
source of inspiration for the Persian fable Kalila wa Dimna), also appeared
for the first time, whilst the theatre knew its first blossoming with the poet
Kalidasa, considered to be one of the greatest dramatists of Indian history,
and the *Navaratna or “Nine Jewels” of Indian tradition.
As for Indian writing, Gupta constitutes the first notation to be individ-
ualised in relation to its Brahmi ancestor. As it became more refined, it gave
INDIAN CIVILISATION
420
birth to Nagari (or Devanagari), in the seventh century CE, which became
the principal style in which Sanskrit and then Hindi were written. From
Nagari came the various styles of northern and central India. Another,
more northern variant of Gupta, evolved into Sharada of Kashmir, or its
derivatives, and also diversified into Siddham, from which the script of
Nepal, Chinese Turkestan and Tibet would be derived (Fig. 24.52). (See
‘Indian numerals).
Thus the Gupta period saw the most spectacular progress in almost all
the fields of learning, and was a veritable “explosion" of Indian culture.
This was also the time when *Lalitavistara Sutra was written, which tells
the legend of Buddha and mentions numbers of the highest orders, follow-
ing very surprising numerical speculation; speculation which grows rapidly
after this period, but for which there is no evidence before this time.
It is doubtless no coincidence that the Gupta era saw the first blossom-
ing of ‘Indian mathematics.
This was also the time of the first developments of trigonometrical
astronomy and “Greek” astrology, which was very different from that
which existed previously in India, both in terms of claims and material, and
which, being in appearance very systematic, already had the scientific foun-
dations of what would soon become Indian astronomy.
Moreover, this was the time when Aryabhata lived. His work would
soon lead to a decisive about-turn in Indian astronomy, breaking once and
for all with the old Greek-Babylonian traditions and developing the cosmic
cycles called *yuga, devoid of physical value but nonetheless based on a
series of unique observations which were more or less precise.
The Lokavibhaga is dated 458 CE. This being the oldest known testi-
mony of the use of zero and of the Indian decimal place-value system, the
latest possible date of this discovery has to be the middle of the Gupta era.
Documents written earlier or at the same time show use of either the ordi-
nary system of the Sanskrit names of the numbers or, as we shall see in the
following chronology, that of the old non-positional system derived from
the Brahmi system (Fig. 24.70).
Thus the earliest possible date of this discovery is the beginning of the
Gupta Dynasty. We must take into consideration, however, the fact that docu-
ments bearing witness to the use of word-symbols or the decimal place-value
system are only found in abundance after the beginning of the sixth century.
Bearing in mind, on the one hand, the perfect understanding of the
place-value system displayed in the Lokavibhaga and the clear desire to pop-
ularise the text, and on the other hand the fact that the text was more than
likely a Sanskrit translation of an earlier document (no doubt written in a
Jaina dialect), it would not be unreasonable to suggest the fourth century CE
as the date of the discovery of zero and the place-value system.
Third to second century BCE. First appearances of Brahmi numerals in the
edicts of Emperor Asoka and the inscriptions of Nana Ghat. These are very
rudimentary. But the first nine figures already constitute the prefiguration of
the nine numerals that we use today (Indian, then Arabic and European).
Sanskrit numerals are already worked out and there are particular
names for the ascending powers of ten up to 10 8 (= 100,000,000) at least.
First century BCE to third century CE. The numerals found in many
inscriptions are derived from Brahmi numerals and constitute a sort of
intermediary between Brahmi numerals and later styles, but the place-value
system is not yet in use.
The Sanskrit system is extended to include powers of ten up to 10 12 (=
1,000,000,000,000). [See ‘Names of numbers)
Fourth to fifth century CE. The numerals derived from Brahmi numerals
begin to diversify into specific styles (Gupta, Pali, Pallava, Chalukya, etc.)
The Sanskrit system is capable of expressing and using powers of ten up
to 10 421 and above, as we see in the * Lalitavistara (before 308 CE).
Discovery of zero and the place-value system
458 CE. (To this day, no document has been found to prove that the nine units
were used at this date according to the place-value system.)
The names of the first nine numbers are used according to the place-value
system, as we shall see in the * Lokavibhaga, dated 458 CE, where the names
of the numbers are sometimes replaced by word-symbols and the word
*shunya (“void”) and its synonyms are used as zeros.
From the sixth century onwards. The use of the place-value system and
zero begin to appear frequently in documents from India and Southeast
Asia (the following list is non-exhaustive):
594 CE. Sankheda charter on copper
628 CE. Brahmasputasiddhanta by Brahmagupta
629 CE. Commentary on the Aryabhatiya by Bhaskara
683 CE. Khmer inscription of Trapeang Prei
683 CE. Malaysian inscription of Kedukan Bukit
684 CE. Malaysian inscription of Talang Tuwo
686 CE. Malaysian inscription of Kota Kapur
737 CE. Charter of Dhiniki on copper
753 CE. Inscriptions of Devendravarmana
760 CE. Javanese inscription of Dinaya
793 CE. Charter of Rashtrakuta on copper
813 CE. Cham inscription of Po Nagar
815 CE. Charter of Buchkala on copper
421
A CULTURE WITH A PASSION FOR HIGH NUMBERS
829 CE. Cham inscription of Bakul
837 CE. Inscription of Bauka
850 CE. Ganitasaramgraha of Mahaviracharya
862 CE. Inscription of Deogarh
875 CE. Inscriptions of Gwalior
877 CE. Balinese inscription of Haliwanghang
878 CE. Balinese inscription of Mamali
880 CE. Balinese inscription of Taragal
917 CE. Charter on copper of Mahipala, etc.
Seventh century CE. Gupta notation gave birth to Nagari numerals, which in
turn were the forerunners of the numerals of northern and central India
( Bengali , Gujarati, Oriya, Kaithi, Maithili, Manipuri, Marathi, Marwari, etc,).
Eighth century CE. First appearance of the stylised numerals of Southeast
Asia (Khmer, Cham, Kawi, etc.).
Ninth century CE. A northern variant of Gupta led to the Sharada numer-
als of Kashmir, ancestors of the numerals of northwest India ( Dogri , Takari,
Multani, Sindhi, Punjabi, Gurumukhi, etc.).
Eleventh century CE. The first appearances of Telugu numerals
(southern India).
A CULTURE WITH A PASSION FOR
HIGH NUMBERS
The early passion which Indian civilisation had for high numbers was a sig-
nificant factor contributing to the discovery of the place-value system, and
not only offered the Indians the incentive to go beyond the "calculable”,
physical world, but also led to an understanding (much earlier than in our
civilisation) of the notion of mathematical infinity itself.
The Indian love for high numbers can be seen in the * Lalitavistara Sutra
or Development of the Games [of Buddha] (a Sanskrit text of the Buddhism of
Mahayana, written in verse and prose, about the life of Buddha, the “Saint
of the Shaka family", as he is said to have told his disciples), where high
numbers are constantly evoked:
Choosing a few random examples, we find in this text a meeting of
ten thousand monks, eighty-four million Apsaras, thirty-two thou-
sand Bodhisattvas, sixty-eight thousand Brahmas, a million Shakras,
a hundred thousand gods, hundreds of millions of divinities, five
hundred Pratyeka-Buddhas, eighty-four thousand sons of gods, then
thirty-two thousand and thirty-six million other sons of gods, sixty-
eight thousand *kotis [= 680,000,000,000] sons of gods and
Bodhisattva, eighty-four hundred thousand *niyuta kotis [=
8,400,000,000,000,000,000,000] of divinities.
The principal signs of Buddha are given the number thirty-two, sec-
ondary signs eighty, signs of his mother thirty-two, those of the
dwelling-place and the family where he is said to have been born eight
and sixty-four. The queen Maya-Devi is served by ten thousand
women; the ornaments of the throne of Buddha are enumerated in
hundreds of thousands; the hundreds of thousands of divinities and
hundred thousand millions of Bodhisattvas and Buddhas pay homage
to this throne which is the result of merits accumulated over one hun-
dred thousand million *kalpas, one kalpa being the equivalent of four
billion, three hundred and twenty million years. The lotus flower that
blossomed the night that Buddha was conceived has a diameter of
sixty-eight million yojana. Two hundred thousand treasures appeared
when Buddha was born; this filled the three thousand great hosts of
worlds, and living creatures came to pay homage to his mother, the
queen Maya-Devi, in throngs of eighty-four thousand and sixty thou-
sand [F. Woepcke (1863)].
Likewise, in The Light of Asia, Edwin Arnold reproduces this passage from
the Lalitavistara Sutra, about the education of Buddha as a child, aged
eight, by the Sage Vishvamitra, who explains, in another passage, that
numeration, numbers and arithmetic constitute the most important dis-
cipline among the seventy-two arts and sciences that the Bodhisattva
must acquire:
And Vishvamitra said: That’s enough [now],
Let us turn to Numbers. Count after me
Until you reach *lakh (- one hundred thousand):
One, two, three, four, up to ten,
Then in tens, up to hundreds and thousands.
After which, the child named the numbers,
[Then] the decades and the centuries, without stopping.
[And once] he reached lakh, [which] he whispered in silence,
Then came *koti, *nahut, *ninnahut, *khamba,
*viskhamba, *abab, *attata,
Up to *kumud, *gundhika, and *utpala
[Ending] with *pundarika [leading]
Towards *paduma, making it possible to count
Up to the last grain of the finest sand
Heaped up in mountainous heights.
Let us interrupt the master for a moment to clarify the numerical values
mentioned in the passage:
INDIAN CIVILISATION
422
lakh
is worth
koti
is worth
nahut
is worth
ninnahut
is worth
khamba
is worth
viskhamba
is worth
abab
is worth
attata
is worth
kumud
is worth
gundhika
is worth
utpala
is worth
pundarika
is worth
paduma
is worth
100,000 = 10 s
10,000,000 = 10 7
1,000,000,000 = 10 9
100,000,000,000 = 10 u
10,000,000,000,000 = 10 13
1,000,000,000,000,000 = 10 15
100,000,000,000,000,000 = 10 17
10,000,000,000,000,000,000 = 10 19
1,000,000,000,000,000,000,000 = 10 21
100,000,000,000,000,000,000,000 = 10 23
10,000,000,000,000,000,000,000,000 = 10 25
1,000,000,000,000,000,000,000,000,000 = 10 27
100,000,000,000,000,000,000,000,000,000 = 10 29
Thus we are dealing with a centesimal scale, the value of each name being
one hundred times bigger than the one preceding it.
But beyond this counting system,
There is the katha which is used to count the stars in the night sky.
The koti-katha for [enumerating] the drops of the ocean,
Ingga, to calculate the circular [movements],
Sarvanikchepa, with which it is possible to calculate
All the sand of a Gunga,
Until we reach antahkapa,
Which is [made up of ten] Gungas.
[And] if a more intelligible scale is required,
The mathematical ascensions, through the *asankhya, which is the sum
Of all the drops of rain which, in ten thousand years,
Would fall each day on all the worlds,
Lead [the arithmetician] to the *mahakalpa,
Which the gods use to calculate their future and their past.
THE LIMITATIONS OF THE (INDIAN) “INCALCULABLE”
The asankhya or *asankhyeya, which was poetically defined as “the sum of
all the drops of rain which, in ten thousand years, would fall each day on all
the worlds”, is actually none other than the Sanskrit term meaning “ incal-
culable". It literally means: “number which is impossible to count” (from
*sankhya or sankhyeya, “number”, accompanied by the privative “a”).
This word is used in Brahman cosmogony, where it is sometimes used to
denote the length of the “*day of Brahma”, in other words 4,320,000,000
human years.
In *Bhagavad Gita, however, “incalculable” corresponds to the entire
length of Brahma’s life, which is 311,040,000,000,000 human years. In
one of the commentaries on the work, it is pointed out that “this incredi-
ble longevity, for us infinite, represents no more than zero in the stream
of eternity.”
Naturally, the value given to “the incalculable” varies considerably
according to the text, the author, the region and the era. Thus, the
*Sankhyayana Shrauta Sutra fixes this limit at 10,000,000,000,000 giving
this number the name *ananta, signifying “infinity” [see Datta and Singh
(1938) p. 10)]. The Tibetans and the Sinhalese pushed the limit of
*asankhyeya much fiirther in giving it a value of one followed by ninety-
seven zeros. In the Pali Grammar of Kachchayana, the same concept is
given a value of 10 14 ° (ten million to the power of twenty), placing this term
at the end of this very impressive nomenclature the scale of which is tens of
millions [see JA 6/17 (1871), p. 411, lines 51-2)]:
A hundred times a hundred times a thousand makes a
A hundred times a hundred times a thousand koti makes a
A hundred times a hundred times a thousand pakoti
A hundred times a hundred times a thousand kotippakoti
A hundred times a hundred times a thousand nahuta
A hundred times a hundred times a thousand ninnahuta
A hundred times a hundred times a thousand akkhobhini
A hundred times a hundred times a thousand bindu
A hundred times a hundred times a thousand abbuda
A hundred times a hundred times a thousand nirabbuda
A hundred times a hundred times a thousand ahaha
A hundred times a hundred times a thousand ababa
A hundred times a hundred times a thousand atata
A hundred times a hundred times a thousand sogandhika
A hundred times a hundred times a thousand uppala
A hundred times a hundred times a thousand kumuda
A hundred times a hundred times a thousand pundarika
A hundred times a hundred times a thousand paduma
A hundred times a hundred times a thousand kathana
A hundred times a hundred times a thousand mahakathana
koti
= 10 7
pakoti
= 10 14
kotippakoti
= 10 21
nahuta
= 10 28
ninnahuta
= 10 35
akkhobhini
= 10 42
bindu
= 10 49
abbuda
= 10“
nirabbuda
= 10 63
ahaha
= 10 7 °
ababa
= 10 77
atata
= 10 84
sogandhika
= 10 91
uppala
= 10 98
kumuda
= 10 105
pundarika
= 10 u2
paduma
= 10 U9
kathana
= 10 126
mahakathana
= 10 133
asankhyeya
= 10 14 °
The extravagant numbers of the legend of Buddha
Thus we can see the extent to which the Indians took their naming of numbers.
We can get an even clearer idea of this if we return to the * Lalitavistara
Sutra, where Bodhisattva (Buddha), now an adult, is almost forced to take
part in a competition:
423
THE LIMITATIONS OF THE (INDIAN) "INCALCULABLE”
When Bodhisattva reached a marriageable age, he was betrothed to
Gopa, the daughter of Shakya Dandapani. But Dandapani refused to let
him marry his daughter, unless the son of the king Shuddhodana
[Bodhisattva] made a public show of his mastery of the arts. Thus a
type of contest, the winner of which would be given Gopa’s hand in
marriage, took place between Bodhisattva and five hundred other
young Shakyas. This contest included writing, arithmetic, wrestling and
archery [F. Woepcke (1863)].
After easily beating all the young Shakyas, Bodhisattva was invited by his
father to pit his wits against the great mathematician Arjuna, who had
judged the contest:
“Young man,” said Arjuna, “do you know how we express num-
bers that are higher than a hundred *kotiV
Bodhisattva nodded, but Arjuna impatiently continued:
“So how do we count beyond a hundred *koti in hundreds?"
Here is Bodhisattva ’s reply, bearing in mind that one *koti is
the equivalent of ten million (= 10 7 ):
“One hundred koti are called an *ayuta, a hundred ayuta make a
*niyuta, a hundred niyuta make a *kankara, a hundred kankara
make a *vivara, a hundred vivara are a *kshobhya, a hundred kshob-
hya make a *vivaha, a hundred vivaha make a * utsanga, a hundred
utsanga make a *bahula, a hundred bahula make a *ndgabala, a
hundred nagabala make a *titilambha, a hundred titilambha make
a *vyavasthdnaprajhapati, a hundred vyavasthanaprajnapati make a
*hetuhila, a hundred hetuhila make a *kamhu, a hundred karahu
make a *hetvindriya, a hundred hetvindriya make a *samapta-
lambha, a hundred samaptalambha make a *ganandgati, a hundred
gananagati make a *niravadya, a hundred niravadya make a
*mudrabala, a hundred mudrabala make a *sarvabala, a hundred
sarvabala make a *visamjhagati , a hundred visamjnagati make a
*sarvajha , a hundred sarvajna make a *vibhutangamd, a hundred
vibhutangama make a *tallakshana.”
Thus, in his reply, Bodhisattva had given the following table:
1 ayuta
= 100 koti
= 10 9
1 niyuta
= 100 ayuta
= 10 u
1 kankara
= 100 niyuta
= 10 13
1 vivara
= 100 kankara
= 10 15
1 kshobhya
= 100 vivara
= 10 17
1 vivaha
= 100 kshobhya
= 10 19
1 utsanga
= 100 vivaha
= 10 21
1 bahula
= 100 utsanga
= 10 23
1 nagabala
= 100 bahula
= 10 25
1 titilambha
= 100 nagabala
= 10 27
1 vyavasthanaprajnapati
= 100 titlambha
= 10 29
1 hetuhila
= 100 vyavasthanaprajnapati
= 10 31
1 karahu
= 100 hetuhila
= 10 33
1 hetvindriya
= 100 karahu
= 10 35
1 samaptalambha
= 100 hetvindriya
= 10 37
l gananagati
= 100 samaptalambha
= 10 39
1 niravadya
= 100 gananagati
Tl<
o
II
1 mudrabala
= 100 niravadya
= 10 43
1 sarvabala
= 100 mudrabala
= 10 45
1 visamjnagati
= 100 sarvabala
= 10 47
1 sarvajna
= 100 visamjnagati
= 10 49
1 vibhutangama
= 100 sarvajna
= 10 51
1 tallakshana
= 100 vibhutangama
= 10 53
In modern terms, the value of the tallakshana corresponds to the following
formula:
1 * tallakshana = (10 7 ) X (10 2 ) 23 = 10 7+46x 1 = 10 53 .
“Having thus reached the *tallakshana, which we would write today as 1
followed by fifty-three zeros, Bodhisattva added that this whole table forms
only one counting system, the *tallakshana counting system, [from the
name of its last term]; but there is, above this system, that of
* dhvajdgravati; beyond that, the counting system * dhvajdgranishdmani,
and beyond that again, six other systems for which he gave the respective
names” [Woepcke (1863)].
The * dhvajdgravati system is also made up of twenty-four terms, and its
first term is the * tallakshana (the largest number in the preceding system,
that is 10 53 ). Since its progression increases geometrically by a ratio equiva-
lent to one hundred, its final term therefore has the value:
1 dhvajdgravati = (10 7+46xl ) X (10 2 ) 23 = 10 7 + 46 x 2 = 10".
As this is the last term in the preceding system, it becomes the first in the
following one, that is to say the third system, the dhvajdgranishdmani, the
final number of which being equal to:
1 dhvajdgranishdmani = (10 7 + 46 x 2 ) x (10 2 ) 23 = 10 7 + 46 x 3 = 10 145 .
Step by step, we thus arrive at the ninth counting system, of which the
name of the last term has the value:
( 10 7 + 46 X 8) X (1 0 2)23 = 10 7 + 46 x 9 = 1() 421
INDIAN CIVILISATION
424
(We write this number as 1 followed by 421 zeros).
Aijuna, full of admiration for the superiority of Buddha’s knowledge,
and wanting nothing more than to learn from him, asked him to explain
how one enters into “the counting system which extends to the particles of
the first atoms ( *Paramanu )” (literally: “first-atom-particle-penetration-
enumeration”) and to teach him and the young Shakyas how many first
atoms there were in a yojana (a unit of length).
Here is Buddha’s reply:
If you want to know this number, use the scale that takes you from the
yojana to four krosha of Magadha, from the krosha of Magadha to a
thousand arcs ( dhanu ), from the arc to four cubits ( hasta ), from the
cubit to two spans ( vitasti ), from the span to twelve phalanges of fin-
gers ( anguli parva), from the phalanx of the finger to seven grains of
barley ( yava ), from the grain of barley to seven mustard seeds (sar-
shapa), from the mustard seed to seven poppy seeds ( liksha raja), from
the poppy seed to seven particles of dust stirred up by a cow (go raja),
from the particle of dust stirred up by a cow to seven specks of dust
stirred up by a ram ( edaka raja), from the speck of dust disturbed by a
ram to seven specks of dust stirred up by a hare ( shasha raja), from the
speck of dust stirred up by a hare to seven specks of dust carried off by
the wind (vdyayana raja), from the speck of dust carried away by the
wind to seven tiny specks of dust (truti), from a tiny speck of dust to
seven minute specks of dust (renu), and from the minute speck of dust
to seven particles of the first atoms (paramanu raja).
In other words, if we use the modern notation of the exponents and if
we use the letter “p” to denote these “first atoms” (paramanu ), this “scale”
can be written in the following manner, starting with the smallest and fin-
ishing with the largest quantity:
1 minute speck of dust
= 7 particles of dust of the first atoms 7 p
1 tiny speck of dust
= 7 minute specks of dust 7 2 p
1 speck of dust carried away by the wind
= 7 tiny specks of dust 7 3 p
1 speck of dust stirred up by a hare
= 7 specks of dust carried away by the wind 7 4 p
1 speck of dust stirred up by a ram
= 7 specks of dust stirred up by a hare 7 5 p
1 speck of dust stirred up by a cow
= 7 specks of dust stirred up by a ram 7 6 p
1 poppyseed
= 7 specks of dust stirred up by a cow 7 7 p
1 mustard seed
= 7 poppy seeds 7 8 p
1 grain of barley
= 7 mustard seeds 7 9 p
1 phalanx of a finger
= 7 grains of barley 7 10 p
1 span
= 12 phalanges of fingers 12 X 7 10 p
1 cubit
= 2 spans 2 x 12 x 7 10 p
1 arc
= 4 cubits 8 x 12 x 7 10 p
1 krosha from Magadha
= 1,000 arcs 1,000 x 8 x 12 x 7 10 p
1 yojana
= 4 krosha from Magadha 4 x 1,000 x 8 x 12 x 7 10 p
Carrying out the multiplication 4 x 1,000 x 8 x 12 x 7 10 which is denoted
by the last term in this scale, Buddha gives the sum by expressing in words the
number of first atoms contained in the “length” of a yojana, namely:
108,470,495,616,000.
From very high numbers to very small numbers
Using the corresponding Sanskrit terms and taking the scale in reverse
order, we have, using the preceding data, the following table which begins
with the phalanges of the digits (anguli parva) and ends with the atoms
( paramanu raja):
1 anguli parva
= 7 yava
1 yava
= 7 sarshapa
1 sarshapa
= 7 liksha raja
1 liksha raja
= 7 go raja
1 go raja
= 7 edaka raja
1 edaka raja
= 7 shasha raja
1 shasha raja
= 7 vatayana raja
1 vatayana raja
= 7 truti
1 truti
= 7 renu
1 renu
= 7 paramanu raja.
425
THE BEGINNINGS OF THESE NUMERICAL SPECULATIONS
Thus:
1 anguli parva = 7™ paramanu raja.
The * paramanu or “highest atom” constitutes, in Indian thought, the small-
est indivisible material particle, which has a taste, a smell and a colour.'
In terms of weight, a *paramanu is the equivalent of one seventh of an
“atom” ( *anu ).
As an *anu is approximately equal to 1/2,707,200 of a tola, which is
itself equal to 11.644 grams, the *paramanu weighs the equivalent of
1/18, 950,400 of 11.644 g; thus:
1 paramanu = 0.000000614 g = 6.14 x 10~ 7 g.
We will now look at the calculation from another angle.
According to the above table, a phalanx of a finger ( anguli parva) corre-
sponds to 7 10 “specks of dust of a supreme atom” ( * paramanu raja) ; thus:
1 * paramanu raja = 7~ 10 anguli parva.
Three phalanges of the fingers make an “inch”; therefore a paramanu raja is
equal to 3.7" 10 inches. As an inch is equal to 27.06995 mm, we have:
1 *paramanu raja = 0.000000287 mm = 2.87 x 10" 7 mm.
These constitute the smallest units of weight and length in India in the
early centuries CE.
Thus we have seen how the Indians could easily deal with both “very
high” and “very small” numbers.
THE BEGINNINGS OF THESE NUMERICAL
SPECULATIONS
The Shakyamuni or “Sage of the Shakyas”, the Indian prince named
Gautama Siddhartha, better known as Buddha, is said to have lived during
the fifth century BCE. Does this mean that the Indian passion for high
numbers began at this time? We do not know, because no work by Buddha
himself has ever been found.
The * Lalitavistara Sutra is a collection of stories and ancient legends
which was actually only compiled relatively recently.
However, a passage of the *Vajasaneyi Samhita enumerates the stones
needed to construct the sacred altar of fire using the following words [see
Weber, in: JSO, XV, pp. 132-40)]:
The *paramami bears no relation to our present-day concept of the “atom", but is more akin to what we
would call a molecule: the smallest particle which constitutes a quantity of a compound body.
*ayuta = 10,000
*niyuta = 100,000
*prayuta = 1,000,000
*arbuda = 10,000,000
*nyarbuda = 100,000,000
*samudra = 1,000,000,000
*madhya = 10,000,000,000
*anta = 100,000,000,000
*parardha = 1,000,000,000,000.
This example, like many others of the same genre, comes from a text
belonging to Vedic literature. We know that the texts of the * Vedas and
most of the literature which derives from this civilisation date far back in
terms of Indian history, but it is impossible to give an exact date to this era;
the texts were first transmitted orally before being transcribed at a later
date. As Frederic explains, “the only chronological order we can give them
is a purely internal one. The Samhita (the three Vedas: Rigveda, Yajurveda,
and Samaveda) seem to have been composed first; next we have the fourth
Veda or Atharvaveda, followed by the Brahmanas, the Kalpasutras and
finally the Aranyakas and the Upanishads.” What we can say with some cer-
tainty is that most of these texts were already in their finished form in the
early centuries CE.
The numerical speculation contained in the legend of Buddha cannot
have appeared later than the beginning of the fourth century CE, as the
* Lalitavistara Sutra was translated into Chinese by Dharmaraksha in the
year 308 CE.
Thus it would not be unreasonable to place the date of the first devel-
opments of these impressive numerical speculations around the third
century CE.
The incredible speculations of the Jainas
The members of the Jaina religious movement figure first and foremost amongst
the Indian scholars to be well acquainted with such numerical speculations.
There are many examples in the text *Anuyogadvara Sutra, where the
sum of the human beings of the creation is given as 2 96 .
There are other, even older Jaina texts, where numbers containing
eighty or even a hundred orders of units are described as "minuscule” in
comparison with those under speculation: these numbers are as high as, or
greater than ten to the power of 250, which we would write today as 1 fol-
lowed by at least two hundred and fifty zeros.
There is also a period of time called *Shirshaprahelika, mentioned in sev-
eral Jaina texts on cosmology, and expressed, according to Hema Chandra
INDIAN CIVILISATION
426
(1089 CE), as “196 positions of numerals of the decimal place- value
system”, and which corresponds, according to the same source, “to the
product of 84,000,000 multiplied by itself twenty-eight times”. Thus:
the Shirshaprahelikd = (84,000,000) 28 » (8 7 ) 28 = 8 7 * 28 = 8 196 .
As for the ages of the world, the Jainas used the Brahman classification.
Thus the fifth age (which we live in) would have begun in 523 BCE and
would be characterised by pain. It would be followed by the sixth and last
“age” of 21,000 years, at the end of which the human race would undergo
horrific mutations, without the world actually coming to an end. According
to Jaina doctrine, the universe is indestructible; this is because it is infinite
in terms of both time and space. It was in order to define their vision of this
impalpable universe, which is both eternal and limitless, that the Jainas
undertook their impressive speculations on gigantic numbers and thus cre-
ated a “science” which was characteristic of their way of thinking.
Their discovery of ‘infinity was doubtless due to the fact that they were
constantly pushing the limits of the *asamkhyeya (the “impossible to
count”, the “innumerable”, the “number impossible to conceive”) further
and further.
THE BIRTH OF MODERN NUMERALS
We can only admire the perfect ease with which the authors and readers of
the texts we have just seen were able to write and pronounce these high
numbers without ever being struck by a feeling of vertigo at the enormous
quantities they were dealing with.
Sanskrit notation had an excellent conceptual quality. It was easy to use
and moreover it facilitated the conception of the highest imaginable num-
bers. This is why it was so well suited to the most exuberant numerical or
arithmetical-cosmogonic speculations of Indian culture.
This spoken counting system had a special name for each of the nine
simple units:
*eka *dvi *tri * *chatur *pahcha *shat *sapta *ashta *nava
123 4 56789
There was an independent name for ten, and for each of its multiples,
which were used alongside other words in the form of analytical combina-
tions to express intermediate numbers. Like all Indo-European spoken
counting systems, the numbers were often expressed - at least in everyday
use - in descending order, from the highest to the smallest units.
However, around the dawn of the Common Era (probably from the
second century BCE), this order was reversed, particularly in learned and
official texts, the numbers being expressed in ascending order, from the
smallest to the highest units. (It has been suggested that this radical trans-
formation was due to the intervention of another civilisation. This idea is
totally without foundation: why and how could this change in direction be
due to an outside influence, bearing in mind that none of the known civili-
sations, Greek, Babylonian and Chinese included, had reached the same
level as the Indians in terms of numerical concepts and expression? As we
shall see later, the reason for this change has absolutely nothing to do with
any outside influence.)
Where we would say “three thousand seven hundred and fifty-nine”,
Indian arithmeticians would have said:
nava panchashat sapta shata cha trisahasra
"nine, fifty, seven hundred and three thousand”.
Apart from saying the numbers in the opposite order, the way that numbers
were said in Sanskrit and the way in which we say them are very similar.
However, there is one fundamental difference. When we say the num-
bers 10,000, 100,000, 10,000,000, 100,000,000, etc., we say ten thousand, a
hundred thousand, ten million, a hundred million, etc. In other words, thou-
sand, million, etc., play the role of auxiliary bases.
There are no such auxiliary bases in the Sanskrit system, at least none
which were used by learned men; each power of ten had a particular name
which was completely independent of all the others.
These names are discussed in c. 1000 by the Muslim astronomer of
Persian origin, al-Biruni, in his Kitabfi tahqiq i ma li'l hind (the book relat-
ing to his experiences of Indian civilisation):
One thing that all nations agree on when it comes to calculations is
the proportionality of the knots of calculation' according to the ratio of
ten [= the decimal base]. This means that there is no order in which the
unit is not worth one tenth of the unit which appears in the following
order and ten times the value of the unit of the preceding order. I care-
fully researched the names of the different orders of numbers used in
different languages to the best of my capabilities. I found that the same
names are repeated once the numbers reach the thousands, as was the
case with the Arabic system, which is the most appropriate method, and
the most fitting to the nature of the subject in question. I have also writ-
ten a whole dissertation on this subject. However, the Indians go beyond
the thousands in their nomenclature, but not in a uniform manner;
some use improvised names, others use names which derive from spe-
cific etymologies; others even mix both these types of names. This
* According to the contemporary Arabic terminology, the knot of calculation is the constituent of a given “order
of units”; thus the knots of units are 1, 2, 3, 4, 5, 6, 7, 8, 9; the knots of tens 10, 20, 30, 40, 50, 60, 70, 80, 90; the
knots of hundreds 100, 200, 300, 400, 500, 600, 700, 800, 900; and so on.
427
THE BIRTH OF MODERN NUMERALS
Order of unit
Corresponding name
Numerical value
Power of ten
1
Atmosphere
1
1
2
Ether
10
10
3
Atmosphere
100
10 2
4
Immensity of space
1,000
10 3
5
Atmosphere
10,000
10 4
6
Point (or Dot)
100,000
10 s
7
Canopy of heaven
1,000,000
10 s
8
Voyage on water
10,000,000
10 7
9
Sky, Atmosphere
100,000,000
10 9
10
Sky, Atmosphere
1,000,000,000
10 9
11
Entire, Complete
10,000,000,000
10 10
12
Hole
100,000,000,000
10 u
13
Void
1,000,000,000,000
10 12
14
Point (or Dot)
10,000,000,000,000
10 13
15
Foot of Vishnu
100,000,000,000,000
10 M
16
Sky
1,000,000,000,000,000
10 15
17
Sky, Space
10,000,000,000,000,000
10 16
18
Path of the gods
100,000,000,000,000,000
10 17
Fig. 24.81. List of Sanskrit names (translated) for powers of ten according to al-Biruni
naming reaches as far as the eighteenth order due to certain subtleties
which were suggested to the people who use these names, by lexicogra-
phers, through the etymologies of these names. I will now describe the
differences [which exist in the Indians' usage of these names]. One differ-
ence is that some people claim that after the *parardha [the name of the
eighteenth order of units] there is a nineteenth order, which is called
bhuri, and that beyond that there is no more need for calculation. But if
calculation stops at a certain point, and there is a limit to the order of
numbers used, this is only a convention; because this could only occur if
one understood nothing besides the names used in the calculations. We
also know [according to the same people] that a unit of this order [the
nineteenth] is one fifth of the biggest nychthemeron. However, in terms
of this method, no mention is made of the influence of any tradition in
the work of those who share this opinion. But traditions do exist which
shall be explained which mention periods made up of the largest
nychthemeron. Adding a nineteenth order is taking the matter to
extremes. Another difference lies in the fact that some people claim that
the furthest limit of calculation is in the *koti [10 7 ] and that beyond this
order we return to multiples of tens, hundreds and thousands because
the number of the divinities ( Deva ) is included in this order. These
people say that the number of divinities is thirty-three *koti [=
330,000,000], and that on each of the three [gods] Brahma, Narayana
and Mahadeva depend eleven koti [= 110,000,000] [of these divinities].
As for the names which come after the eighth order, they were created by
the grammarians for reasons we shall give below. A further difference is
due to the fact that in everyday usage, the Indians use *dasha sahasra
[“ten thousand”] for the fifth order, and *dasha laksha for the seventh
[the tens of millions], because the names of these two orders are hardly
ever used. In the work entitled Arjabhad [the Arabic name for * Aryab-
hata] from the town of Kusumapura, the names of the orders, from the
tens of thousands to the tens of *kotis, are as follows: *ayuta, *niyuta,
*prayuta, *koti, *padma, parapadma. Yet another difference lies in the
fact that some people create names out of pairs. Thus they call the sixth
order *niyuta to follow the name of the fifth [*ayuta], and they call the
eighth *arbuda so that the ninth order [*vyarbuda\ can follow on, as the
twelfth [*nikharva] follows the eleventh [*kharva\. They also call the
thirteenth order *shankha, and the fourteenth *mahdshankha [the “big
shankha ’]; and according to this rule the *mahapadma [the thirteenth
order] was preceded by the *padma [twelfth]. These are the differences it
is worthwhile knowing. But there are many more which are of no use to
us, and only exist because the numbers are taught without the slightest
regard for their proper order, or because some people [use them but]
claim that [they do not] know [their exact meaning]. This [knowing the
precise meanings of all the names] would be difficult for tradesmen.
According to the Pulisha Siddhanta, after * sahasra, which is the fourth
order, the fifth is *ayuta, the sixth *niyuta, the seventh *prayuta, the
eighth *koti, the ninth *arbuda, the tenth *kharva. The names which
follow are the same as the ones above [in Fig. 24.81].
These differences apart, the Sanskrit spoken counting system shows the
remarkable spirit of organisation of the Indian scholars who, being the
good arithmeticians and lexicographers that they were, sought, at an early
stage, to give this system an impeccably ordered structure.
This fact is even more remarkable given that the Greeks got no further
than ten thousand. As for the Romans, they only had specific names for
numbers up to a hundred thousand. In his Natural History (XXXIII, 133),
Pliny explains that the Romans, scarcely able to name the powers of ten
superior to a hundred thousand, contented themselves with expressing
“million” as: decks centena milia, “ten hundred thousand”.
The French had to wait until the thirteenth century for the introduction
of the word million in their vocabulary which took place c. 1270 [O. Bloch
and W. von Wartburg (1968)], and until the end of the fifteenth century for
the names of numbers higher than that.
In 1484, Nicolas Chuquet invented the very first set of names for high
numbers above a million, using the million 10 6 as the multiplier: “byllion” =
10 12 , “tryllion” = 10 18 , “quadrillion” = 10 24 , “quyllion” = 10 30 , “sixlion” = 10 36 ,
INDIAN CIVILISATION
428
“septyllion” = 10 42 , “octyllion” = 10 48 , and “nonyllion” = 10 54 . Chuquet's work
was never published, so that it was not until the middle of the seventeenth
century that words like billion, trillion, etc. were found at all commonly.
Nowadays, US English has the most regular naming system, using 10 3 as the
multiplier, as follows: 10 6 million, 10 9 billion, 10 12 trillion, 10 15 quadrillion.
10 18 quintillion, 10 21 sextillion, 10 24 septillion.
In British English, however, the term “billion” is used for 10 12 (10 9 being
just “a thousand million”), and the multiplier used remains 10 6 , so that tril-
lion = 10 18 and quadrillion = 10 24 . Despite this, the American sense of
billion is now used in all financial calculations, and is rapidly displacing the
dictionary meaning in British English. French officially uses the same
system as the US, except that the older term “milliard” is commonly used
for 10 9 ; “billion”, officially given the value of 10 12 in 1948, is rarely used,
and 10 12 is most often expressed (as in US English) by “trillion”.
A comparison between the Arabic, Greek, Chinese and current British
systems of expressing high numbers will give a better idea of the impressive
conceptual quality of the Sanskrit system.
To make this even clearer, we will use the following number, which will
be expressed successively according to the above systems:
523 622 198 443 682 439.
As we know, in their nomenclature of the powers of ten, the ancient Arabs
always stopped at one thousand, then superposed thousand upon thousand,
whilst still using the names of the inferior powers of ten. In other words, in
their language, the above number would be expressed rather like this:
Five hundred thousand thousand thousand thousand thousand and twenty-
three thousand thousand thousand thousand thousand and six hundred
thousand thousand thousand thousand and twenty-two thousand thousand
thousand thousand and a hundred thousand thousand thousand and
ninety-eight thousand thousand thousand and four hundred thousand
thousand and forty-three thousand thousand and six hundred thousand
and eighty-two thousand and four hundred and thirty-nine.
Equally, in their nomenclature of powers of ten, the ancient Greeks and the
Chinese always stopped at the myriad (ten thousand); from there, they
superposed myriads on top of myriads, mixed with the names of the infe-
rior powers of ten. In other words, in these languages, the above number
would have been expressed rather like this [see Daremberg and Saglio
(1873); Dedron and Itard (1974); Guitel (1975); Menninger (1957); Ore
(1948); Woepcke (1863)]:
Fifty-two myriads of myriads of myriads of myriads and three thousand
six hundred and twenty-two myriads of myriads of myriads and one
thousand nine hundred and eighty-four myriads of myriads and four
thousand three hundred and sixty-eight myriads and two thousand
four hundred and thirty-nine.
In the United States this would be expressed as:
Five hundred and twenty-three quadrillion, six hundred and twenty-
two trillion, one hundred and ninety-eight billion, four hundred and
forty-three million, six hundred and eighty-two thousand, four hun-
dred and thirty-nine.
In British English, this number would be expressed as:
Five hundred and twenty-three thousand six hundred and twenty-two
billion, one hundred and ninety-eight thousand four hundred and
forty-three million, six hundred and eighty-two thousand four hun-
dred and thirty-nine.
All the above methods are rather complicated, and it is difficult to get a
clear idea of the positional value of the number.
Since around the time of the *Vedas, the Sanskrit system was much
clearer; it possessed names for all the powers of ten up to 10 8 (=
100,000,000). Later, this was extended to 10 12 (1,000,000,000,000) (proba-
bly at the start of the first millennium CE). When the powers of ten were
named up to 10 17 (and sometimes even further, as we saw in the *Jaina texts
and in the *legend of Buddha) around 300 CE, it is likely that this was due
to the development of the language itself.
Thus the following would have sufficed to express the above number in
Sanskrit, using as an example for the base the nomenclature reported by al-
Biruni (Fig. 24.81):
nava cha trimshati cha chaturshata cha dvisahasra cha ashtayuta cha
shatlaksha cha triprayuta cha chaturkoti cha chaturvyarbuda cha ash-
tapadma cha navakharva cha ekanikharva cha dvimahapadma cha
dveshahka cha shatsamudra cha trimadhya cha dvantya cha pahcha-
parardha.
In semi-translation, the number reads something like this:
429
THE BIRTH OF MODERN NUMERALS
Nine and three dasha and four shata and two sahasra and eight ayuta
and six laksha and three prayuta and four koti and four vyarbuda and
eight padma and nine kharva and one nikharva and two mahapadma
and two shankha and six samudra and three madhya and two antya and
five parardha.
In giving each power of ten an individual name, the Sanskrit system gave
no special importance to any number.
Thus the Sanskrit system is obviously superior to that of the Arabs (for
whom the thousand was the limit), or of the Greeks and the Chinese
(whose limit was ten thousand) and even to our own system (where the
names thousand, million, etc. continue to act as auxiliary bases).
Instead of naming the numbers in groups of three, four or eight orders
of units, the Indians, from a very early date, expressed them taking the
powers of ten and the names of the first nine units individually. In other
words, to express a given number, one only had to place the name indicat-
ing the order of units between the name of the order of units that was
immediately below it and the one immediately above it.
That is exactly what is required in order to gain a precise idea of the
place-value system, the rule being presented in a natural way and thus
appearing self-explanatory. To put it plainly, the Sanskrit numeral system
contained the very key to the discovery of the place-value system.
In order to grasp this idea, the names of the powers of ten need not
always be the same.
In fact, if the mathematical genius of the Indians could embrace vari-
ations on the names of the numbers whilst maintaining a clear idea of
the series of the ascending powers of ten, this only made it more dis-
posed to understanding the place-value system.
These names need not necessarily have been in everyday use in
India. They need only have been familiar to those who were capa-
ble of developing the potential ideas behind them, namely to
learned men.
We can understand al-Biruni’s surprise at seeing grammarians
creating these names, and being practically the only ones to use
them, for, in the scientific development of Arabic civilization,
grammar, lexicography and literature were completely separate
movements from the mathematical, medical and philosophical sci-
ences [F. Woepcke (1863)].
However, grammar and interpretation in ancient India were closely linked
to the handling of high numbers. Studies relating to poetry and metrics ini-
tiated "scientists” to both arithmetic and grammar, and grammarians were
just as competent at calculations as the professional mathematicians.
Thus we can see the importance of the role of Indian “scientists”,
philosophers and cosmographers who, in order to develop their arithmetical-
metaphysical and arithmetical-cosmogonical speculations concerning ever
higher numbers, became at once arithmeticians, grammarians and poets,
and gave their spoken counting system a truly mathematical structure
which had the potential to lead them directly to the discovery of the deci-
mal place-value system.
In fact, since a time which was undoubtedly earlier than the middle of
the fifth century CE, all mention of the names indicating the base and its
diverse powers was suppressed in the body of the numerical expressions
expressed by the names of the numbers.
In other words, the Indian scholars quite naturally arrived at the idea of
writing numbers without the names * dasha (= 10), *shata (= 10 2 ), *sahasra
(= 10 3 ), *ayuta (= 10 4 ), *laksha (= 10 5 ), *prayuta (= 10 6 ), *koti (= 10 7 ),
*vyarbuda (= 10 8 ), *padma (= 10 9 ), *kharva (= 10 10 ), *nikharva (= 10 u ),
*mahapadma (= 10 12 ), *shankha (= 10 13 ), *samudra (= 10 14 ), *madhya (=
10 15 ), *antya (= 10 16 ), *parardha (= 10 17 ), etc. From that time on, they
simply wrote, in strict order, the names of the units which acted as multi-
plying coefficients in their numerical expression, according to the order of
the ascending powers of ten. Thus they expressed numbers using nothing
more than the names of the units.
Instead of writing the number 523 622 198 443 682 439 using the names
of the numbers according to the ordinary principle of the Sanskrit language
(the complete form of the ‘Sanskrit numeral system), they only retained the
names of the units forming the coefficients of the diverse consecutive powers
(abridged form, characteristic of the ‘simplified Sanskrit numeral system):
Complete form
Nine and three dasha and four shata and two sahasra and eight ayuta and
six laksha and three prayuta and four koti and four vyarbuda and eight
padma and nine kharva and one nikharva and two mahapadma and two
shanka and six samudra and three madhya and two antya and five parardha.
Mathematical breakdown
= 9 + 3 x 10 + 4 x 10 2 + 2 x 10 3 + 8 x 10 4 + 6 x 10 5
+ 3 x 10 6 + 4 x 10 7 + 4 x 10 8 + 8 x 10 9 + 9 xl0 10
+ 1 x 10 u + 2 x 10 12 + 2 x 10 13 + 6 x 10 14 + 3 x 10 15
+ 2 x 10 16 + 5 x 10 17
= 523,622,198,443,682,439
INDIAN CIVILISATION
430
Abridgedform
Nine, three. four. two. eight. six. three. four.four.
eight.nine.one.two.two.six.three.two.five
The numbers in the *Jaina text, the *Lokavibhaga, (the first document that
we know of to make regular use of the place-value system) were expressed
in a very similar manner.
In other words, the Indian system of numerical symbols (or at least the
ancestor of this unique system) was born out of a simplification of the
Sanskrit spoken numeral system.
Such a simplification is not at all surprising when we consider the con-
sistency and potential of the human mind, as well as humankinds
intelligence, actions and thoughts upon such matters. When two human
beings or two cultures have the same needs and methods due to identical
basic (social, psychological, intellectual and material) conditions, they
inevitably follow the same paths to arrive at similar, if not identical results.
This is exactly what happened amongst the priest-astronomers of Maya
civilisation. Due to a need to abbreviate increasingly high numerical
expressions, and also because the units in their system of expressing
lengths of time were presented in an impeccable order which was always
rigorously followed, they discovered a place-value system, to which they
added a sign which performed the function of zero.
As with the Maya, this simplification held no ambiguity for the Indians.
The fact that the successive names of the powers of ten had always fol-
lowed an invariable order which was firmly established in the mind made
the simplification even more comprehensible.
The actual reason for the simplification was doubtless a need for abbre-
viation. This would have become increasingly necessary as the Indian
mathematicians gradually dealt with higher and higher numbers.
To write numbers containing dozens of orders of units according to
their names would have taken up whole “pages” of writing. Even expressing
one single number could take up several “sheets".
The scholars would have also wanted to be economical with their writ-
ing materials. They had to go and pick palm leaves themselves which they
used for writing upon. These had to be picked just at the right time, before
they opened out entirely, then dried and smoothed out in order to make
them fit for the writing of manuscripts. (See ‘Indian styles of writing). The
scholars wanted to give themselves as much time as possible to devote to
the more noble task of contemplation, for example studying sacred texts or
practising the physical, spiritual and moral exercises of yoga.
The simplification brought about an authentic place-value system which
had the Sanskrit names of the nine units as its base symbols. Their value
varied according to their relative position in a numerical expression.
Thus three, two, one gave the value of simple units to three, the value of a
multiple of ten to two and the value of a multiple of a hundred to one.
However, as we can see in the following example, this method could pre-
sent certain difficulties.
Given that the Sanskrit name for the number three is tri, in order to
express the number 33333333333, it would be written thus:
tri. tri. tri. tri. tri. tri. tri. tri. tri. tri. tri
Three.Three.Three.Three.Three.Three.Three.Three.Three.Three.Three
33333333333
(= 3 + 3 x 10 + 3 x 10 2 + 3 x 10 3 + 3 x 10 4 + 3 x 10 5 + 3 x 10 6 +
3 x 10 7 + 3 x 10 8 + 3 x 10 9 + 3 x 10 10 ).
This expression, which involves the repetition of tri eleven times, nei-
ther sounds pleasant nor is conducive to the memorisation of the number
in question.
Moreover, this number only has eleven orders of units. It would be
much worse if it had thirty or a hundred, or even two hundred orders
of units.
To avoid this repetition of the same word, the Indian astronomers used
synonyms for the Sanskrit names of the numbers. They used all kinds of
ideas from traditions, mythology, philosophy, customs and other charac-
teristics of Indian culture in general. This is how they gradually replaced
the ordinary Sanskrit names of numbers with an almost infinite synonymy.
Thus the above number would have been expressed by the following
kind of symbolic expression:
agni.murti.guna.loka.jagat.dahana.kala.hotri.vdchana.Rama.vahni
Fire. Shape. Quality. World. World. Fire. Time. Fire. Voice. Wdma. Fire.
33 3 3 3333333
This substitution of the ordinary names of numbers marked the birth
of the representation of numbers by ‘numerical symbols in the place-
value system.
Why did Indian astronomers favour this use of numerical symbols over
the nine numerals and the sign for zero?
The fact is that they had excellent reasons for this choice.
First and foremost, the concept of zero and the decimal place-value
system is totally independent of the chosen style of expressing the numbers
(be it conventional graphics, letters of the alphabet or words with or with-
out evocative meaning). All that matters is that there is no ambiguity and
that the chosen system of representation contains a perfect concept of zero
and the place-value system.
431
THE BIRTH OF MODERN NUMERALS
There are other reasons which are specific to the field of Indian astron-
omy and mathematics, which were generally written in Sanskrit, as were all
important erudite Indian texts. The first thing to remember is that in India
and Southeast Asia Sanskrit played, and still does play, a role comparable
with that of Latin or Greek in Western Europe, with the added virtue of
being the only language capable of translating, at the time of the medita-
tions, the mystical transcendental truths said to have been revealed to the
Rishi of Vedic times. Bearing in mind the power given to the language (and
thus to its written expression), Sanskrit is considered to be the “language of
the gods”; whoever masters the language is said to possess divine con-
sciousness and the divine language (see ‘Mysticism of letters). This
explains why the Sanskrit inscriptions of Cambodia, Champa and other
indianised civilisations of Southeast Asia do not contain “numerals” for the
expression of the Shaka dates. These inscriptions were nearly always in
verse. As far as the stone-carvers of these regions were concerned, the intro-
duction of numerical signs (considered “vulgar”) in Sanskrit texts in verse
would have constituted a sort of heresy, not only from an aesthetic point of
view, but also and moreover in terms of mysticism and religion. This is why
the dates were firstly written in the names of the numbers and then usually
in numerical symbols as well. Moreover, the actual name “Sankskrit" is
rather significant in this respect, as the word *samskrita (Sanskrit) means
“complete”, “perfect” and “definitive”. In fact, this language is extremely
elaborate, almost artificial, and is capable of describing multiple levels of
meditation, states of consciousness and psychic, spiritual and even intellec-
tual processes. As for vocabulary, its richness is considerable and highly
diversified [see L. Renou (1959)]. Sanskrit has for centuries lent itself
admirably to the diverse rules of prosody and versification. Thus we can see
why poetry has played such a preponderant role in all of Indian culture and
Sanskrit literature.
This explains why the Indian astronomers preferred to use numerical
symbols instead of numerals.
Numerical tables, Indian astronomical and mathematical texts, as well
as mystical, theological, legendary and cosmological works were nearly
always written in verse:
Whilst making love a necklace broke.
A row of pearls mislaid.
One sixth fell to the floor.
One fifth upon the bed.
The young woman saved one third of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?
This is a mathematical problem posed in the *Lilavati, a famous mathe-
matical work in the form of poems, written by ‘Bhaskaracharya (in 1150),
the title of which is the name of the daughter of the mathematician. Here is
another example:
Of a cluster of lotus flowers,
A third were offered to *Shiva,
One fifth to * Vishnu,
One sixth to *Surya.
A quarter were presented to Bhavani.
The six remainingflowers
Were given to the venerable tutor.
How many flowers were there altogether?
From this type of game, the Indian scholars went on to use imagery to
express numbers; the choice of synonyms was almost infinite and these
were used in keeping with the rules of Sanskrit versification to achieve the
required effect. Thus the transcription of a numerical table or of the most
arid of mathematical formulae resembled an epic poem.
We need only look at the following lines from a text recording astro-
nomical data to see how poetic and elliptical such documentation could be:
The apsids of the moon in ayuga
fire. void, horsemen. Vasu. serpent, ocean
and of its waning node
Vasu. fire, primordial couple, horsemen, fire, horsemen.
However, aesthetic refinement was not the only motive. This method also
offered enormous practical advantages. Billard provides us with the precise
fundamental reasons as to why the Indian astronomers chose to use word-
symbols to express numbers:
Indian astronomical texts were always written in Sanskrit. They con-
tained little historical information, were totally devoid of discussions
and demonstrations and of the kind of observations which we recog-
nise the value of today - except for the occasional commentary, which
was always written in prose - yet they possessed remarkable, even
exceptional qualities. The astronomical data is not only always explicit,
but moreover the numerical values are still perfectly conserved after all this
time and after so many copies have been made. Although expressed in a
very elliptical manner in the text, where the tradition of versification,
used here for mnemonic purposes, led to a synonymy which was often
infinite within the technical language - a rather unusual occurrence in
the history of astronomy - the astronomical data is very precise and
unrivalled in terms of reliability.
INDIAN CIVILISATION
432
The importance of numerical data in the Indian astronomical texts is so
great because the texts contain so little direct information. All we know of
their ‘astronomical canons, for example, is the terminology by which they
were described (average elements, apsids, nodes, eccentricities, exact longi-
tudes, average longitudes, etc.), the terminology being the word-symbols.
It is precisely the study of the numerical data which led to the finding of a
given canon in various different texts from very different eras, as well as facil-
itating the distinction between different canons (see ‘Indian astronomy).
Thus we can appreciate just how reliable this numerical data had to be
in order for it to be transmitted from one generation to the next.
Although initially it might seem puerile, the use of word-symbols was
in fact extremely efficient in conserving the exact value of the numbers
they expressed, and it was doubtless to this end that the word-symbols
were invented. “This conservation of the value of numbers in Sanskrit
texts”, writes Billard, “is even more impressive when one considers that
Indian manuscripts, in material terms, generally do not survive more
than two or three centuries [due to climate and above all vermin, which
render the conservation of manuscripts extremely difficult], and had the
numerical data been recorded in numerals, it would no doubt have
reached us in an unusable state."
And Guitel observes that “from a purely mathematical point of view, the
use of many different words to express each of the numbers presented no
ambiguity; a text written in word-numbers could easily be translated into
numerals [and vice versa]. All one would have needed was a glossary of all
the words possessing a numerical value, which could be used like a dictio-
nary of rhymes.” Whatever the benefits of this system, however, it could
not be used to carry out calculations.
The reason for this is obvious: numbers could be expressed using the
place-value system with nine numerals and zero, and this system was doubt-
less invented at the same time as the positional system of the word-symbols.
However, no one would have dreamt of adding *fire, * arrows, *planets
and * serpents, or of subtracting * oceans, * orifices or *naga from * elephants,
or multiplying the * faces of Kumara by the *eyes of Shiva or dividing the
*arms of Vishnu by the * great sacrifices'.
Since no later than the fifth century, Indian arithmeticians used the
place-value system with the nine numerals and zero to carry out compli-
cated mathematical operations. They avoided the use of numerals for
recording numerical data, but used them in their rough calculations.
On the other hand, it was very difficult for the Indian astronomers to
express their numerical data in numerals, because numerals were far less
reliable than the word-symbols. This is because, graphically speaking, the
numerals varied according to the style of writing of each region (Fig. 24.3 to
52), and also according to the era and the author or transcriber. A shape
which represented the number 2 to one person might well have repre-
sented a 3, a 7, or even a 9 to another.
The situation is completely different for us in the twentieth century,
because the shapes of the numerals we use and their respective values are
the same the world over. For the Indian astronomers, however, the use of
numerals could cause confusion. The use of verse and word symbols, on
the other hand, was very reliable, because the slightest error could break
the rhythm of the verse or verses in question, and therefore would not
escape notice. This is why Indian astronomers favoured word-symbols for
many centuries.
There is also another, equally fundamental reason. As we have seen,
Indian astronomical texts were always in verse: a prosody of long or short
syllables was used, as in Graeco-Latin metrics, except that the metre and
the number of syllables used in the Indian texts were always perfectly clear
and very systematic. Thus the word-symbols not only guaranteed the con-
servation of the values of the numbers expressed, but also served a
mnemonic function. “As well as allowing the writer to find a synonym
which gave the required scansion, the word-symbol formed part of the
metre, and the number that it expressed was at once firmly established in
the text and in the mathematician’s memory, who recited the verses as he
worked out his calculations” (Billard). The method facilitated and rein-
forced the Indian scholars’ memory: it allowed them to make the best use
of their memory through associations of ideas or images contained within
rhythms determined by the metre which was dictated by the rules of
Sanskrit versification.
When we consider the above conditions, we can understand why the
Indian astronomers developed the Sanskrit word-symbols, and continued
to use them for such a prolonged length of time.
The same conditions led the astronomer ‘Aryabhata to develop his
famous alphabetical numeral system. He was no doubt familiar with the
Sanskrit word-symbols, but needed a system which was more concise
whilst meeting the requirements of certain versified Sanskrit compositions.
It is likely that he found the word-symbols to be lacking in brevity and per-
haps also precision, especially when he wrote his famous sine table.
Similar reasons led astronomers such as ‘Haridatta or ‘Shankaranarayana,
at a later date, to use an alphabetical numeral system which was even more effi-
cient then the *katapaya system.
433
THE BIRTH OF MODERN NUMERALS
The coexistence of different methods of achieving the same goals is
one of the characteristics particular to the highly inventive genius of
the Indians, which enjoyed both the finest distinctions and determina-
tions, and the fluctuating wave of an abundant production, and was
little inclined towards that precise and rather dry sobriety of the
ancient Semites [F. Woepcke (1863)].
The discovery of the place-value system required another, equally basic
progression. As soon as the place-value system was rigorously applied to
the nine simple units, the use of a special terminology was indispensable to
indicate the absence of units in a given order.
The Sanskrit language already possessed the word *shunya to express
“void” or “absence”. Synonymous with “vacuity”, this word had for several
centuries constituted the central element of a mystical and religious philos-
ophy which had become a way of thinking.
Thus there was no need to invent a new terminology to express this new
mathematical notion: the term *shunya (“void”) could be used. This is how
the word finally came to perform the function of zero as part of this excep-
tional counting system.
A number such as 301 could now be expressed:
eka.shunya.tri
one. void, three.
1 0 3
The Sanskrit language, however, being an unrivalled literary instrument in
terms of wealth, possessed more than just one word to express a void: there
was a whole range of words with more or less the same meaning: words
whose literal meaning was connected, directly or indirectly, with the world
of symbols of Indian civilisation.
Thus words such as *abhra, *ambara, *antariksha, *gagana, *kha,
*nabha, *vyant or *i yoman, which literally meant the sky, space, the atmos-
phere, the firmament or the canopy of heaven, came to signify not only a
void, but also zero. There was also the word * akasha, the principal mean-
ing of which was “ether”, the last and the most subtle of the “five elements”
of Hindu philosophy, the essence of all that is believed to be uncreated and
eternal, the element which penetrates everything, the immensity of space,
even space itself.
To the Indian mind, space was the “void” which had no contact with
material objects, and was an unchanging and eternal element which defied
description; thus the association between the elusive character and very
different nature of zero (as regards numerals and ordinary numbers) and
the concept of space was immediately obvious to the Indian scholars. The
association between ether and “void” is also obvious because akasha (to the
Indian mind) is devoid of all substance, being considered the condition of
all corporal extension and the receptacle of all substance formed by one of
the other four elements (earth, water, fire and air). In other words, once
zero had been invented and put into use, it brought about the realisation
that, in terms of existence, akasha played a role comparable with the one
which zero performed in the place-value system, in calculations, in mathe-
matics, and the sciences.
The following are other Indian numerical symbols for zero: *bindu,
“point”; *ananta, “infinity”; *jaladharapatha, “voyage on water”; *vishnu-
pada, “foot of Vishnu”; *purna, “fullness, wholeness, integrity,
completeness”; etc. (See also *Zero.)
The use of one of these words prevented any misunderstanding. Later
than the Babylonians, and most likely before the Maya, the Indian scholars
invented zero, although for the time being it was little more than a simple
word which formed part of everyday vocabulary.
So just how did the place-value system come to be applied to the nine
Indian numerals?
Let us now go back to the numeral system of ancient India: the Brahmi
system, which, as we have already seen, constituted the prefiguration of the
nine basic numerals that we use today (Fig. 24.29 to 52 and 24.61 to 69).
Current documentation suggests that the history of truly Indian numer-
als began with the Brahmi inscriptions of Emperor Asoka (in the middle of
the third century BCE). But the numerals were invented before the Maurya
Dynasty, by which time the numerals were highly developed graphically
speaking, and widespread throughout the Indian territory.
In fact, as we have already seen, the first nine Brahmi numerals which
appear in Asoka ’s edicts are probably vestiges of an old indigenous system
(no doubt older than Brahmi writing itself), where the nine units were rep-
resented by the necessary number of vertical lines, similar to the
arrangements in Fig. 24.59.
We will now sum up the evidence we have compiled in this chapter on
the early stages in the history of these numerals.
Like all the other civilisations of the world, the Indians initially used the
required number of vertical lines to write the first nine numbers. However,
as a row of vertical lines was not conducive to rapid reading and compre-
hension, this system was gradually abandoned, at least for the numbers 4
to 9. To overcome the problem, the lines representing the units were split
into two groups (two lines on top of two others for 4, three lines on top of
two others for 5, etc.; see Fig. 1.26), or a ternary arrangement was used
(three lines on top of one line for 4, three above two for 5, etc.; see
INDIAN CIVILISATION
Fig. 1. 27). This was how the Sumerians, Cretans and Urarteans proceeded,
as well as the Egyptians, Assyrians, Phoenicians, Aramaeans and Lydians.
In the long run, however, such groupings of lines did not allow for rapid
writing, or time-saving, which was the main preoccupation of the scribes.
Thus - due to a combination of circumstances imposed by the very nature
of the writing materials used (the scribes wrote upon tree bark or palm
leaves with a brush or calamus) depending on the region - a numeral
system evolved which was unique to each civilisation and the numerals no
longer visually represented their respective values. In each civilisation the
change was brought about by both the nature of the writing materials and
the desire to save time. Signs were invented which could be written in one
single stroke or in short, quick strokes. Ligatures were exploited wherever
possible, so that the brush need not be lifted, allowing several lines to be
grouped together in one single sign. The initial representations of the num-
bers were radically modified, as we can see with the Brahmi numerals for
the numbers four to nine (Fig. 24.57, 58 and 60).
At the outset, these nine signs were not used in conjunction with the
place-value system: the Brahmi system relied on the principle of addition
and a specific sign was given to each of the nine units in each decimal order,
up to and including tens of thousands (Fig. 24.70).
Mathematical operations, even simple addition, were almost impossi-
ble without the invention of a device. The ancestors of our modern
numerals remained static for some time before acquiring the dynamic
and workable nature of the current numerals. Like certain other systems
of the ancient world, this system was also rudimentary whilst it was only
used to express numbers.
Mathematicians, philosophers, cosmographers and all others who, for
one reason or another, were handling high numbers at that time, resorted
to using the Sanskrit names of the numbers.
However, like all the mathematicians of the ancient world, Indian arith-
meticians, before discovering the place-value system, used their fingers or,
more often, concrete mathematical devices.
It seems that the most common was the abacus: from right to left, the
first column represented the units, the next the tens, the third the hun-
dreds, and so on.
Unlike the Greeks, Romans or Chinese, however, who then went on to
use pebbles, tokens or reeds, the Indian mathematicians had the idea of
using the first nine numerals of their counting system, tracing them in fine
sand or dust, inside the column of the corresponding decimal order.*
* This information was obtained from descriptions given by various Indian authors, and later accounts
from many Arabic, Persian, European and even Chinese authors [see Allard (1992); Cajori (1980); Datta
and Singh (1938); Iyer (1954); Kaye (1908); Levey and Petruck (1965); Waeschke (1878)1.
434
Thus a number such as 7,629 would have been represented in the fol-
lowing manner, with nine in the units column, two in the tens column, six
in the hundreds and seven in the thousands:
Of course, when a unit within an order of units was missing, one only
needed to leave the appropriate column empty; thus the representation of
10,267,000:
1 0 2 6 7 0 0 0
The mathematical operation would be carried out by successively erasing
the results of the intermediary calculations. (There is a simple example of
this in Chapter 25.)
Like us today, the “Pythagorean” tables had to be known by heart,
which give the results of the four elementary operations of the nine signifi-
cant numerals.
Before the beginning of the fifth century BCE, then, all the necessary
“ingredients” for the creation of the written place-value system had been
amassed by the Indian mathematicians:
• the units one to nine could be expressed by distinct numerals,
whose forms were unrelated to the number they represented, namely
the first nine Brahmi numerals;
• they had discovered the place-value system;
• they had invented the concept of zero.
A few stages, however, were still lacking before the system could attain
perfection:
• the nine numerals were only used in accordance with the additional
principle for analytical combinations using numerals higher than or
equal to ten, and the notation was very basic and limited to numbers
below 100,000;
• the place-value system was only used with Sanskrit names for numbers;
• and zero was only used orally.
435
THE BIRTH OF MODERN NUMERALS
In order for the “miracle” to take place, the three above ideas had to
be combined.
By using the nine Brahmi numerals in the appropriate columns of the
“dust” abacus, the Indian mathematicians had already reached the stage
which would inevitably lead them to this major discovery.
This becomes dear when we imagine the Indian mathematicians at work,
recording the result of a calculation they had carried out by drawing their
abacus in the dust, bearing in mind that they had two methods of expressing
numbers: the Brahmi numerals and the Sanskrit names of the numbers.
In the abacus, they would have drawn the numerals in a contemporary
style (those from the inscriptions of Nana Ghat, for example, dating from
the second century BCE; see Fig. 24.30 and 71), and a calculation might
have given the following result:
* 1 \ 1
The figure obtained is 4,769.
As we know, from this time on, the numbers were expressed in their Sanskrit
names in the order of ascending powers of ten, from the smallest to the highest.
Therefore this result would have been expressed as follows:
nava shashti saptashata cha chatursahasra
“nine sixty seven hundred and four thousand”.
In numerals, however, the numbers would have been written in the oppo-
site order, reading from left to right:
■pf H? H ?
4,000 700 60 9
>
We have evidence of these methods of expressing figures in Indian inscrip-
tions, since the third century BCE, from those of Asoka, Nana Ghat and
Nasik to those of the Shunga, Shaka, Kusana, Gupta and Pallava dynasties.
The corresponding numerical notations, all issuing from the original
Brahmi system, possess a different numeral for each unit of each decimal
order (Fig. 24.70 and 71).
When we examine the signs used, we discover that these numerals are
not all independent of one another; the only numerals which are really
unique are the following:
123456789
10 20 30 40 50 60 70 80 90
100 1,000
The numerals for 200 and 300, as well as those for 2,000 and 3,000, derive
from the sign for 100 and 1,000 respectively, with the simple addition of
one or two horizontal lines (Fig. 24.70 C).
In other words, the four numerals in question conformed to the follow-
ing mathematical rules:
200 = 100 + 1 x 100 2,000 = 1,000 + 1 X 1,000
300 = 100 + 2 x 100 3,000 = 1,000 + 2 x 1,000
As for the remaining multiples of one hundred and one thousand, they
were represented using the principle of multiplication and placing the
numeral for the corresponding unit to the right of the sign for one hundred
or one thousand:
400 = 100 x 4
500 = 100 x 5
600 = 100 X 6
700 = 100 x 7
800 = 100 X 8
900 = 100 x 9
4.000 = 1,000 x 4
5.000 = 1,000 x 5
6.000 = 1,000 X 6
7.000 = 1,000 x 7
8.000 = 1,000 X 8
9.000 = 1,000 x 9
It is visibly evident that this rule also applied to the notation of tens of thousands
by placing the corresponding number of tens next to the sign for a thousand:
10.000 = 1,000 x 10
20.000 = 1,000 x 20
30.000 = 1,000 x 30
40.000 = 1,000 x 40
Thus, the number 4,769 was written:
W -i?
1,000 X 4 + 100 X 7 + 60 + 9
This corresponds exactly, but in the opposite order, to the above Sanskrit
expression of the figure:
nava shashti saptashata cha chatur sahasra
9 + 60 + 7 x 100 + 4 x 1,000
If we look at either way of expressing the sum in the opposite direction
from the way it would have been spoken or written, we obtain the
INDIAN CIVILISATION
436
arithmetical breakdown of the other. This is what the Indian arithmeti-
cians expressed in the phrase *ankanam vamato gatih, which literally
means the “principle of the movement of numerals from the right to the
left”, which applies to the reading of numbers from the smallest unit to
the highest in ascending order of powers of ten.
The inscriptions of Nana Ghat provide the earliest known significant
evidence of this principle. Thus we know that from at least as early as the
second century BCE, the numbers were expressed in ascending powers of
ten in Brahmi numerals; in other words, they were expressed in the oppo-
site order than from left to right. This means that the structure of Brahmi
notation had been copied exactly from the Sanskrit system.
Since the highest Brahmi numeral expressed the number 90,000, it was
impossible to use this system to express a number that was higher than 99,999.
As the Brahmi numerals constituted an abbreviated written form of the
spoken numeration, it had been developed to avoid having to express fre-
quently used numbers through the long-winded Sanskrit names of the
numbers.
In other words, the result of a calculation which was equal to or higher
than 100,000 could only be written down in the Sanskrit names of the
numbers.
The abacus traced in the dust could be used to carry out calculations
involving extremely high numbers: each column represented a power of ten,
and there was no limit to the number of columns which could be drawn.
Thus there was a very close link between the ability to carry out calcula-
tions on this abacus and the level of conception of high numbers and the
capacity to express them orally or through writing.
In Indian calculation, the successive columns of the abacus always rigor-
ously corresponded to the consecutive powers of ten. As the Sanskrit
counting system possessed the same mathematical structure, these
columns corresponded exactly to the impeccable succession of names
which the Sanskrit system possessed for the various powers of ten. Thus
each system constituted the mirror image of the other.
This is exactly where the potential to discover the place-value system of
the nine numerals lay. As with the Sanskrit names of the numbers, the struc-
ture of the abacus contained the key to the discovery of the decimal
place-value system. This is why the Sanskrit notation was perfect for record-
ing the results of the calculations which were carried out on the abacus.
This becomes even clearer when we take the number 523,622,198,443,
682,439, as it would have appeared when written on the abacus using the
nine Brahmi numerals (see adjacent column).
We can see how the close relationship between the representation of the
numbers on the abacus and the Sanskrit system led to the change in direc-
x
The numerals are written in descending powers
often
The direction in which the numbers would have
been read (in ascending powers of ten)
<
tion, before the second century BCE, of numerical expressions given using
the Sanskrit names of the numbers.
Whilst the numerals read from left to right on the abacus in descending
powers of ten, they came to be read from right to left, from the smallest
number to the highest.
Bearing in mind the conditions imposed by the very nature of the calcu-
lating instrument, the Indian mathematicians had no other choice but to
adopt the expression of numbers in the direction described by *ankanam
vamato gatih : "the principle of the movement of the numerals from the
right to the left”. How could they know how to write a high number on the
abacus if they began with the highest order? They would have had to work
out which column each order had to be placed in by counting each corre-
sponding column beginning with the column for the simple units. This
would have wasted time. Thus the best solution was to always start with the
column for the simple units.
Thus the old system was abandoned. By beginning with the highest
power of ten, the arithmetician immediately knew the size of the number
he was dealing with, but this did not facilitate the drawing, on the columns
of the abacus, the successive numerals of a number which possessed more
than four or five orders of units.
This is why the opposite direction was adopted, the advantage of which
being that, no matter how high a number might be, there could be no mis-
take as to which column each numeral must be written in. It was for the same
reason that this direction of expressing the numbers was conserved later on
when the positional notation was invented using numerical symbols:
We must not forget that the numbers which appear in the scientific
poems [such as the numerical data given by the Indian astronomers]
were destined for mathematical use. Certain lists contain numbers
proportional to the differences of sines of angles which ascend in
mathematical order; these enabled, with the aid of additions, an
almost instant reconstruction of the numbers proportional to the
sines of these angles.
437
THE BIRTH OF MODERN NUMERALS
The pandit dictated the poetic text which the scribe wrote in
numerals. How could an addition be carried out if the data consisted
of wing (= 2) and fire (= 3)? If only one number had to be reproduced,
even a very high number, it would have been easy to translate it
directly onto “paper”, but if a series of numbers was involved, how
could they be correctly placed on the counting table if they were read
out in descending orders of units?
It would have been impossible to transcribe a number in this way
unless it was known in advance. The only solution was to read out the
numbers in ascending orders of units.
However, when the pandit was reading out a high number, the
scribe needed to know its highest order; this is why the pandit
started with the highest powers of the base; this is not possible if one
uses the spoken positional numeration, yet this system did enable
one to place the number correctly on the counting table, and then
one could plainly see the powers of the base which had been
recorded [G. Guitel, (1966)].
If we look again at the representation of the number given above, on the
“dust” abacus, its mathematical breakdown according to *ankandm vamato
gatih was as follows:
= 9 + 3 x 10 + 4 x 10 2 + 2 x 10 3 + 8 x 10 4 + 6 x 10 5
+ 3 x 10 6 +4 x 10 7 +4 x 10 8 +8 x 10 9 +9 x 10“
+ 1 x 10 u +2 x 10 12 +2 x 10 13 + 6 x 10 14 + 3 x 10 15
+ 2 x 10 16 + 5 x 10 17
= 523,622,198,443,682,439
This corresponds exactly to the following Sanskrit expression:
“Nine and three dasha and four shata and two sahasra and eight ayuta and
six laksha and three prayuta and four koti and four vyarbuda and eight padma
and nine kharva and one nikharva and two mahapadma and two shankha and
six samudra and three madhya and two antya and five parardha."
Once the Sanskrit numeration was simplified, this number could be
expressed in the following manner:
nine.three.four.two.eight.six.three.four.four.
eight.nine.one.two.two.six.three.two.five
Why was the number written in the names of the numbers instead of in
Brahmi numerals, to which the place-value system could have been applied?
This is surely what the Indian mathematicians asked themselves one
day, in their continuing desire to economise with time and materials.
They carried out calculations involving high numbers, for which even
the intermediate results constituted very high numbers, and which could
be difficult to memorise. The results were first recorded in rough. As they
needed to be sparing with materials and time, the Indian mathematicians
sought a way to write faster and in a more abridged form than the Sanskrit
system, even in its simplified form. They realised that the nine Brahmi
numerals could provide the “stenography” that they required.
However, bearing in mind the position of the numerals on the abacus, it
was necessary to revert to the descending order, thus going from the high-
est orders of units to the smallest, so as not to cause confusion in the
numeral representations; for the results of calculations carried out on the
abacus, the numerals had to be placed in the columns in the same positions
as they appeared when written in rough.
Thus the number in question acquired the following notation, the num-
bers reading from left to right in descending order of powers of ten,
constituting a faithful reproduction, minus the columns, of its representa-
tion on the abacus, as well as the reflection of the abridged form of the
corresponding Sanskrit expression:
52362219844368243 9.
Whence came the decimal position values which were given to the first nine
numerals of the old notation which originated at the time of the reign of
Emperor Asoka.
This was the birth of modern numerals, which signalled the death of the
abacus and its columns.
The introduction of a new symbol proved indispensable in order to
convey the absence of units in a given decimal order; whilst this sign was
not needed when the abacus was used, it was of utmost necessity in the new
positional numeral notation.
The Indians, never lacking in resources in these matters, again turned to
their unique symbolism.
As we saw earlier, the word-symbols that the Sanskrit language used to
express the concept of zero conveyed concepts such as the sky, space, the
atmosphere or the firmament.
In drawings and pictograms, the canopy of heaven is universally repre-
sented either by a semi-circle or by a circular diagram or by a whole circle.
The circle has always been regarded as the representation of the sky and of
the Milky Way as it symbolizes both activity and cyclic movements [see
J. Chevalier and A. Gheerbrant (1982)].
Thus the little circle, through a simple transposition and association of
ideas, came to symbolise the concept of zero for the Indians.
Another Sanskrit term which came to mean zero was the word *bindu,
which literally means "point”.
INDIAN CIVILISATION
438
The point is the most insignificant geometrical figure, constituting as it
does the circle reduced to its simplest expression, its centre.
For the Hindus, however, the bindu represents the universe in its non-
manifest form, the universe before it was transformed into the world of
appearances ( rupadhatu ). According to Indian philosophy, this uncreated
universe possessed a creative energy, capable of generating everything and
anything: it was the causal point.
The most elementary of all geometrical figures, which is capable of cre-
ating all possible fines and shapes ( rupa ) was thus associated with zero,
which is not only the most negligible of quantities, but also and above all
the most fundamental of all abstract mathematics.
The point was thus used to represent zero, most notably in the Sharada
system of Kashmir, and in the vernacular notations of Southeast Asia
(Fig. 24.82).
From the fifth century CE, the Indian zero, in its various forms, already
surpassed the heterogeneous notions of vacuity, nihilism, nullity, insignifi-
cance, absence and non-being of Greek-Latin philosophies. *Shunya
embraced all these concepts, following a perfect homogeneity: it signified
not only void, space, atmosphere and “ether”, but also the non-created, the
non-produced, non-being, non-existence, the unformed, the unthought,
the non-present, the absent, nothingness, non-substantiality, nothing
much, insignificance, the negligible, the insignificant, nothing, nil, nullity,
unproductiveness, of little value and worthlessness (see *Shunyata, *Zero,
and Fig. 24.D10 and Dll of the latter entry in the Dictionary).
It was also, and above all, an eminently abstract concept: in the simplified
Sanskrit system, as well as in the positional system of the numerical symbols,
the word *shunya and its various synonyms served to mark the absence of
units within a given decimal order, in a medial position as well as in an initial
or final position; the point or the little circle were used in the same way.
This zero was also a mathematical operator: if it was added to the end of a
numerical representation, the value of the representation was multiplied by ten.
By freeing the nine basic numerals from the abacus and inventing a sign
for zero, the Indian scholars made significant progress, primarily simplify-
ing quite considerably the rules of a technique which would lead to the
birth of our modern written calculation.
The Indian people were the only civilisation to take the decisive step
towards the perfection of numerical notation. We owe the discovery of
modern numeration and the elaboration of the very foundations of written
calculations to India alone.
It is very likely that this important historical event took place around
the fourth century CE. It is thanks to the genius of the Indian arithmeti-
cians that three significant ideas were combined:
LIST OF SANSKRIT WORDS FOR ZERO
SYMBOLS
THEIR MEANINGS
*Abhra
Atmosphere
* A kasha
Ether
*Ambara
Atmosphere
*Ananta
Immensity of space
*Antariksha
Atmosphere
*Bindu
Point (or Dot)
*Gagana
Canopy of heaven
*Jaladharapatha
Voyage on water
*Kha
Space
*Nabha
Sky, Atmosphere
*Nabhas
Sky, Atmosphere
*Purna
Entire, Complete
*Randhra (rare)
Hole
*'Shunya
Void
* Vindu
Point (or Dot)
* Vishnupada
Foot of Vishnu
*Vyant
Sky
* Vyoman
Sky, Space
Ref.: AI-Biruni; Biihler; Burnell; Dana and
Singh; Fleet, in: CIIN, III;Jacquet, in: JA,
XVI, 1935; Renou and Filliozat; Sircar;
Woepcke (1863).
GRAPHICAL SIGNS FOR ZERO
First sign:
the Little circle
o
Nowadays used in nearly all the nota-
tions oflndia and Southeast Asia
(Nagari, Marathi, Punjabi, Sindhi,
Gujarati, Bengali, Oriyd, Nepali,
Telugu, Kannara, Thai, Burmese,
Javanese, etc.). There is evidence of the
use of this sign which dates back
many centuries for nearly all these
systems.
Second sign:
the point, or dot
0
Formerly used in the regions of
Kashmir ( Sharada numerals).
There is also evidence of the use of this
sign in the Khmer inscriptions of
ancient Cambodia and the vernacular
inscriptions of Southeast Asia.
Fig. 24.3 to 51, 24.78 to 80. See also
'Numerals “0”, •Zero and
Fig. 24.D11.
Fig. 24 . 82 . The various representations of the Indian zero'
• nine numerals which gave no visual clue as to the numbers they rep-
resented and which constituted the prefiguration of our modern
numerals;
• the discovery of the place-value system, which was applied to these
nine numerals, making them dynamic numerical signs;
• the invention of zero and its enormous operational potential.
Thus we can see that the Indian contribution was essential because it
united calculation and numerical notation, enabling the democratisation of
calculation. For thousands of years this field had only been accessible to the
privileged few (professional mathematicians). These discoveries made the
domain of arithmetic accessible to anyone.
It still remained for the Indian scholars to perfect the concept of zero
and enrich its numerical significance.
Beforehand, the *shunya had only served to mark the absence of units in
a given order. The Indian scholars, however, soon filled in the gap. Thus, in
* The Arabs acquired their signs for zero, as well as the place- value system, from the Indians. This is why we
find the point and the little drde used to express zero in Arabic texts. The circle was the sign to prevail in the
West, after the Arabs transmitted it to the Europeans some time after the beginning of the twelfth century.
439
a short space of time, the concept became synonymous with what we now
refer to as the “number zero" or the “zero quantity".
The *shunya was placed amongst the *Samkhya, which means it was
given the status of a “number".
In c. 575, astronomer ‘Varahamihira, in Panchasiddhantika, mentioned
the use of zero in mathematical operations, as did ‘Bhaskhara in 629 in his
commentary on the Aryabhatiya.
In 628, in Brahmasphutasiddhanta, ‘Brahmagupta defined zero as the
result of the subtraction of a number by itself (a - a = 0), and described its
properties in the following terms:
When zero ( *shunya ) is added to a number or subtracted from a
number, the number remains unchanged; and a number multi-
plied by zero becomes zero.
Moreover, in the same text, Brahmagupta gives the following rules con-
cerning operations carried out on what he calls “fortunes” ( dhana ), “debts”
( rina ) and “nothing” (kha) [see S. Dvivedi (1902), pp. 309-10, rules 31-5)]:
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero ( *shunya ) minus zero is nothing ( *kha ).
A debt subtracted from zero is a fortune.
So a fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or a fortune is zero.
The product of zero multiplied by itself is nothing.
The product or the quotient of two fortunes is one fortune.
The product or the quotient of two debts is one debt.
The product or the quotient of a debt multiplied by a fortune is a debt.
The product or the quotient of a fortune multiplied by a debt is a debt.
Modern algebra was born, and the mathematician had thus formulated
the basic rules: by replacing “fortune” and “debt” respectively with “posi-
tive number” and “negative number”, we can see that at that time the
Indian mathematicians knew the famous “rule of signs” as well as all the
fundamental rules of algebra.
It is clear how much we owe to this brilliant civilisation, and not only in the
field of arithmetic; by opening the way to the generalisation of the concept of the
number, the Indian scholars enabled the rapid development of algebra, and thus
played an essential part in the development of mathematics and exact sciences.
The discoveries of these men doubtless required much time and imagi-
nation, and above all a great ability for abstract thinking. The reader will
not be surprised to leam that these major discoveries took place within an
environment which was at once mystical, philosophical, religious, cosmo-
logical, mythological and metaphysical.
THE BIRTH OF MODERN NUMERALS
Sarasvati, goddess of knowledge and music. From Moor’s Hindu Pantheon
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
440
CHAPTER 24 PART II
DICTIONARY OF THE
NUMERICAL SYMBOLS OF
INDIAN CIVILISATION
As we have seen in the course of this chapter, India has always dominated
the world in the field of arithmetic, and indeed the art of numbers plays a
leading role in Indian culture.
It is precisely this skill in the field of mathematics which led Indian
scholars, at a very early stage, to develop their astonishing * arithmetical
speculations which could involve numbers comprising hundreds of decimal
places, whilst maintaining a clear idea of the order of the ascending powers of
ten within a nomenclature which contained a highly diversified terminology,
based as it was upon both specific etymologies and improvised terms born
out of a highly creative symbolical imagination.
This same arithmetical genius led to their inevitable discovery of zero
and the place-value system and even enabled them to come within touching
distance of the concept of mathematical infinity.
Therefore one should not be surprised to learn that, a thousand years
earlier than the Europeans, Indian mathematicians already knew that zero
and infinity were inverse concepts. They realised that when any number is
divided by zero the result is infinity: a / 0 = this “quantity” undergoing
no change if it is added to or subtracted from a finite number.*
We should not forget that these crucial discoveries were not the fruit of
just one genius’s individual inspiration nor even that of a group of
“mathematicians” as we understand the term today. They were, of course,
learned Indians. However, there should be no ambiguity about the meaning
of this term: to be termed as learned at that time meant that one was a
thinker, a little like the scholarly gentlemen of sixteenth-century Europe,
except that an Indian’s way of thinking would have been very different from
that of the European scholars. The Indian scholar would have been a man of
constant reflection whose studies covered the most diverse topics. Moreover,
mystical, symbolical, metaphysical and even religious considerations came
first and foremost in his learning:
As India knew nothing of the work of either Aristotle or Descartes, and was
ignorant of Jewish or Christian ethics, it would be futile to attempt to draw
a parallel between these vastly different civilisations. The foundations are
* In other words, Indian scholars knew the following properties: (a/0) ± k = k ± (a/0) = (a/0), which is to
say: « ± k = k ± » = »
completely different, as are the customs and ways of thinking. It is
impossible to make any comparisons, even if some aspects of Indian
culture do seem to coincide with our own. [L. Frederic, Didionnaire de la
civilisation indienne (1987)]
It would also be futile to try and make any comparisons between Indian
mathematics and modern mathematics, modern mathematics being the
very refined product of contemporary western civilisation: a highly abstract
science that has been stripped of any mystical, philosophical or religious
influence.
Moreover, the following pages prove that the main preoccupations of
Indian scholars had nothing to do with what we in the West today refer to
as “hard sciences”. In fact, we will see that these major discoveries stem
from the incessant study of astronomy, poetry, metric theory, literature,
phonetics, grammar, philosophy or mysticism, and even astrology,
cosmology and mythology all at once.
In India, an aptitude for the study of numbers and arithmetical research
was often combined with a surprising tendency towards metaphysical
abstractions: in fact, the latter is so deeply ingrained in Indian thought and
tradition that one meets it in all fields of study, from the most advanced
mathematical ideas to disciplines completely unrelated to ‘exact’ sciences.
In short, Indian science was born out of a mystical and religious culture
and the etymology of the Sanskrit words used to describe numbers and the
science of numbers bears witness to this fact. + Together, the discoveries in
question represent the culmination of the uniqueness, wealth and
incredible diversity of Indian culture.
To give the reader a clearer idea of the circumstances and conditions
under which these major discoveries were made, it seems useful now to
reiterate the principal notions that have already been explained in this
chapter in the form of a Sanskrit and English dictionary and so to define (if
need be) these ideas in a more analytical form. This dictionary can serve as
a glossary to the numerous ideas which have been covered, each term being
marked with an asterisk.
* The term for the “Science of numbers" in Sanskrit is *samkhyana (also spelt sankhyana) which means
“arithmetic": and, by extension, ‘astronomy” (from the time when the science of the stars was not consid-
ered to be a separate discipline from arithmetic). The word is frequently used in this sense in *Jaina
literature, where the science of numbers was considered to be one of the fundamental requirements for the
full development of a priest; likewise in later Buddhist literature it was considered to be the most noble of
the arts, the word “number” itself is *samkhya or *samkhyeya. One should note that this term not only
applies to the concept of number but also to the actual numerical symbol. Arithmetician or mathematician
is denoted by *sdmkhya. But *sdmkhya is also the term used for one of six orthodox (and most ancient) sys-
tems of the Hindu philosophy of the six *darshana ("contemplations"), which teaches “number” as a way of
thinking which is connected to the liberation of the soul, and according to which the universe was born out
of the union of *prakriti (nature) and purusha (the conscience). It is significant that the word *samkhyd,
which also means a follower of this philosophy, is the term used to refer to the “calculator", but in a mystical
sense in this context.
441
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
Thus the dictionary can help the reader through the maze of obscure
rubric of the Sanskrit language and the complex concepts of Indian science
and philosophy.
The dictionary is not only aimed at specialists: the entries, recorded with
careful clarity and precision, are concise and easily accessible to the layman.
It is not even necessary to have read Chapter 24 (or the preceding chapters)
to grasp the concepts it explains.
Its entries are recorded alphabetically, regardless of whether the
terminology is in English or Sanskrit.
This dictionary also serves as a thematic index which can clarify the ideas
presented in this chapter through an effective reference system of general or
specific rubric, giving not only references to Chapter 24 but also to those of
the forthcoming Chapters 25 and 26. For example, the reader has only to turn
to the entries *Chhedi, *Shaka or *Vikrama to find out about each era.
Under *Asankhyeya, the Sanskrit for “incalculable”, we learn that the
same term was also sometimes used to express the rather more modest sum
of ten to the power 140.
Similarly, the term *Padma, or *Paduma, reveals that the poetic name
“pink lotus” was used to denote the number ten to the power of 14 (or 29)
as well as ten to the power of 119. The lotus flower was used to represent
various numbers in Indian mathematics, the values of which depended on
the colour, the number of petals and whether the flower was open, just
flowering or still in bud. Thus *white lotus came to mean ten to the power
27 or ten to the power 112, *pink lotus ten to the power 21 or 105, and *blue
lotus (half-open) ten to the power 25 or 98. (See *Utpala, *Pundarika,
*Kumud and *Kumuda that can all mean “lotus”, according to slight
characteristic differences of the flower.)
Under entries entitled *High numbers, there are numerous other
examples of the unique symbolism which without a doubt was the source of
inspiration for the names of these large figures. The same entries also
demonstrate how in ancient India, grammar and interpretation were
inextricably linked to the handling of high numbers to the extent to which
the study of poetry and the Sanskrit metric system inevitably initiated the
Indian scholars into the art of arithmetic as well as grammar. Consequently
poets, grammarians and astronomers, in fact all learned men, were as
skilled at calculation as the arithmeticians and the teachers themselves.
Under the entries *Ananta, we see that the Sanskrit name for infinity
was not only used to denote the sum of ten thousand million (ten billion in
US English) but also, curiously, it was used as the symbol for zero.
The entries * Infinity and * Serpent will allow the reader to understand the
relationship between infinity as we understand the term, and the
mythological world of Hinduism, and that *Ananta, often represented as
coiled up in a sort of sleeping “8” (like our symbol °°), is none other than
the immense serpent of infinity and eternity, which is linked to the ancient
myths of the original serpent.
This dictionary provides an insight into the circumstances under which the
Indian scientists invented zero and the place-value system. See *Namcs of
numbers, *Sanskrit, *Place-value system, *Pnsition of numerals, *Numerical
symbols (Principle of the numeration of), *Shunya and *Zero.
Under the entries *Shunya and *Shunyatd, the Indian philosophical
notions of "void” or “emptiness” are very briefly explained, and we can see
how these early Indian concepts went far beyond corresponding but very
heterogeneous notions of contemporary Graeco-Latin philosophies.
It also shows how, from a very early stage, *shunya meant zero as well as
emptiness, the central element of the deeply religious and mystical
philosophy, *shunyatd. The word came to represent zero when the place-
value system was discovered, the two concepts fitting together naturally.
The dictionary also explains how, through the use of symbolism, this
concept finally came to be graphically represented as the little circle that we
all recognise as zero.
The entries *Yuga and *Kalpa tell of Indian cosmic cycles, and the
speculations developed about them, both in Indian cosmogony and in the
learned astronomy introduced by ‘Aryabhata at the beginning of the
sixth century. See also * Aryabhata, *Cosmic cycles, *Day of Brahma and
* Indian astronomy.
The entry * Aryabhata’s number-system serves as further proof that it was
the Indians who discovered zero and who are responsible for our current
written number-system. In fact, we will see that this scholar, whilst
constructing his own numerical notation (a very clever alphabetical
number-system), could not have failed to have known of zero and the place-
value system, given the very structure of the system.
If we look at the entries beginning with the expression *numeral
alphabet, we can find information about Indian alphabetical numbering
systems. This will also give the reader some idea of the practices which were
quite naturally born out of their usage: the composition of chronograms,
the invention of secret codes, the preparation of talismans (closely linked to
the Indian mysticism of letters and numbers), the development of
homiletic or symbolic interpretations, predictive calculations and magical
and divinatory practices.
Under the appropriate headings, one can similarly find short
biographical notes about the great Indian scientists such as ‘Aryabhata,
‘Bhaskara, ‘Bhaskaracharya, ‘Brahmagupta or ‘Varahamihira, often
accompanied by very precise accounts of the numerical notations they
adopted (including bibliographical references).
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
442
If we consult the entries *Brdhmi numerals, *Gupta numerals, *Nagari
numerals, *Sharada numerals, etc., we can also find out all about each style
and see the impressive diversity of *Indian written numeral systems.
Alternatively, the reader can consult * Indian numerals.
Extra details can be found about * Indian arithmetic, the different *ages
of the sub-continent and * Indian astrology, * Indian astronomy and the
* Indian mathematics of this civilisation.
However, it is not necessary to look up all of the references given here:
entries are accompanied by references to similar terms.
Entries such as * Algebra, * Arithmetic, * High numbers, *Names of numbers,
*Numcrical notation etc., include either an alphabetical or a numerical list
of terms relating to each of the ideas in question.
As for references which seem to have little to do with arithmetic,
see: * Astronomical speculations, *Buddhism, *Brahmanism, *Cosmogonic
speculations, *Hinduism, * Indian cosmogonies and cosmologies, *Indian divinities,
* Indian mythologies, * Indian thought, * Indian religions and philosophies, *)aina, etc.
The main aim behind creating this dictionary has been, however, to give
the reader a better idea of the subtle and complicated world of Indian
numbers: a world which is largely unknown to Western readers and which
is closely linked to Indian legends and cosmogonies. Its symbols, rather
than being ordinary graphic signs and names of numbers, derive from a
huge wealth of synonyms inspired by nature, human morphology, everyday
activities, social conventions and traditions, legends, religion, philosophy,
literature, poetry, the attributes of the divinities and by traditional and
mythological ideas.
Thus, depending on the context, the idea of* wind can evoke the number
5, the number 7 or the number 49. This demonstrates a subtlety which
Westerners would not grasp if it was not shown from the correct
perspective. The reasoning behind these diverse meanings offers a
fascinating example of a logic and a way of thinking which is highly
characteristic of the Indian mentality, and will help the reader to
understand Cartesianism, which can often, due to the very nature of its
rationalism, seem in total contradiction. To find out about the logic behind
the above values of* wind, see *Vayu, *Pdvana and * Mount Meru.
Other examples include: the term *anga, which literally means “limbs”
or “parts” and is often used as the numerical symbol for the number six; the
word *rasa, “sensations", is frequently used to denote the same number;
the name of *Rudra, the ancient Vedic god of the Sky, was used as the
numerical symbol for 11, etc. Similarly, the *God of carnal love, whose name
is *Kdma, was a symbol for the number 13; the *God of water and oceans,
*Varuna, was the symbol for the 4; *Agni, the *God of sacrificial fires meant
“three”, etc.
These examples (along with many others) show the subtlety of the
Indian symbolic system as well as demonstrating one of the most
characteristic traits of the Indian cast of mind.
This dictionary contains a considerable amount of symbolism. A term with
a symbolic meaning is denoted by an [S], an abbreviation of the actual Indian
numerical symbol, and is defined firstly by its numerical value and literal
meaning in Sanskrit and then, where possible, its symbolism is explained.
To gain a better understanding of the world in which the Indians created
such symbols, the reader might find it useful to read the entries * Symbols
and * Numerical symbols.
To find the Indian numerical symbol for a given number, one only has to
look up the English (or Sanskrit) equivalent: *One, *Two, *Three, etc. (or
*Eka, *Dva, *Tri, etc.) Under *Ashta, for example, the normal Sanskrit
word for the number 8, there is a list of terms in which the word appears
because of a direct link with the idea of the number 8 (for example
*ashtadiggaja, the “eight elephants”, guardians of the eight horizons of
Hindu cosmogony). But if the reader wishes to know about words that have
a more symbolical relationship with the same number, he should refer to
the entry * Eight, where there is not only a list of numerical symbols which
are synonymous with this number, but also a summary explanation of its
different symbolical meanings: the serpent ( *Ahi , *Naga, *Sarpa), the
serpent of the deep ( *Ahi ), the elephant ( *Dantin , *Dvipa, *Gaja), the eight
elephants ( *Diggaja ), a sign that augurs well ( *Mangala ), etc.
Of course, one could also consult the more detailed rubric either in
Sanskrit: *Hastin, *Lokapdla, *Murti, *Tanu, etc., or the English translation
of the concepts behind Indian numerical symbols, such as: *Sky, *Space,
* Elephant, *Moon, * Earth, *Sun, *Zodiac, *Serpent, etc.
The entry * Numerical symbols (general alphabetic list) contains all the
word-symbols of the Sanskrit language that are included in this dictionary,
whilst the entry *Symbolism of words with a numerical value gives an
alphabetical list of English words which correspond with associated ideas
contained within the Sanskrit symbols.
The entry * Symbolism of numbers provides a list of ideas (in arithmetical
order) found in the symbolism of ordinary numbers, in high numbers and
in the concept of infinity or zero.
This dictionary is the first of its kind to attempt to understand the thought
process of the symbolic mind that characterises Indian numerical thinking.
Through a multidisciplinary process, mainly concerned with numbers
and the symbols which represent them, the following is a kind of “vertical
reading” of literary, philosophical, religious, mystical, mythological,
cosmological, astronomical, and of course mathematical elements of this
incredibly rich and subtle civilisation: elements which can be found in
443
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
“horizontal” presentation in a great many wide-ranging publications in the
most specialised of libraries.
This dictionary could be said to complement L' I ride Classique by L. Renou
and J. Filliozat (1953), and also the monumental Dictionnaire de la civilisation
indienne by L. Frederic (1987) (the first of its kind to condense, in a
remarkably simple yet well-documented manner, the essential facts about
the India of both yesterday and today from a historical, geographical,
ethnographic, religious, philosophical, literary and linguistic perspective). It
also supplements the Dictionnaire de la sagesse orientate by K. Friedrichs, I.
Fischer-Schreiber, F. K. Erhard and M. S. Diener (1989), which constitutes a
vast yet accessible range of references and a very enriching insight to those
who are interested in philosophy, mysticism and meditation, or in a general
introduction to the doctrines of Hinduism, Buddhism, Taoism and the
religion of Zen. It is also the perfect companion to the Dictionnaire des
symboles by J. Chevalier and A. Gheerbrant (1982), which not only explains
the history of symbolic language through the ages and in different
civilisations and the indelible yet hidden imprint it has left in our minds, but
also opens the doors of the imagination and invites the reader to meditate
on the symbols, just as Gaston Bachelard invited us to muse on our dreams
in order to discover within them the taste and feel of a living reality. These
works have all influenced the writing of this book; without them the
following dictionary could not have been compiled because the required
research would have taken an inordinate amount of time and would have
involved reading analytical works which are inaccessible to the public, and
which, moreover, are devoid of any synthesis.
The author warmly thanks Billard, Frederic, Chevalier and Jacques for
their invaluable personal correspondence, especially Billard and Frederic
for reading the rubrics of this dictionary and who offered pertinent and
constructive remarks.
The writing of this dictionary had to be handled with utmost caution
(especially considering that this field of study was completely new to the
author) for several reasons:
• The author was in danger of being carried away by his own enthusiasm;
a justified enthusiasm, yet capable of leading to erroneous interpretations.
• The vertiginous world of Indian symbols is highly complex.
• Moreover, the culture in question (whose diverse aspects were stud-
ied, notably the countless symbols which are often multivalent) is not
only incredibly complex but is also full of pitfalls. See * Indian docu-
mentation ( Pitfalls of).
• Finally, Indian astronomy has played a significant role in this histo-
riography. (The available documentation only offered a relatively
simple insight into the literature of Indian astronomy. However, C.
Jacques obligingly recommended Billard’s Astronomic Indienne (1971)
which is an unprecedented publication on the subject.)
Suffice to say that embarking upon this domain was rather like coming
face to face with one of the many-headed dragons of the legends of
Indian mythology: it was merciless and threatened to devour the author
at any moment, as it had those who had set foot in this territory before
without the necessary amount of precaution and vigilance. Now tamed,
however, the appeased monster offers the reader all the delights of
Eastern subtlety, and the wonder of this ingenious civilisation and its
inestimable contributions.
A
ABAB. Name given to the number ten to the
power seventeen (= a hundred trillion). See
Abhabagamana, Names of numbers, High
numbers.
Source: *I.alitavistara Sutra (before 308 CK).
ABABA. Name given to the number ten to the
power seventy-seven. See Abhabagamana,
Names of numbers, High numbers.
Source: *\ /ydkarana (Pali grammar) by Kachchayana
(eleventh century CE).
ABBUDA. Name given to the number ten to
the power fifty-six. See Names of numbers,
High numbers.
Source: *Vyakarana (Pali grammar) by Kachchayana
(eleventh century CE).
ABDHI. [SI. Value = 4. “Sea”. Four seas were
said to surround *Jambudvipa (India). See
Sagara, Four. See also Ocean.
ABHABAGAMANA. “Beyond reach”. Sanskrit
term used to express the uncountable and
unlimited. It is possible that the words *abab
(ten to the power seventeen) and *ababa (ten
to the power seventy-seven) were abbreviations
of this word. See Names of numbers, High
numbers, Infinity. See also Asamkhyeya.
ABHRA. [S]. Value = 0. “Atmosphere”. The
atmosphere represents “emptiness”. See Shunga,
Zero.
ABJA. Literally: “Moon”. Name given to the
number ten to the power nine (= a thousand
million). See Names of numbers. For an expla-
nation of this symbolism: see High numbers
(Symbolic meaning of).
Source: *Ulavati by Bhaskaracharya (1150 CE);
*Trishatika by Shridharacharya (date uncertain).
ABJA. [SI. Value = 1. “Moon”. The moon is
unique. Another reason for this symbolism
could be that in Indian tradition the moon is
considered to be the source and symbol of
fertility. It is likened to the primordial waters
from whence came the revelation: it is the
receptacle of seeds of the cycle of rebirth. It is
thus the unity as well as the starting point.
See One.
ABLAZE. [S]. Value = 3. See Shikhin and Three.
ABSENCE, ABSENT. See Shunyata, Zero.
ABSOLUTE. As a symbol representing a large
quantity. See High numbers (Symbolic mean-
ing of), Infinity.
ADDITION. [Arithmetic]. *Samkalita in Sanskrit.
ADI. [SJ. Value = 1. “Commencement, primor-
dial principle”. In Hindu and Brahman
philosophy, this principle is said to be found in
all things before the creation; thus it is the
unity as well as the starting point. See One.
ADITYA. [S]. Value = 12. “Children of Aditi". In
Brahman and Vedic cosmogony, Aditi is the infi-
nite sky, the original space. The Aditya are its
children. In Vedic times, there were five, then
they became seven, and finally twelve and were
consequently identified by the twelve months of
the year and the course of the sun during this
period of time. The same word also signifies
Surya, the sun god of the * Vedas. As Surya = 12,
the children of Aditi = 12. See Surya.
ADRI. [S|. Value = 7. “Mountain”. Allusion to
‘Mount Meru, sacred mountain which, according
to ancient Indian cosmological representation,
was situated at the centre of the universe and con-
stituted a meeting and resting place for the gods: a
representation where we know seven played a sig-
nificant role. See Seven.
AGA. [Sj. Value = 7. “Mountain”. See Adri.
Seven.
AGES (The four). See Chaturyuga.
AGNI. [S] . Value = 3. “Fire”. In Brahman
mythology, Agni is the god of sacrificial fire
(the three Vedic fires), which is represented as a
man with three bearded heads who appears in
three different forms: in the sky as the sun, in
the air as lightning and on the earth as fire.
Hence: “fire” = 3. See Fire, Three.
AGNIPURANA. See Purana and positional
numeration.
AHAHA. Name given to the number ten to
the power seventy. See Names of numbers,
High numbers.
Source: *Vydkarana (Pali grammar) by Kachchayana
(eleventh century CE).
AHAR. [S]. Value = 15. “Day”. See Tithi, Fifteen.
AHI. [SI. Value = 8. Probably an allusion to
Ahirbudhnya (or Ahi Budhnya) who, in Vedic
mythology, designates the serpent of the
depths of the ocean, born of dark waters. Thus:
Ahi = 8, because the serpent corresponds sym-
bolically to the number eight. See Ndga, Eight.
See also Serpent (Symbolism of).
AHIRBUDHNYA. See Ahi.
AKASHA. [S]. Value = 0. “Ether”, the “element
which penetrates everything", “space". It was
considered as emptiness which could not mix
with material things, immobile and eternal,
beyond description. The association of ideas
with the “void" or “emptiness" ( shunya ) was
established well before shunya was identified
with the concept of zero. In Indian thought
ether was not only the void; it was also and
above all the most subtle of the five elements of
the revelation. It is certainly devoid of sub-
stance, but akasha is regarded as the condition
of all corporeal extension and the receptacle of
all matter formed by one of the other four ele-
ments (earth, water, fire or air). The association
of ideas with zero became even more evident
when this fundamental discovery was made:
zero not only signified a void and “that which
has no meaning”, but also played an important
role in the place-value system, and in terms of
an abstract number, an equally essential role in
mathematics and all the other sciences. Hence
the symbolism: akasha = “space” = “void” =
“ether” = "element which penetrates every-
where" = 0. See Shunya, Shunyata, Zero.
AKKHOBHINI. Name given to ten to the
power forty-two. See Names of numbers. High
numbers.
Source: M 'ydkarana (Pali grammar) by Kachchayana
(eleventh century CE).
AKRITI. [SJ. Value = 22. In terms of the poetry
of Sanskrit expression, Akriti means the metre
of four times twenty-two syllables per verse.
See Indian metric.
AKSHARA. [Sj. Value = 1. “Indestructible". A
Sanskrit word which, in Hindu philosophy,
denotes the “undying" part of the vocal sound
corresponding to the revelation of the
Brahman. This is a direct reference to the word
*ekakshara, the “Unique and undying” which is
often expressed by the Sacred Syllable *AUM.
See Trivarna, Mysticism of letters. One.
AKSHARAPALLf. Prakrit word meaning
“letter-phoneme, syllable". It denotes a numer-
ical notation of the alphabetical type
frequently used in *Jaina manuscripts. See
Numeral alphabet.
AKSHITI. Name given to the number ten to the
power fifteen (= trillion). See Names of numbers,
High numbers.
Source: *Panchavimsha Brahmana (date uncertain).
AL-BIRUNI (Muhammad ibn Ahmad Abu’l
Rayhan) (973-1048). Muslim astronomer
and mathematician of Persian origin. After
having lived in India for nearly thirty years,
and having been initiated into the Indian sci-
ences, he wrote many works, including Kitab
al arqam (“Book of numerals"), Tazkirafi’l
hisab wa’l mad bi'l arqam al Sind wa’l hind
(“Arithmetic and counting systems using
numerals in Sind and the Indias”), and above
all Kitab fi tahqiq i ma li’l hind which consti-
tutes one of the most important accounts of
India in mediaeval times. Al-Biruni described
the system of Sanskrit numerical symbols in
minute detail, and stressed the importance
of the place-value system and zero. He also
went into much detail about the Sanskrit
counting system, attaching particular impor-
tance to the Indian nomenclature of high
numbers (see Fig. 24. 81). Here is a list of the
principal names of numbers mentioned in
Kitab fi tahqiq i ma li’l hind (see Woepcke
(1863) p. 2791 :*Eka (= 1). *Dasha (= 10).
* Shat a (= 10 2 ). *Sahasra (= 10 3 ). *Ayuta
(= lO* 1 ). *Laksha (= 10 5 ). *Prayuta (= 10 6 ).
*Koti (= 10 7 ). *Vyarbuda (= 10 8 ). *Padma
(= 10 s ). *Kharva (= 10 10 ). *Nikharva (= 10 u ).
*Mahapadma (= 10 12 ). *Shankha (= 10 13 ).
*Samudra (= 10 14 ). * Madhya (= 10 15 ). *Antya
(= 10 16 ). *Pardrdha (= 10 17 ). See Indian
numerals, Nagari numerals, Names of
numbers, High numbers, Sanskrit,
Numerical symbols.
ALGEBRA. Alphabetical list of the words relat-
ing to this discipline, to which a rubric is
dedicated in this dictionary: * Avyaktaganita.
*Bija. *Bijaganita. *Ghana. ‘Indian mathematics
(History of). *Samikarana. * Varga. *Varga-Varga.
* Varna. *Vyavahara. *Ydvattavat.
ALPHABETICAL NUMERATION. See
Aksharapalli, Aryabhata’s numeration,
Katapayadi numeration, Numeral alphabet,
Varnasamjha and Varnasankhya.
AMARA. [S]. Value = 33. “Immortal”. Allusion
to the thirty-three gods. See Deva, Thirty-three.
AMBARA. [S]. Value = 0. “Atmosphere". See
Abhra, Zero.
AMBHODHA (AMBHODHI). [SJ. Value = 4.
"Sea". It was said that four seas surrounded
*Jambud\>ipa (India). See Sagara, Four. See
also Ocean.
AMBHONIDHI. [Sj. Value = 4. “Sea”. See
Sagara , Four. See also Jala.
AMBODHA (AMBODHI, AMBUDHI). [S].
Value = 4. “Sea”. See Sagara. Four. See also
Ocean.
AMBURASHI. [S]. Value = 4. “Sea”. See
Sagara. Four. See also Ocean.
AMRITA. Nectar of “Immortality". See Soma,
Serpent (Symbolism of the).
ANALA. [S]. Value = 3. “Worlds". See Loka.
Three.
ANANTA. Literally “Infinity". Name given to
the number ten to the power thirteen (= ten
billion). See Asamkhyeya, Names of numbers.
For an explanation of this symbolism see High
numbers (symbolic meaning of).
Source: *Sankhyayana Shrauta Sutra (date
uncertain), which defines this number as the '‘limit
of the calculable".
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
446
ANANTA. Word which literally means “infin-
ity”. In Hindu mythology, the ananta denotes a
huge serpent representing eternity and the
immensity of space. It is shown resting on the
primordial waters of original chaos (Fig. D. 1).
Vishnu is lying on the serpent, between two
creations of the world, floating on the “ocean of
unconsciousness". The serpent is always repre-
sented as coiled up, in a sort of figure eight on
its side (like the symbol ®o), and theoretically
has a thousand heads. It is considered to be the
great king of the *ndgas and lord of hell
( *patdla ). Each time the serpent opens its
mouth it produces an earthquake because
there is a belief that the serpent also supported
the world on its back. It is the serpent that at
the end of each *kalpa, spits the destructive fire
over the whole of creation. See Infinity. See
also Serpent.
ANANTA. [S]. Value = 0. “Infinity”. It seems
paradoxical, yet this symbolism comes from
the association of Ananta, the serpent of infin-
ity, with the immensity of space. As “space” =
0, the name of the serpent became a synonym
of zero. See Ananta (the second of the above
entries). Zero.
ANCESTOR. [S], Value = 1. See Pitamaha. One.
ANDHRA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary of
Shunga, Shaka and Kushana numerals, used in
the contemporary inscriptions of the Andhra
dynasty (second - third century CE). These signs
are notably found in the inscriptions of
Jaggayapeta. The corresponding system did not
function according to the place-value system
and moreover did not possess zero. See Indian
written numeral systems (Classification of)
See also Fig. 24.34, 36, 24.52. and 24.61 to 70.
ANGA. IS]. Value = 6. “Limb". The human
body consists of six “limbs”, or members: the
head, the trunk, two arms and two legs. This
is not the only reason, however, for this sym-
bolism: there are six appended texts of the
Veda (a group of Vedic texts called Vedanga
which deal mainly with Vedic rituals, of their
conservation and their perfect transmission).
As Vedanga means the “members of Veda”, we
can see how the idea of “member" or “limb”
came to signify the number six. See Veda,
Vedanga, Six.
ANGULI. IS]. Value = 10. “Digit”, because we
have ten fingers. See Ten.
ANGULI. IS]. Value = 20. “Digit”, because we
have ten fingers and toes. See Twenty.
ANKA. Literally “mark, sign”. The term means
“numeral”, “sign of numeration”. See Anka [S].
See also all entries beginning with Numeral.
ANKA. [S]. Value = 9. “Numerals”. Allusion to
the nine significant numerals of the Indian
place- value system. This symbol was in use no
later than the time of *Bhaskara I (629 CE). See
Anka. Ankasthana. Nine.
ANKAKRAMENA. Expression which literally
means “in the order of the numerals", and
alludes to the principle which the numerals are
subjected to in the place-value system. See
Anka. Sthana.
Source: * l.okavibhdga (458 CF.).
ANKANAM VAMATO GATIH. Expression
which means “principle of the movement of
the figures from the right to the left”. The num-
bers were read out in ascending order, from the
smallest units to the highest multiple of ten.
This was the reverse of how the numbers were
presented in Indian numerical notations (from
left to right).
ANKAPALLI. Prakrit term which literally means
“numerals, representation”. It is applied to any
system of representing numbers using actual
numerals. Thus it denotes “numerical notation”.
ANKASTHANA. Literally “Numerals in position".
The Sanskrit name for positional numeration.
Source: *Lokavibhdga (458 CE).
ANKLE. IS]. Value = 2. See Gulpha and Two.
ANTA. Name given to the the number ten to
the power eleven (= a hundred thousand mil-
lion). See Names of numbers, High numbers.
Sources: * Vajasaneyi Samhitd, *Taittiriya Samhitd and
*Kdthaka Samhitd (from the start of the first millen-
nium CE); * Pahchavimsha Rrahmana (date uncertain).
ANTARIKSHA. [S]. Value = 0. “Atmosphere”.
See Abhra. Zero.
ANTYA. Literally “(the) last”. Name given to
the number ten to the power twelve (= a bil-
lion). See Names of numbers, High numbers.
Source: *Sankhyayana Shrauta Sutra (date uncer-
tain). An allusion is made to the highest order of
units of the ancient Sanskrit numeration at the time
of the * Vajasaneyi Samhitd, *Taittiriya Samhitd and
*Kathaka Samhitd (from the start of the first millen-
nium CE), where the nomenclature stopped at ten
to the power twelve.
ANTYA. Literally “(the) last”. Name given to
the number ten to the power fifteen (= a tril-
lion). See Names of numbers, High numbers.
Sources: *I.ilavati by Bhaskaracharya (1150 CE);
*Ganitakaumudi by Narayana (1350 CE),
*Trishalika by Shridharacharya (date uncertain). At
this later date, when the Sanskrit names for num-
bers by far surpassed the simple power of fifteen,
the name of this number still retained a vestige of
the limitation of the spoken numeration of ancient
times. See the first entry under Antya.
ANTYA. Literally “(the) last”. Name given to the
number ten to the power sixteen (ten trillion).
See Antya (the above entries, Sources), Names
of numbers, High numbers.
Source: *Kitab ji tahqiq i ma li'l hind by al-Biruni
(c. 1030).
ANU. Sanskrit name for “atom”. See Paramdnu.
ANUSHTUBH. IS]. Value = 8. Name given to
certain groups of verses of Vedic poetry. This is
an allusion to the eight syllables which made
up each one of the four elements which consti-
tuted the stanzas which were called anushtubh.
See Eight, Indian metric.
ANUYOGADVARA SUTRA. Title of a Jaina
cosmological text giving countless examples of
extremely high numbers, the corresponding
speculations reaching (and even surpassing)
easily as high as ten to the power two hundred
(one followed by two hundred zeros). Thus, the
figure said to express the total number of
human beings of the creation is described as
the “quantity obtained by multiplying the sixth
power of the square of two ( = (2 2 ) 6 = 2 12 ) by
the third power of two [ = 2 3 = 8], which is
equal to the number which can be divided by 2
ninety-six times ( = (2 12 ) 8 = 2 12 * 8 = 2 96 ]” [see
Datta and Singh (1938), p.12]. There is also the
period of time called *Shirshaprahelika,
expressed, according to Hema Chandra (1089
CE), by “196 places of the place-value system”
and which corresponds approximately “to the
product of 84,000,000 multiplied by itself
twenty-eight times” (see Datta and Singh,
op.cit). This text, amongst many others, shows
that the Jainas were amongst the Indian schol-
ars who were most familiar with such
arithmetical-cosmogonical speculations. See
Names of numbers, High numbers, Infinity.
APA. Sanskrit term meaning “water”. See Jala.
APHORISM. [S]. Value = 3. See Vdchana. Three.
APTYA. [S]. Value = 3. “Spirit of the Waters”.
Allusion to the Vedic divinity named Trita Aptya,
the “Third Spirit of the Waters”, who killed
Vishvarupa, the three-headed demon. See Three.
ARABIC NUMERATION (Positional systems
of Indian origin). See “Hindi” numerals and
Ghubar numerals.
ARAMAEAN-INDIAN NUMERALS. See
Kharoshthi numerals.
ARAMAEAN-INDIAN NUMERATION. See
Kharoshthi numerals.
ARBUDA. Name given to the number ten to
the power seven (= ten million). See Names of
numbers. High numbers.
Sources: * Vajasaneyi Samhitd (beginning of Common
Era); * Taittiriya Samhitd (beginning of Common
Era); *Kdthaka Samhitd (beginning of Common Era);
*Pahchavimsha Brdhmana (date unknown);
*Sankhyayana Shrauta Sutra (date unknown);
*Aryabhatiya (510 CE).
ARBUDA. Name given to the number ten to
the power eight (= one hundred million). See
Names of numbers. High numbers.
F i g . 2 4 D . i . Vishnu with Lakshmi and the serpent Ananta and Brahma sitting on a lotus flower which
grows out of Vishnu’s navel. From Dubois dejancigny, L'Univers pittoresque, Hachette, Paris, 1846
447
ARBUDA
Source: *Liidvati by Bhaskaracharya (1150 CE);
*Ganitakaumudi by Narayana (1350 CE);
*Trishatikd by Shridharacharya (date unknown).
ARBUDA. Name given to the number ten to
the power ten (= ten billion). See Names of
numbers, High numbers.
Source: *Ganitasdrasamgraha by Mahaviracharya
(850 CE).
ARITHMETIC. Here is an alphabetical list
of terms relating to this discipline which
appear as headings in this dictionary:
*Abhabaganiana, ‘Addition, *Algebra,
*Anka, * Ankakramena, *Ankasthana, ‘Arith-
metical operations, ‘Aryabhata, ‘Aryabhata
(Numerical notations of), ‘Aryabhata’s
numeration, *Asamkhyeya, ‘Base 10, ‘Base of
one hundred, ‘Bhaskara, ‘Bhaskaracharya,
*Bija, *Bijaganita, ‘Brahmagupta, ‘Buddha
(Legend of), ‘Calculation (The science of),
‘Calculation (Methods of), ‘Calculation on
the abacus, ‘Calculator, ‘Cube, ‘Cube root,
*Dashaguna, * Dashagunasamjhd, ‘Day of
Brahma, *Dhulikarma, ‘Digital calculation,
‘Divi-dend, ‘Division, ‘Divisor, ‘Equation,
‘Fractions, ‘High numbers, ‘Indeterminate
equation, ‘Indian mathematics (The history
of), ‘Infinity, *Kaliyuga, *KaIpa, *Katapayddi
numeration, *Khachheda, *Khahdra,
* Mahaviracharya, ‘Mathematician, ‘Math-
ematics, ‘Mental arithmetic, ‘Multiplication,
‘Names of numbers, ‘Narayana, ‘Numeral
alphabet, ‘Numerals, ‘Numerical symbols,
‘Numerical symbols (Principle of the numer-
ation of), *Pati, ‘Quotient, ‘Remainder,
‘Rule of five, ‘Rule of eleven, ‘Rule of nine,
‘Rule of seven, ‘Rule of three, ‘Sanskrit,
*Shatottaraganana, *Shatottaraguna, *Shato-
ttarasamjha, ‘Shridharacharya, ‘Square root,
*Sthana, ‘Subtraction, ‘Total, *Yuga, ‘Zero.
ARITHMETICAL OPERATIONS. See Calcu-
lation, Dhulikarma, Indian methods of
calculation, Parikarma, Pali, Pdtiganita,
Square roots (how Aryabhata calculated his).
ARITHMETICAL SPECULATIONS. See
Anuyogadvara Sutra, Asamkhyeya, Calcula-
tion, Day of Brahma, Yuga, Kalpa, Jaina,
Names of numbers, High numbers, and
Infinity.
ARITHMETICAL-COSMOGONICAL SPECU-
LATIONS. See Anuyogadvara Sutra, Asamkhyeya,
Calculation, Cosmic cycles, Day of Brahma,
Yuga (Definition), Yuga (Systems of calculation
of), Yuga (Cosmogonic speculations on),
Kalpa, Jaina, Names of numbers, High
numbers, Infinity.
ARJUNAKARA. [SI. Value = 1,000. “Hands of
Arjuna”. Allusion to the mythical sovereign
Arjunakartavirya, leader of the Haihayas and
king of the “seven isles”, who, according to one
of the legends of the Mahabhdrata, had a thou-
sand arms. See Thousand.
ARKA. [Si. Value = 12. “Bright”. An epithet
given to Surya, the sun god, who, symbolically,
represents the number twelve. See Surya.
Twelve.
ARMS. [SI. Value = Two. See Baku and Two.
ARMS OF ARJUNA. [S]. Value = 1,000. See
Arjunakara and Thousand.
ARMS OF KARTTIKEYA. [SI. Value = Twelve.
See Shanmukhabdhu and Twelve.
ARMS OF RAVANA. [SJ. Value = Twenty. See
Rdvanabhuja and Twenty.
ARMS OF VISHNU. [SJ. Value = Four. See
Haribdhu and Four.
ARNAVA. [SJ. Value = 4. “Sea”. Four seas were
said to surround *Jambudvipa (India). See
Sdgara. Four. See also Ocean.
ARROW. [SI. Value = 5. See Bdna, Ishu,
KaJamba, Morgana, Say aka, Shara, Vishikha
and Five.
ARYABHATA. A veritable pioneer of Indian
astronomy, Aryabhata is without doubt one of
the most original, significant and prolific schol-
ars in the history of Indian science. He was
long known by Arabic Muslim scholars as
Arjabhad, and later in Europe in the Middle
Ages by the Latinised name of Ardubarius. He
lived at the end of the fifth century and the
beginning of the sixth century CE, in the town
of Kusumapura, near to Pataliputra (now
Patna, in Bihar). His work, known as Aryab-
hatiya was written c. 510 CE. It is the first
Indian text to record the most advanced
astronomy in the history of ancient Indian
astronomy. The work also involves trigonome-
try and gives a summary of the main
mathematical knowledge in India at the begin-
ning of the sixth century, bearing witness to
the high level of understanding that had been
reached in this field at this time. The following
rapturous declamation by ‘Bhaskara (one of
Aryabhata’s disciples and most fervent of
admirers), taken from the Commentary which
he wrote on the Aryabhatiya in 629 CE gives
some indication of the high level of abstract
thought achieved by the scholar way ahead of
his time [see Billard, IJHS. XII, 2, p. Ill J;
“Aryabhata is the master who, after reaching
the furthest shores and plumbing the inmost
depths of the sea of ultimate knowledge of
mathematics, kinematics and spherics, handed
over the three [sciences] to the learned world.”
See Indian astronomy (History of).
Indian mathematics (History of).
ARYABHATA. (Numerical notations of).
When referring to numerical data, Aryabhata
often used the Sanskrit names of the numbers:
at least this is the impression we get if we look at
the sections of his work respectively entitled
Ganitapada, (which deals with “mathematics”),
Kdlakriyd (which talks of “movements", in par-
ticular his system of exact longitudes in his
* Astronomical canon) and Golapada (which
relates to “spherics" and other three-dimen-
sional problems). Here is a list of the principal
names of numbers mentioned in the Aryab-
hatiya [see Arya, II, 2j:
*Eka (= 1). * Dasha (= 10). *Shata (= 10 2 ).
*Sahastra (= 10 3 ). *Ayuta (= 10 4 ). *Niyuta (=
10 5 ). *Prayuta (= 10 6 ). *Koti (= 10 7 ). *Arbuda (=
10 8 ). * Vrinda (= 10 9 ).
See Names of numbers and High numbers.
Aryabhata also used a method of recording
numbers which he invented himself: it was a
clever (if not terribly practical) alphabetical
system. However, he certainly knew the system
of ‘numerical symbols, as we can see if we look
at the Ganitapada, which contains two exam-
ples of numbers expressed in this way [see
Arya, II, line 20; Billard, p. 88]:
sarupa, “added to the form”, and: rashiguna,
“multiplied by the zodiac”.
*Samkalita means addition (literally: “put
together”) and *gunana means multiplication.
These words can be abbreviated to sa (“plus”)
and guna (“multiplied by”). *Rupa and *rashi
are the respective numerical symbols for
“shape” (or “form”) and “zodiac”, the numeri-
cal values for which are one and twelve. Thus
the two above expressions can be translated as
follows; sarupa, “added to one”, and: rashiguna,
“multiplied by twelve”. This is concrete proof
that Aryabhata was familiar with the method
of recording numbers using the numerical
symbols. These are the only two examples that
have been found in his work; however, Billard
(pp. 88-89) shows that the Aryabhatiya, in its
present state, is in fact two different works put
together, or rather the result of reorganisation
carried out on the original version. Some parts
were left unaltered, some were slightly modi-
fied, and others still were radically changed in
terms of numerical data, basic constants and
metre. The text we have today consists of noth-
ing more than the reworked parts, as the
original has not been found. It is probable that
Aryabhata used the numerical symbols in the
first version of his work and later changed his
method of representing numbers as he re-
organised his work. Finally, it is extremely
likely that Aryabhata knew the sign for zero
and the numerals of the place-value system.
This supposition is based on the following two
facts: first, the invention of his alphabetical
counting system would have been impossible
without zero or the place-value system; sec-
ondly, he carries out calculations on square
and cubic roots which are impossible if the
numbers in question are not written according
to the place-value system and zero. See Indian
written numeration (Classification of),
Sanskrit numeration, Numerical symbols
(Principle of the numeration of), Aryabhata’s
numeration. See also Square roots (How
Aryabhata calculated his).
ARYABHATA'S NUMBER-SYSTEM. This is an
alphabetical numerical notation invented by
the astronomer Aryabhata c. 510 CE. It is a
system which uses thirty-three letters of the
Indian alphabet and is capable of representing
all the numbers from 1 to 10 18 .
Aryabhata, it appears, was the first man in
India to invent a numerical alphabet. He devel-
oped the system in order to express the constants
of his * astronomical canons, and his surprising
astronomical speculations about *yugas, and the
system is more elegant and also shorter than that
which uses numerical symbols. See Aryabhata
(Numerical notations of), Numeral alphabet,
Numerical symbols, Numeration of numerical
symbols, Yuga (Systems of calculating) and
Yuga (Astronomical speculations about).
The use of this notation is found through-
out his work entitled Dashagitikapada,
where he describes it in the following way:
Vargaksharani varge ’vargc ’vargasha rani kat
nmauyah Khadvinavake svara nava varge “ varge
navantya varge vd.
Translation: “The letters which are [said to
bej classed (varga) [are], from [the letter] ka,
[those which are placed! in odd rows (varga);
the letters which are [called] unclassed (avarga)
[are those which are placed] in even rows
(avarga); [thus, one] jot is equal to nmau (= na +
ma 1; the nine vowels [are used to record] the
nine pairs of places (kha) in even or odd [rows].
The same [procedure] can be repeated after the
last of the nine even rows”.
Ref.: JA, 1880, II. p. 440; JRAS, 1863, p.
380; TLSM, I, 1827, p. 54; ZKM, IV, p. 81;
Datta and Singh, (1938), p. 65; Shukla and
Sarma, Ganita Section, (1976), pp. 3ff.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
448
To put it plainly, Aryabhatas method consists
of assigning a numerical value to the consonants
of the Indian alphabet in the following manner:
1. For the first twenty-five consonants, the
order of normal succession is followed for
whole numbers starting with 1.
2. The twenty-sixth represents five units
more than the twenty-fifth.
3. For the remaining seven, the progression
grows by tens.
4. The last consonant of the alphabet
receives the value of one hundred.
Thus this notation contains a number of pecu-
liarities which are unique to Indian syllables
(which are transcribed below in modern Nagari
characters). For a better understanding of this
principle, it should be remembered that this
writing uses thirty-three different characters,
which represent the consonants, and many
other signs representing the vowels in an iso-
lated position (a, a, i, /, u, u, ri, ri, l, e, o, ai, au).
An isolated consonant is always pro-
nounced with a short a, but when it is
combined with another vowel, a special sign is
added which graphically has nothing in
common with the sign representing this vowel
in an isolated position (a vertical line to the
right of the consonant, a line above, a loop
below the letter, a horizontal line above the
letter, with a loop, and so on).
As for the writing of the word, it is done
with a continuous horizontal line called mdtra
(see Fig. D. 2).
It must not be forgotten that the essential
phonetic elements of this syllable system are
constituted by the association of a consonant
with a following vowel ( which is either short
like a, i, u, ri, la, or long like a, i, u, ri), or a
diphthong (e, o, ai, au), which is always long by
definition. Thus to a given consonant, a short
vowel a, or a long vowel a, or even any one of
remaining vowel or diphthongs can be joined
(/, u, ri, la, /, u, ri, la, e, o, ai, au). Therefore, the
following syllables correspond to the conso-
nant ma (m) [for example]: ma, md, mi, mi, mu,
mu, mri, mri, mla, mid, me, mo, mai.
Conversely, the thirty-three consonants can
be joined to any vowel. Take, for example, a:
5 gutturals ka kha ga gha na
5 palatals cha chha ja jha na
5 cerebrals ta tha da dim na
5 dentals ta tha da dha na
5 labials pa pha ba bha ma
4 semivowels}'/? ra la va
3 sibilants sha shasa
1 aspirated ha
The alphabetical notation of numbers
invented by this astronomer/phonetician/
mathematician is based upon precisely this
structure. Starting from the first vowel (a):
• the five guttural consonants (ka, kha, ga,
gha, na) receive the values 1 to 5;
• the five palatals (cha chha ja jha h) those
of 6-10;
• the 5 cerebrals (ta tha da dha na) those
of 11 to 15;
• the 5 dentals (ta ths da dha na) those of
16 to 20; the five labials (pa pha ba bha ma)
those of 21 to 25
• the 4 semivowels (ya ra la va) receive the
values 30, 40, 50 and 60;
• the 3 sibilants (sha sha sa) those of 70,
80 and 90;
• and the last letter of the alphabet (the
aspirated ha) that of 100.
However, if a vertical line is added to the right
of a devandgari consonant (thus vocalising the
consonant with a long a), the value remains the
same: ka = ka; kha = kha; ta = ta, etc.
In other words, this numeration does not
distinguish between long and short vowels
when attributing numerical values to the let-
ters (ka = ka, mi - mi, hu - hu, pri - pri, etc.).
Thus, from this point on, to avoid con-
fusion, only the consonants accompanied by a
short vowel will be referred to.
To record numbers above one hundred,
Aryabhata came up with the idea of using the
rules of the vocalisation of consonants.In keep-
ing with the order of the letters of the alphabet,
the first thirty-three consonants with the vowel
a represent the numbers from 1 to 100 accord-
ing to the above rule. But if these consonants
are vocalised using an i or an /, ( which follow a
in the Indian syllable system), the value of each
is multiplied by a hundred (Fig. D. 3). If they
are then accompanied by a u or a u, their initial
values are multiplied by 10,000 (= 10 4 ). Thus,
when either ri or ri are attached to the succes-
sive consonants, they represent the initial
gutturals
sr
t=r
IT
3
ka = 1
kha = 2
&a = 3
gha = 4
ha = 5
palatals
$
F
ft
3T
cha = 6
chha - 7
ja - 8
jha - 9
ha = 10
cerebrals
7
z
T
S
W
ta = 11
tha = 12
da = 13
dha = 14
na = 15
dentals
ft
ST
z
F
F
ta = 16
tha = 17
da = 18
dha = 19
na = 20
labials
TT
TR
F
*
pa = 21
pha - 22
ha = 23
bha = 24
ma = 25
semivowels
F
$
F
ya = 30
ra = 40
la = 50
va = 60
sibilants
ST
F
FT
sha = 70
sha = 80
sa = 90
aspirates
ha = 100
Fi g . 2 4 D . 2 . Alphabetical numeration of Aryabhata: numerical value of consonants in isolated position
( vocalisation using a short “a'). Ref : NCEAM, p. 257. Datta and Singh (1937); Guitel (1966); Jacquet
(1835); Pihan (I860); Rodet
fa?
fa
fa
ft
2r*
ii
o
o
khi = 200
gi = 300
ghi = 400
hi = 500
fa
fa
fa
ffa
chi = 600
chhi = 700
ji = 800
jhi = 900
hi = 1,000
ft
fs
fr
fa
fa!
ti = 1,100
thi = 1,200
di = 1,300
dhi = 1,400
ni = 1,500
fir
ftr
ft
fa
fa
ti = 1,600
thi = 1,700
di = 1,800
dhi = 1,900
ni = 2,000
fa
fa?
fa
fa
fa
pi = 2,100
phi = 2,200
bi = 2,300
bhi = 2,400
mi - 2,500
fa
n
fa
fa
yi = 3,000
ri = 4,000
li = 5,000
ftr
fa
fa
shi = 7,000
shi = 8,000
si = 9,000
*
hi = 10,000
F i g . 2 4 D . 3 . Alphabetical numeration of Aryabhata; numerical value of consonants vocalised by a
short "i")
449
Aryabhata’s number-system
values multiplied by 1,000,000 (= 10 d ) . And so
on with each of the consecutive vowels of the
alphabet, multiplying by successive powers of
100 (10 2 , 10\ 10 6 , etc.). Using all the possible
phonemes, this rule enables numbers to be
expressed up to the value of the number that
we recognise today in the form of a 1 followed
by eighteen zeros (10 18 ).
If we look at the question from another
angle, Aryabhata’s alphabetical notation fol-
lows the successive powers of a hundred: it is
thus an additional numeration with a base of
100, where the units and tens (units of the first
centesimal order) are expressed by the first
thirty-three successive consonants in an iso-
lated position, according to the vocalisation
with an a (long or short). The units of the
second centesimal order (units and tens, multi-
plied by a hundred = 10 z ) are expressed by the
same consonants, this time vocalised by an i
(short or long). Those of the third centesimal
order (units and tens multiplied by ten thou-
sand = 10 4 ) are expressed by the consonants
accompanied by the vowel u (or li). And so on
until the units of the ninth centesimal order
(units and tens multiplied by 10 16 ) using the
thirty-three consonants with au (which corre-
sponds to the last vowel. This is how the values
for the consonant ka are obtained, using the
successive vocalisations (Fig. D. 4). The first
four orders of Aryabhata’s numeration are pre-
sented in Fig. D. 5.
As the numbers were set out according to
the ascending powers of a hundred, starting
with the smallest units, the representation was
carried out - at least theoretically - within a
rectangle subdivided into several successive
rectangles, where the syllables were written
from left to right according to the centesimal
order (Fig. D. 6). The number 57,753,336 cor-
responds to the number of synodic revolutions
of the moon during a *chaturyuga.
In Aryabhata’s language (Sanskrit), this
number is expressed as follows (see Arya, II, 2):
Shat thimshati trishata trisahasra pahchayuta
saptaniyuta saptaprayuta pahchakoti.
This can be translated as follows, where the
numbers are expressed in ascending order,
starting with the smallest unit:
"Six [=shat],
three tens [= trimshati],
three hundreds [= trishata],
three thousands [= trisahasra],
five myriads [= pahchayuta],
seven hundred thousand [= saptaniyuta],
seven millions [= saptaprayuta],
five tens of millions [= pahchakoti]" .
See Aryabhata (Numerical notations of),
Ankanam vamato gatih, Names of numbers,
Sanskrit.
Aryabhata’s notation conformed rigorously to
this order, the only difference being that it
functioned according to a base of 100 and not
Centesimal order
<- 2" d — »
<- 3 rd — »
<- 4' 1 ' ->
Syllable
Row
odd even
odd even
odd even
odd even
Fig. 24D.6.
of 10. Thus it was necessary to break down the
expression of the number in question (at least
mentally) into sections of two decimal orders,
as follows:
l sl centesimal order: six, three tens
2 nd centesimal order: three hundreds, three
thousand,
3 rd centesimal order: five myriads, seven hun-
dred thousand,
4'h cen tesimal order: seven million , five tens of
millions.
For the first centesimal order, it was necessary
to take the consonants, vocalised by a, which
correspond respectively to the values 6 and 30
(six and three tens), which gives the syllables
cha and ya (see Fig. D. 6A).
For the second centesimal order, the conso-
nants were vocalised by i, corresponding
respectively to the values 300 and 3,000, the
syllables beingg/ and yi (see Fig. D. 6B).
For the third centesimal order, the conso-
nants were vocalised by u, and corresponded
respectively to the values 50,000 and 700,000,
the syllables being nu and shu (see Fig.
D.6C).
Finally, for the fourth centesimal order, the
consonants were vocalised with ri, correspond-
ing respectively to the values 7,000,000 and
50,000,000, the syllables being chhri and Iri
(see Fig. D.6D).
6 30
cha
ya
odd even
Fig. 24D.6A.
Centesimal order
of units
I*
2 nd
3 rd
4th
5 th
6 .h
7th
8 lh
glh
fat
Syllable
ka
ki
ku
kri
kli
ke
kai
ko
kau
Value
i
10 2
10 4
]
10 6
10*
10 10
10 12
10 14
10 16
3,336 -»
300 3,000
cha
ya
gi
y‘
odd even
Fig.24D.6B.
Fig,24D.4. Consecutive orders of units in Aryabhata's alphabetical numeration (successive values of
syllables formed beginning with "ka ’).
Vocalisation
Associated centesimal order
Power of ten
Row of syllable
with an a
with an i
with a u
with an r
i si
2 nd
3 rd
4 Ih
1
10 2
10 4
10 6
odd even
odd even
odd even
odd even
753,336
50,000 700,000
cha
ya
gi
y‘
nu
1
shu
odd even
Fig.24D.6C.
57,753,336
7,000,000 50,000,000
cha
ya
gi
>'i
nu
shu
chhri
lri
odd
even
Fig.24D.5.
Fig.24D.6D.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
450
Thus, the notation for the number in question
would be: chayagiyinushuchhrilri.
Fig. I). 6E shows the main breakdowns for
this value (where the value of a syllable is called
absolute when the vocalisation accompanying
the consonant is ignored, and it is called rela-
tive where the opposite is the case).
The number 4,320,000 is expressed in the
same manner, this figure corresponding to the
total number of years in a *chaturyuga (Fig. D.
6F): khuyughri. This notation is proof that
inventive genius does not always go hand in
hand with simplicity.
Thus, contrary to the opinion of several
authors, this notation was not based on the place-
value system, and certainly did not use zero. It is
in fact an additional numeration of the third cate-
gory of the classification given in Chapter 23.
However, it is very likely that Aryabhata
knew about zero and the place-value system.lt is
precisely because he already knew about these
concepts that he was able to achieve the degree
of abstraction that was needed to develop a
numerical notation such as this, which is unique
in the whole history of written numerations.
Whilst this system is additional in principle, its
mathematical structure is full of the purest con-
cepts of zero and the place-value system. This is
made clear in Fig. D. 4. The consonant ka is the
chosen graphical sign upon which everything
else is based. Going from left to right, the rule of
numerical vocalisation invented by Aryabhata
can be resumed as follows: by adding / to this
sign, in reality two zeros are being added to the
decimal representation of the value of the letter
ka (in other words, the unit); but by adding a u,
a ri, a //, an e, an ai, an o, or an au, four, six,
eight, ten, twelve, fourteen or sixteen zeros
would be added.
The Jews, the Syrians and the Greeks cer-
tainly used similar conventions, but only for
highly specialised usage and without perceiv-
ing them from the same angle as Aryabhata. By
adding an accent, a dot or even one or two suf-
fixes to a letter, they multiplied its value by 100
or 1,000, but they never managed to generalise
their convention from such an abstract angle
as Aryabhata.
Syllables
Absolute values
Relative total values
for each column
Total value
57,753,336
Fig. 24D.6E.
Syllables
Absolute values
Relative total values
for each column
Total value
57,753,336 ->
1
cha |
ya
g'
y>
nu
shu
chhri
in
6
30
3
30
5
70
7
50
36
33
X 10 2
75 >
: 10 4
57 X
10 6
36
f 33
X 10 2
+ 75 >
: 10 4 -
h 57 X
10 6
cha
ya
g '
y>
nu
shu
chhri
hi
||
2 30
4
32 X 10 4
4 X 10 6
32 X 10 4 + 4 X 10 6
khu yu ghri
Fig. 24D.6F.
Aryabhata had an advantage, because the
phonetic structure which characterises the
Indian syllable system is almost mathematical
itself. This is confirmed by *Bhaskara I, a
faithful disciple separated by a century from
Aryabhata. In his Commentary on the Aryab-
hatiya (629 CE), he gives this brief explanation
of the rule in question: nyasashcha sthananam
oooooooooo. Translation: “By writing in the
places (*sthana), we have: oooooooooo [= ten
zeros]”.
[See: commentary on the Ganitapada, line
2; Shukla and Sarma, (1976), pp. 32-4; Datta
and Singh, (1938), pp. 64-7].
In his text, the commentator uses not only
the word sthana which means “place” (which
the Indian scholars often used in the sense of
“positional principle”), but also and above all
the little circle, which is the numeral “zero” of
the Indian place-value system. See Sthana,
Numeral 0 and Zero.
Later on in his commentary, Bhaskara
I writes the following: khadvinavake svara
nava varge ’ varge khani shunyani, khanam
dvinavakam khadvinavakam tasmin khadvinavake
ashtadashasu shunyopalakshiteshu . . . Translation:
“The nine vowels ( nava varge ) [are used to
note] the nine pairs of zeros ( khadvinavake );
[because] *kha means zero (*shunya). In the
nine pairs of places, that is, in the eighteen
(ashtadashasu) [places] marked by zeros
(shunyopalakshiteshu ) ..."
The use of the term *kha as one of the des-
ignations of zero is explained as one of
the synonyms of *shunya, a word meaning
“void” which Indian mathematicians and
astronomers have always used, since at least
the fifth century CE, in the sense of zero.
This leaves no doubt: even if the master
was not very loquacious on the subject, his
disciple and commentator explains Aryab-
hata’s system and uses the Indian symbol for
zero (the little circle), and also the three fun-
damental Sankrit terms (*sthana, *kha,
*shunya). See Zero.
The Sanskrit term *kha, literally “space”, sig-
nifies “sky” and “void”, and thus by extension
“zero” in its mathematical sense. As for “place”
(* sthana), Aryabhata gave it the meaning of the
place occupied by a given syllable; thus, by
extension, it meant “order of unit” in his alpha-
betical numeration. This is due to the “row”
occupied by the syllable in a square which is
formed by the structure of his notation system
(Fig. D. 5). To his mind, it was a completely sepa-
rate “place”, one for the even row, and one for
the odd row, within a unit of the centesimal
order (the odd row having a value of a simple
unit and the even row a value of a multiple of
ten). It is due to the fact that such a “place” can
be “emptied” if the units or tens of the corre-
sponding “order” are absent that “place” came to
mean both “position” and void. As for the
expression khadvinavake, for Aryabhata this
meant the “nine pairs of zeros" , the eighteen
zeros added to the decimal positional representa-
tion of the initial value of a given consonant, at
the end of the successive vocalisation operations.
Moreover, in Golapdda, Aryabhata alludes
to the essential component of our place-value
system when he states that “from place to place
(*sthana), each [of the numerals] is ten times
[greater] than the preceding one” (see Clark
(1930), p. 28]. What is more, in the chapter of
the Ganitapada on arithmetic and methods of
calculation, he gives rules for operations in dec-
imal base for the extraction of square roots and
cube roots. Neither of these two operations can
be carried out if the numbers are not expressed
in writing, using the place-value system with
nine distinct numerals and a tenth sign which
performs the function of zero. See Pdtiganita,
Indian methods of calculation and Square
roots (How Aryabhata calculated his).
This is mathematical proof that, at the
beginning of the sixth century CE, Aryabhata
had perfect knowledge of zero and the place-
value system, which he used to carry out
calculations. The question remains: why did he
invent such a complicated system when he
could have used a much simpler one? It seems
that the alphabet offered an almost inex-
haustible supply when it came to creating
mnemonic words, especially for complicated
numbers, and this facilitated the readers’ mem-
orising of them. He always wrote in Sanskrit
verse, and thus he had found a very convenient
way not only to write numbers in a condensed
form, but also to meet the demands of the metre
and versification of the Sanskrit language.
Luckily, Aryabhata was the only one to
make use of the system that he had invented.
His successors, including those that referred to
his work, generally adopted the method of
*numerical symbols. Even those that opted for
an alphabetical numerical notation did not
choose to use his system: they used a radically
transformed form which was much simpler.
See Katapayadi numeration.
When Aryabhata’s alphabetical numera-
tion became widespread in the field of Indian
astronomy, it no doubt was disastrous for the
preservation of mathematical data. Worse yet,
it caused the Indian discoveries of the place-
451
value system and zero, which took place
before Aryabhata’s time, to be irretrievably
lost to history.
ARYABHATIYA. Title given to Aryabhata’s
work by his successors.
ASAMKHYEYA (or ASANKHYEYA). Literally:
“number impossible to count" (from *sam-
khyeya or *sankhyeya, “number", the “highest
number imaginable”. See High numbers and
Infinity.
ASANKHYEYA. Literally: “non-number”. Term
designating the “incalculable". See Asamkhyeya.
ASANKHYEYA. Literally: “impossible to
count". Name given to the number ten to the
power 140. See Names of numbers. For an
explanation of this symbolism, see High num-
bers (Symbolic meaning of).
Source: *Vydkarana (Pali grammar) by
Kachchayana (eleventh century CE).
ASHA. [SJ. Value = 10. “Horizons”. See Dish,
Ten.
ASHITI. Ordinary Sanskrit name for the
number eighty.
ASHTA (or ASHTAN). Ordinary Sanskrit
name for the number eight. It is used in
the composition of several words which
have a direct relationship with the idea
of this number. Examples: *Ashtadanda ,
*Ashtadiggaja, *Ashtamangala, *Ashtamurti,
* Ashtanga and *Ashtavimoksha.
For words which have a more symbolic
relationship with this number, see Eight and
Symbolism of numbers.
ASHTACHATVARIMSHATI. Ordinary Sanskrit
name for the number forty-eight. For words
which have a more symbolic relationship with
this number, see Forty-eight and Symbolism
of numbers.
ASHTADANDA. “Eight parts”. These are the
eight parts of the body that we use to conduct a
profound veneration by stretching out face
down on the ground. See Ashtanga.
ASHTADASHA. Ordinary Sanskrit name
for the number eighteen. For words which
have a more symbolical relationship with
this number, see Eighteen and Symbolism
of numbers.
ASHTADIGGAJA. “Eight elephants”. Collective
name for the guardians of the eight “horizons”
of Hindu cosmogony (these being: Airavata,
*Pundarika, Vdmana, *Kumuda, Anjana,
Pushpadanta, Sarvabhauma and Supratika).
See Diggaja.
ASHTAMANGALA. “Eight things that augur
well”. This concerns the “eight jewels" that
Buddhism considers as the witnesses of the
veneration of Buddha. See Mangala.
ASHTAMURTI. “Eight shapes (or forms)”. The
name of the most important forms of Shiva.
See Ashta and Murti.
ASHTAN. A synonym of * Ashta.
ASHTANGA. “Eight limbs (or members)”.
This term denotes the eight limbs of the
human body which are used in prostration (the
head, the chest, the two hands, the two feet
and the two knees).
ASHTAVIMOKSHA. “Eight liberations”. This
refers to a Buddhist meditation exercise, which
has eight successive stages of concentration,
the aim of which being to liberate the individ-
ual from all corporeal and incorporeal
attachments.
ASHTI. [SI. Value = 16. In terms of Sanskrit
poetry, this refers to the metre of four times
sixteen syllables per line. See Sixteen and
Indian metric.
ASHVA. IS]. Value = 7. “Horse". Allusion to the
seven horses (or horse with seven heads) of the
chariot upon which *Surya, the Brahmanic god
of the sun, raced across the sky. See Seven.
ASHV1N. [S]. Value = 2. “Horsemen”. Name of
the twin gods Saranyu and Vivashvant (also
called *Dasra and *Nasatya) of the Hindu pan-
theon. They symbolise the nervous and vital
forces, and are supposed to respectively repre-
sent the morning star and the evening star.
They are the offspring of horses, hence their
name (from *Ashva). These divinities are con-
sidered as the “Primordial couple” who
appeared in the sky before dawn in a horse-
drawn golden chariot. See Two.
ASHVINA. [S]. Value = 2. “Horsemen”. See
Ashvin and Two.
ASHVINAU. [S]. Value = 2.”Horsemen". See
Ashvin and Two.
ASTRONOMICAL CANON. A group of ele-
ments conceived as a whole by the author of an
astronomical text. These elements are always
presented together in a text, commentary or
quotation, being effectively (astronomically),
or supposedly, interdependent. Often, how-
ever, except for historical information, and
even though complete, the canons are only in
the form of the game of *bija. Thus, mathemat-
ically, a given canon can be placed in any era or
represent any unit of time. See Indian astron-
omy (The history of).
ASTRONOMICAL SPECULATIONS. See Yuga
(Astronomical speculation on).
ASURA. “Anti-god". Name given to the Titans
of Indian mythology.
ATATA. Name given to the number ten to the
power eighty-four. See Names of numbers and
High numbers.
Source: *Vydkarana (Pali grammar) by Kachchayana
(eleventh century CE).
ATIDHRITI. [S]. Value = 19. The metre of four
times nineteen syllables per verse. See Indian
metric.
ATMAN. [SJ. Value = 1. In Hindu philosophy,
this term describes the “Self”, the “Individual
soul”, the “Ultimate reality”, even the
* Brahman himself, who is said to possess all the
corresponding characteristics. The uniqueness
of the “Self” and above all the first character of
the Brahman as the “great ancestor” explain the
symbolism. See *Pitamaha and One.
ATMOSPHERE. [SJ. Value = 0. See Infinity,
Shunya and Zero.
ATRI. [S]. Value = 7. Proper noun designating
the seventh of the Saptarishi (the “Seven Great
Sages” of Vedic India), considered to be
the founder of Indian medecine. See Rishi
and Seven.
ATRINAYANAJA. [SI. Value = 1, “Moon". See
Abja and One.
ATTATA. Name given to the number ten to the
power nineteen (= ten British trillions). See
Names of numbers and High numbers.
Source: * I.alitavistara Sutra (before 308 CE).
ATYASHTI. [S]. Value = 17. The metre of four
times seventeen syllables per line in Sanskrit
poetry. See Indian metric.
AUM. Sacred symbol of the Hindus. See
Mysticism of letters and Ekakshara.
AVANI. [S], Value = 1. “Earth”. See Prithivi and
One.
AVARAHAKHA. Generic name of the five ele-
ments of the revelation. See Bhuta and
Mahabhuta.
AVATARA. [S], Value = 10. “Descent”. The
incarnation of a Brahmanic divinity, birth
through transformation, the aim being to carry
out a terrestrial task in order to save humanity
from grave danger. The allusion here is made
to the Dashavatara, the “ten Avatara" , or major
incarnations of *Vishnu, attributed to the four
“ages” of the world (*yugas), according to
Hindu cosmogony. See Dashavatara and Ten.
ARYABHATIYA
AVYAKTAGANITA. Name given to algebra
(literally: “science of calculating the
unknown”), as opposed to arithmetic, called
vyaktaganita. See Vyaktaganita, Algebra,
Arithmetic.
AYUTA. Name for the number ten to the
power four (= ten thousand). See Names of
numbers and High numbers.
Source: *Vajasaneyi Sarnhita (beginning of
Common era): *Taittiriya Sarnhita (beginning of
Common era): *Kathaka Sarnhita (beginning of
Common era): * Pahchavimsha Brdhrnana (dale
uncertain); *Sankhyayana Shrauta Sutra (date
uncertain); *Aryabhatiya (510 CE); *Kitah fi tahqiq i
ma li'l hind by al-Biruni (1030 CE); *Uldvati by
Bhaskaracharya (1150 CE); *Canilakaumudi by
Narayana (1350 CE); *Trishatika by
Shridharacharya (date uncertain).
AYUTA. Name given to the number ten to the
power nine (= a thousand million). See Names
of numbers and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
B
BAHU. [SJ. Value = 2. “Arms”, due to the sym-
metry of the two arms. See Two.
BAHULA. Name given to the number ten to
the power twenty-three (= a hundred thousand
[British] trillion). See Names of numbers and
High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
BAKSHALI’S MANUSCRIPT. See Indian doc-
umentation (Pitfalls of).
BALINESE NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
ofShunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali”, Vatteluttu
and Kawi numerals. Currently in use in Bali,
Borneo and the Celebes islands. The corre-
sponding system functions according to the
place-value system and possesses zero (in the
form of a little circle). For ancient numerals,
see Fig. 24.50 and 80. For modern numerals,
see Fig. 24.25. See Indian written numeral
systems (Classification of). See also Fig. 24.52
and 24.61 to 69.
BANA. IS]. Value = 5. “Arrow". See Shara and
Five. See also Pahchabana.
BASE OF ONE HUNDRED. See
Shatottaraganana, Shatottaraguna and
Shatottarasamjha.
BASE TEN. See Dashaguna and
Dashagundsamjha.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
452
BEARER. (S], Value = 1. See Dharani and One.
BEGINNING. [S]. Value = 1. See Adi and One.
BENGALI NUMERALS. Signs derived from
*Brahmi numerals, through the intermedi-
ary of Shunga, Shaka, Kushana, Andhra,
Gupta, Nagari and Kutila numerals.
Currently used in the northeast of India, in
Bangladesh, Bengal and in much of the
centre of Assam (along the Brahmaputra
river). The corresponding system functions
according to the place-value system and pos-
sesses zero (in the form of a little circle).
See Indian written numeral systems
(Classification of). See also Fig. 24.10, 52
and 24.61 to 69.
BENGALI SAL (Calendar). See Bengali San.
BENGALI SAN (Calendar). The solar era
beginning in the year 593 CE. It is still used
today in Bengal. To obtain the corresponding
date in Common years, add 593 to a date
expressed in this calendar. It is also called
Bengali Sal. See Indian calendars.
BHA. [S]. Value = 27. “Star”. Allusion to the
twenty-seven *nakshatra. See Twenty-seven.
BHAGAHARA. Term used in arithmetic to
denote division, although the word is most
often used to denote the divisor (which is also
called bhajaka). See Chhedana.
BHAGAVAD GITA. “Song of the Lord". A long
philosophical Sanskrit poem containing the
essence of* Vedanta philosophy, explained by
Krishna and Arjuna in a dialogue about action,
discrimination and knowledge. It is a relatively
recent text (c. fourth century CE) and is found
in the sixth book of the *Mahabharata [see
Frederic; Dictionnaire (1987)].
BHAJAKA. Term used in arithmetic to denote
the divisor. See Bhdgahara.
BHAJYA. Term used in arithmetic to denote
the dividend. Synonym: hdrya. See Bhdgahara
and Chhedana.
BHANU. [S]. Value = 12. An epithet of *Surya,
the Sun-god. Bhanu = Surya = twelve. See
Twelve.
BHARGA. [SI. Value = 11. One of the names of
*Rudra. See Rudra-Shiva and Eleven.
BHASKHARA. Indian mathematician, disciple
of * Aryabhata (a century after his death). He
was born in the first half of the seventh cen-
tury. He is known mainly for his Commentary
on the ‘Aryabhatiya, in which examples of the
use of the place-value system expressed by
means of the Sanskrit numerical symbols are
found in abundance. The translation of the
numbers expressed in this manner is often
given using the nine numerals and zero (also
according to the rules of the place-value
system) [see Shukla and Sarma (1976)1. He is
usually called “Bhaskara I" so that he is not
confused with another mathematician of the
same name (‘Bhaskaracharya). See Aryabhata
(Numerical notations of), Aryabhata’s
numeration, Numerical symbols, Numerical
symbols (Principle of the numeration of),
and Indian mathematics (The history of).
BHASKARA I. See Bhaskara.
BHASKARA II. See Bhaskaracharya.
BHASKARACHARYA. Indian mathematician,
astronomer and mechanic, usually referred to
as Bhaskara II. He lived in the second half of
the twelfth century CE. He is famous for his
work, the Siddhantashiromani, an astronomical
text accompanied by appendices relating to
mathematics, amongst which we find the
*Lildvati (the “Player”), which contains a whole
collection of problems written in verse. He fre-
quently uses zero and the place-value system,
which are expressed in the form of Numerical
symbols. He also describes methods of calcula-
tion which are very similar to our own and are
carried out using the nine numerals and zero.
Moreover, he explains the fundamental rules of
algebra where the zero is presented as a mathe-
matical concept, and defines Infinity as the
inverse of zero [see Sastri (1929)].
Here is a list of the main names of numbers
given in the Lilavati (Lil, p. 2) [see Datta and
Singh (1938), p.13]:
*Eka (= 1). *Dasha (= 10). *Shata (= 10 2 ).
*Sahasra (= 10 3 ). *Ayuta (= 10 4 ). *Laksha (=
10 5 ). *Prayuta (= 10 6 ). *Koti (= 10 7 ). *Arbuda (=
10 8 ). *Abja (= 10 9 ). *Kharva (= 10 l °). *Nikharva
(= 10 u ). *Mahapadma (= ,10 12 ). *Shanku (=
10 13 ). *Jaladhi (= 10 14 ). *Antya (= 10 15 ).
* Madhya (= 10 16 ). * Pa rdrdh a (= 10 17 ).
See Names of numbers, High numbers,
Positional numeration, Numerical symbols
(Principle of the numeration of), Zero,
Infinity, Arithmetic, Algebra, and Indian
mathematics (The history of).
BHASKARIYABHASYA. See Govindasvdmin.
BHATTIPROLU NUMERALS. Signs derived
from ‘Brahmi numerals, through the interme-
diary of Shunga, Shaka, Kushana, Andhra,
Pallava, Chalukya, Ganga and Valabhi numer-
als. Used since the eighth century CE by the
Dravidians in southern India. Kannara, Telugu,
Grantha, Malayalam Tamil, Sinhalese, etc.
numerals derived from these numerals. The
corresponding system does not use the place-
value system or zero. See Indian written
numeral systems (Classification of). See also
Fig. 24.52 and 24.61 to 69.
BHAVA. [S J . Value = 11. “Water”. One of the
names of *Rudra, the etymological meaning of
which is related to the tears. See Rudra, Rudra-
Shiva and Eleven.
BHAVISHYAPURANA. See Parana.
BHINNA. [Arithmetic). Sanskrit term used to
denote “fractions” in general (literally “broken
up”). It is synonymous with bhaga, amsha, etc.
(literally “portion”, “part”, etc.).
BHOJA. Indian astronomer who lived in the
eleventh century CE. He is known as the author
of a text entitled Rdjamrigdnka, in which there
are many examples of the place-value system
expressed through Sanskrit numerical symbols
[see Billard (1971), p. 10]. See Numerical sym-
bols, Numerical symbols (Principle of the
numeration of), and Indian mathematics
(The history of).
BHU. [S ] . Value = 1. “Earth”. See Prithivi and
One.
BHUBHRIT. [S]. Value = 7. “Mountain”.
Allusion to *Mount Meru. See Adri and Seven.
BHUDHARA. [Sj. Value = 7. “Mountain".
Allusion to *Mount Meru. See Adri and Seven.
BHUMI. [S ] . Value = 1. “Earth”. See Prithivi
and One.
BHUPA. [S]. Value = 16. “King”. See Nripa and
Sixteen.
BHUTA. [S]. Value = 5. “Element”. In Brahman
and Hindu philosophy, there are five elements
(or states) in the manifestation: air (*vayu), fire
(*agni), earth ( *prithivi ), water (*apa) and
ether ( *akdsha ). See Pahchabhuta and Five.
See also Jala.
BHUVANA. [S]. Value = 3. “World". The “three
worlds” ( *triloka ). See Loka, Triloka, and Three.
BHUVANA.[S]. Value = 14. “World”.
According to Mahayana Buddhism, the thir-
teen “countries of election" or “heavens" of Jina
and Bodhisattva, to which was added
*Vaikuntha. See Fourteen.
BijA. Word meaning “letters” in terms of
mathematical symbols (letters used to express
unknown values). In algebra, the word is also
used in the sense of “element” or even “analy-
sis”. See Algebra and Bijaganita.
BIJA. Word meaning “letters” in terms of reli-
gious symbols (which generally represent the
divinities of the Brahman pantheon or the
Buddhist tan trie) and esoteric symbols (accord-
ing to a power w'hich is believed to be creative or
evocative). See Mysticism of letters.
BijA. Word used in astronomy to denote correc-
tive terms expressed numerically and applying
to the elements of a given text, modifying those
of the corresponding ‘astronomical canon. See
Indian astronomy (The history of).
BIJAGANITA. Word denoting algebraic
science or science of analytical arithmetic
and the calculation of elements (from *bija\
“letter-symbol”, “element”, “analysis” and from
*ganita : “science of calculation”). The word
was used in this sense since Brahmagupta’s
time (628 CE). However, Indian mathemati-
cians only ever used the first syllable of the
word denoting a given operation as their alge-
braic symbols. See Indian mathematics (The
history of).
BILLION. (= ten to the power twelve. US, ten to
power of nine). See Antya, Kharva, Mahabja,
Mahapadma, Mahasaroja, Pardrdha, and
Shankha.
BINDU. Literally “point”. This is the name
given to the number ten to the power forty-
nine. See Names of numbers. For an
explanation of this symbolism, see High num-
bers (The symbolic meaning of).
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
BINDU. [SI. Value = 0. This word literally
means “point”. This is the symbol of the uni-
verse in its non-manifest form, before its
transformation into the world of appearances
( rupadhatu ). The comparison between the
uncreated universe and the point is due to the
fact that this is the most elementary mathemat-
ical symbol of all, yet it is capable of generating
all possible lines and shapes (*rupa). Thus the
association of ideas with “zero”, which is not
only considered to be the most negligible
quantity, but also and above all it is the most
fundamental of mathematical concepts and the
basis for all abstract sciences. See Zero.
BIRTH. [S]. Value = 4. See Gati, Yoni and Four.
BLIND KING. [Si. Value = 100. See
Dhdrtarashtra and Hundred.
BLUE LOTUS (half-open). This has repre-
sented the number ten to the power
453
BLUE LOTUS
twenty-five. See Utpala and High numbers
(The symbolic meaning of).
BLUE LOTUS (half-open). This has repre-
sented the number ten to the power
ninety-eight. See Vppala and High numbers
(The symbolic meaning of).
BODY. [S]. Value = 1. See Tanu and One.
BODY. [S], Value = 6. See Kaya and Six.
BODY. [SI. Value = 8. See Tanu and Eight.
BORN TWICE. [S]. Value = 2. See Dvija and
Two.
BOW WITH FIVE FLOWERS. See
Pahchabana.
BRAHMA. Name of the “Universal creator", the
first of the three major divinities of the Brahman
pantheon (Brahma, "Vishnu, "Shiva). See
Pitamaha. Atman and Parabrahman.
BRAHMA. [SJ. Value = 1. See Atman,
Pitamaha, Parabrahman and One.
BRAHMAGUPTA. Indian astronomer who
lived in the first half of the seventh century
CE. His best-known works are
Brahmasphutasiddhanta and Khandakhadyaka,
where there are many examples of the place-
value system using the nine numerals and zero,
as well as the "Sanskrit numerical symbols. He
also describes methods of calculation which
are very similar to our own using the nine
numerals and zero. Moreover, he gives basic
rules of algebra where zero is presented as a
mathematical concept, and he defines Infinity
as the number whose denominator is zero [see
Dvivedi (1902)]. See Numerical symbols
(Principle of the numeration of), Zero,
Infinity, Arithmetic, Algebra, and Indian
mathematics (The history of).
BRAHMAN. See Atman, Pitamaha,
Parabrahman, Paramdtman, Day of Brahma
and High numbers (The symbolic meaning of).
BRAHMANICAL RELIGION. See Indian
philosophies and religions.
BRAHMANISM. See Indian religions and
philosophies.
BRAHMASPHUTASIDDHANTA. See
Brahmagupta and Indian mathematics (The
history of).
BRAHMASYA. [S[. Value = 4. “Faces of
"Brahma”. In representations, this god gener-
ally has four faces. He also has four arms and
he is often depicted holding one of the four
* Vedas in each hand. See Four.
BRAHMI ALPHABET. See Fig. 24. 28.
BRAHMI NUMERALS. The numerals from
which all the other series of 1 to 9 in India Central
and Southeast Asia are derived. These are found
notably in Asoka’s edicts and in the Buddhist
inscriptions of Nana Ghat and Nasik. The corre-
sponding system does not function according to
the place-value system, nor does it possess zero.
See Fig. 24.29 to 31 and 70. For notations derived
from Brahmi, see Fig. 24.52. For their graphic
evolution, see Fig. 24.61 to 69. See Indian writ-
ten numeral systems (Classification of).
BREATH. [S], Value = 5. See Prana, Parana
and Five.
BRILLIANT. [S], Value = 12. See Arka and
Twelve.
BUDDHA (The legend of). Legend recounted
in the * Lalitavistara Sutra, which is full of
examples of immense numbers. See Indian
mathematics (The history of).
BUDDHASHAKARAJA (Calendar). Buddhist
calendar which is hardly used outside of Sri
Lanka and the Buddhist countries of South-
east Asia. It generally begins in 543 BCE, thus
by adding 543 to a date in this calendar we
obtain the corresponding date in our own cal-
endar. See Indian calendars.
BUDDHISM. Here is an alphabetical list of
all the terms related to Buddhism which
can be found as headings in this dictionary:
*Ashtamangala, *Ashtavimoksha, *Bhuvana,
"Buddha (Legend of), *Chaturmukha,
*Chaturyoni, * Dashabala , * Dashabumi , *Dharma,
*Dvddashadvarashastra, * Dvatrimshadvaratak-
shana, *Gati, "High numbers (The symbolic
meaning of), *Indriya, *Jagat, *Kaya, *Loka,
*Mangala, *Pahchdbhijha, *Pahcha Indryani,
Pahchachaksus , * Pahchaklesha , * Pahchanan -
tar a, *Ratna, *Saptabuddha, * Shunya ,
*Shunyatd, *Tallakshana, *Trikaya, *Tripitaka,
*Vajra and "Zero.
BUDDHIST RELIGION. See Buddhism and
Indian religions and philosophies.
BURMESE NUMERALS. Signs derived from
"Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali”, Vatteluttu
and Mon numerals. Used since the eleventh
century CE by the people of Burma. The corre-
sponding system uses the place-value system
and zero (in the form of a little circle). For
ancient numerals, see Fig. 24.51. For modern
numerals, see Fig. 24.23. See Indian written
numeral systems (Classification of). See also
Fig. 24.52 and 24.61 to 69.
c
CALCULATING BOARD. See Pdti, Pdtiganita.
CALCULATING SLATE. See Pdtiganita.
CALCULATION (Methods of). See
Dhulikarma, Pdti, Pdtiganita and Indian
methods of calculation.
CALCULATION (The science of). See Ganita.
CALCULATION ON THE ABACUS. See
Dhulikarma.
CALCULATOR. [Arithmetic]. See Samkhya.
CANOPY OF HEAVEN. [SJ. Value = 0. See
Zero, Zero (Indian concepts of) and Zero and
Infinity.
CARDINAL POINT. [SI. Value = 4. See Dish
and Four.
CAUSAL POINT. See Paramabindu and
Indian atomism.
CELESTIAL YEAR. See Divyavarsha.
CENTESIMAL NUMERATION. See
Shatottaraganana, Shatottaraguna and
Shatottarasamjha.
CHAITRA. Lunar-solar month corresponding
to March / April.
CHAITRADI. “The beginning of Chaitra”. This
is the name of the year beginning in spring
with the month of *Chaitra, the first lunar-
solar month.
CHAKRA. [SI. Value = 12. “Wheel”. This refers
to the zodiac wheel. See Rdshi and Twelve.
CHAKSHUS. [S]. Value = 2. “Eye”. See Netra
and Two.
CHALUKYA (Calendar). Calendar of the
dynasty of the eastern Chalukyas, beginning in
the year 1075 CE. This calendar was used until
the middle of the twelfth century (until c.
1162). To obtain the corresponding date in our
own calendar, add 1075 to a date expressed in
this calendar. See Indian calendars.
CHALUKYA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra and
Pallava numerals. Contemporaries of the
“Vatapi” dynasty of the Chalukyas of Deccan
(fifth to seventh century CE). The correspond-
ing system does not use the place-value system
or zero. See Fig. 24.45 and 70. See Indian writ-
ten numeral systems (Classification of). See
also Fig. 24.52 and 24.61 to 69.
CHAM NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary of
Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali” and Vatteluttu
numerals. Used from the eighth to the thir-
teenth century CE to express dates of the Shaka
calendar in the vernacular inscriptions of
Champa (in part of Vietnam). The correspond-
ing system uses the place-value system and
zero. See Indian written numeral systems
(Classification of). See Fig. 24.79 and 80. See
also Fig. 24.52 and 24.61 to 69.
CHANDRA. [S]. Value = 1. “Luminous”.
An attribute of the *Moon as a (male) divinity
of the Brahmanic pantheon. See Abja, Soma
and One.
CHARACTERISTIC. [S]. Value = 5. See
Puranalakshana and Five.
CHATUR. Ordinary Sanskrit name for the
number four, which forms part of the composi-
tion of many words which have a direct
relationship with the idea of this number.
Examples: *Chaturananavadana, *Chaturyuga,
*Chaturanga, *Chaturangabalakdya, *Chaturash-
rama, *Chaturdvipa, *Chaturmahdraja,*Chat-
urmdsya, *Chaturmukha, *Chaturvarga, *Chat-
uryoni. For words which have a more symbolic
relationship with the number four, see Four and
Symbolism of numbers.
CHATURANANAVADANA. [S]. Value = 4.
The “four oceans". See Chatur, Sdgara, Four.
See also Ocean.
CHATURANGA. “Four parts”. Name given to
an ancient Indian game, the ancestor of chess:
there were four players and the board consisted
of eight by eight squares and eight counters
(the king, the elephant, the horse, the chariot
and four soldiers). See Chatur.
CHATURANGABALAKAYA. “Four corps”.
Name given to the ancient organisation of the
Indian army, which consisted of elephants
(, hastikaya ), the cavalry ( ashvakaya ), the chari-
ots (rathakaya) and the infantry (pattikaya ).
See Chatur.
CHATURASHRAMA. “Four stages”. According to
Hindu philosophy, there were four stages to a
man s life, in keeping with the Vedic concept: in
the first (called brahmacharya ), intellectual capaci-
ties are developed, profane and religious
instruction are received and the virtues of spiritual
life are cultivated; in the second (grihastha ), mar-
riage and home-making take place; in the third
(hanaprastha), having fulfilled his role as master of
the house and having served his community, he
goes alone into the forest to devote himself to
intensive meditation, philosophical studies and
the Scriptures; finally, in the fourth stage (san-
nydsa ), he gives up all his possessions and no
longer cares for earthly things. See Chatur.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
454
CHATURDASHA. Ordinary Sanskrit name for
the number fourteen. For words with a sym-
bolic relationship with this number, see:
Fourteen and Symbolism of numbers.
CHATURDViPA. “Four Islands". In Brahmanic
mythology and Hindu cosmology this is the
name given to the four island-continents said
to surround India (Jambudvipa ). See Chatur.
For an explanation of this choice of number,
see Ocean, which gives the same explanation
about the four seas (*chatursdgara).
CHATURMAHARAJA. “Four great kings”.
These are the four guardian divinities of the
cardinal points, who are said to live on the
peaks of *Mount Meru (Vaishravana in the
North, Virupaksha in the West, Virudhaka in
the South and Dhritarashtra in the East).
See Chatur.
CHATURMASYA. “Four months”. Name of an
Indian ritual which takes place every four
months, once at the start of spring, once at the
start of the rain season, and once at the start of
autumn. See Chatur.
CHATURMUKHA. “Four faces”. Name given
to all the Brahmanic or Buddhist divinities who
are represented as having four faces (*Brahma,
*$hiva, etc.). See Chatur.
CHATURSAGARA. “Four oceans”. These are
the four seas said to surround India
( Jambudvipa ). See Sagara. For an explanation
of this choice of number, see Ocean.
CHATURVARGA. “Four aims”. These are the
*trivarga of Hindu philosophy (the three
objectives of human existence), namely:
artha, (material wealth), *kdma (passionate
love), and *dharma (duty), to which some-
times a fourth is added, moskha, the liberation
of the soul. See Chatur.
CHATURVIMSHATI. Ordinary Sanskrit name
for the number twenty-four. For words which
have a symbolic relationship with this number,
see: Twenty-four and Symbolism of numbers.
CHATURYONI. The "four types of reincarna-
tion”. According to Hindus and Buddhists,
there are four different ways to enter the cycle
of rebirth ( *samsdra ): either through a vivipa-
rous birth (jarayuva), in the form of a human
being or mammal; or an oviparous birth
(i andaja ), in the form of a bird, insect or reptile;
or by being born in water and humidity
(, samsvedaja ), in the form of a fish or a worm;
or even through metamorphosis ( aupapaduka ),
which means there is no “mother” involved,
just the force of Karma [see Frederic (1994)].
See Chatur.
CHATURYUGA. The “four periods”. Cosmic
cycle of 4,320,000 human years, subdivided
into four periods. Synonymous with
*mahayuga. See Chatur and Yuga (Definition).
CHATURYUGA. (Astronomy). According to
speculations about *yugas, the chaturyuga, or
cycle of 4,320,000 years, is defined as the
period at the beginning and end of which the
nine elements (namely the Sun, the Moon,
their apsis and node and the planets) are in
average perfect conjunction at the starting
point of the longitudes. See Chaturyuga (previ-
ous entry) and Yuga (Astronomical
speculation on).
CHATVARIMSHATI. Ordinary Sanskrit name
for the number *forty.
CHHEDANA. [Arithmetic]. Term meaning
division (literally: "to break into many pieces”).
Synonyms: bhagahara, bhajana, etc.
CHHEDI (Calendar). Calendar beginning
5 September, 248 CE, which was used in the
region of Malva and in Madhya Pradesh. To
obtain the corresponding date in our own
calendar, add 248 to a given Chhedi date.
Sometimes called kalachuri, it was in use
until the eighteenth century CE. See Indian
calendars.
CHHIDRA. IS]. Value = 9. “Orifice”. The nine
orifices of the human body (the mouth, the two
eyes, the two nostrils, the two ears, the anus
and the sexual orifice). See Nine.
CHRONOGRAM. A short phrase or sentence
whose numerical value amounted to the date of
a given event. There are many methods of com-
posing chronograms in India.
CHRONOGRAMS (Systems of letter numer-
als). One of the processes of composing
chronograms involves the use of a *numeral
alphabet. The hidden date is revealed by evalu-
ating the various letters of each word of the
sentence in question, then totalling the value of
each word. This requires a mixture of mathe-
matical and poetical skill, using the
imagination to create sentences which have
both literal and mathematical meaning. These
types of chronograms (for which the system of
evaluation clearly varies according to the
system of attribution of numerical values to the
letters of the alphabet) were not only written in
Sanskrit, but also in various *Prakrits (local
dialects). Many examples have been found
throughout India, from Maharashtra, Bengal,
Nepal or Orissa to Tamil Nadu, Kerala or
Karnataka. They were also used by the
Sinhalese, the Burmese, the Khmers, and in
Thailand, Java and Tibet. Many other examples
also exist in Muslim India and in Pakistan, but
these are many chronograms which employ
numeral letters of the Arabic-Persian alphabet
using a process called Abjad. See Numeral
alphabet and composition of chronograms.
CHRONOGRAMS (Systems of numerical
symbols). Another method of composing
chronograms is only used for expressing the
dates of the Shaka calendar: the language used
is always Sanskrit and the dates are always
expressed metaphorically, using Indian
•numerical symbols ruled by the place-value
system. This process was used for many cen-
turies in India and in all the Indianised
civilisations of Southeast Asia (Khmer, Cham,
Javanese, etc., kingdoms). See Numerical sym-
bols, Numerical symbols (principle of the
numeration of).
CIRCLE. As a symbolic representation of the
sky. See Serpent (Symbolism of the).
CIRCLE. As the graphic representation of zero.
See Shunya-chakra, The numeral 0, Zero.
CITY-FORTRESS. [S]. Value = 3. See Pura.
Tripura and Three.
COBRA (Cult and symbolism of). See Serpent
(Symbolism of) and Naga.
CODE (secret writing and numeration). See
Numeral alphabet and secret writing.
COLOUR. [S]. Value = 6. See Raga and Six.
COMPLETE. As a synonym of a large quantity.
See High numbers (Symbolic meaning of).
COMPLETE. As a synonym of zero. See
Purna.
CONSTELLATION. [S]. Value = 27. See
Nakshatra and Twenty-seven.
CONTEMPLATION. [S]. Value = 6. See
Darshana and Six.
COSMIC CYCLES. The division and length of
cosmic cycles has always been of great impor-
tance in terms of Brahmanism: These periods
represented the successive sections of cosmic
life, conceived as cyclical and eternally revolv-
ing. The divisions of time were naturally the
key elements of these cycles. The temporal
dimension was meant to correspond to the
duration of the creative and animating power
of the cosmos, the “Word” (*vachana), which
was uttered by the supreme progenitor of the
world, Brahman-Prajapati, and that which
assimilates “knowledge" par excellence, the
Veda. Thus the progenitor resembles the year
which is taken as a unit of measurement of its
cyclical activity, and the * Veda, a collection of
lines, is divided into as many metric elements as
there are moments in the “year” (see HGS, I, pp.
157-8). Of course, the “year” in question here is
a “divine” year as opposed to a human year. See
Divine Year, Yuga (Definition), Yuga (Systems
of calculation of), Yuga (Cosmogonic specula-
tions on), Kalpa, Day of Brahma.
COSMIC ERAS. See Cosmic cycles and Yuga
(Definition).
COSMOGONIC SPECULATIONS. See Yuga
(Cosmogonic speculations on), Kalpa, Jaina.
COURAGE. [SI. Value = 14. See Indra and
Fourteen.
COW. IS]. Value = 1. See Go and One.
COW. [S). Value = 9. See Go and Nine.
CUBE ROOT. [Arithmetic]. See Ghanamula.
CUBE. [Arithmetic]. See Ghana.
D
DAHANA. IS]. Value = 3. “Fire”. See Agni and
Three.
DANTA. [S]. Value = 32. “Teeth”. Humans
have thirty-two teeth. See Thirty-two.
DANTIN. [SI. Value = 8. “Elephant”. See
Diggaja and Eight.
DARSHANA. [S]. Value = 6. “Vision", “con-
templation”, “system”, and by extension
“demonstration” and “philosophical point of
view”. This concerns the six principal systems
of Hindu philosophy: mental research
( mimamsa) ; method (nyaya); the study and
description of nature (vaisheshika): number as
a way of thinking applied to the liberation of
the soul {*samkhya)\ the philosophies and prac-
tices of the liberation of the spirit from
material ties ( yoga); and studies based on the
Vedanta Sutras which deal with the basic iden-
tity of the soul and the *Brahman ( vedanta ).
See Shaddarshana and Six.
DASHA (or DASHAN). Ordinary Sanskrit
name for the number ten, which appears in
the composition of many words which
have a direct relationship with the idea
of this number. Examples: *Dashabala,
*Dashabhumi, *Dashaguna, * Dashagundsamjhd,
* Dashagunottarasamjha, *Dashahara,
*Dashdvatdra.
For words which have a more symbolic
relationship with this number, see Ten and
Symbolism of numbers.
DASHABALA. “Ten powers”. This refers to the
ten faculties possessed by a Buddha, which give
455
DASHABHUMI
him ten powers, namely: the intuitive knowl-
edge of the possible and the impossible,
whatever the situation; the development of
actions; the superior and inferior faculties of
living beings; the diverse elements of the
world; the paths which lead to purity or impu-
rity; contemplation, concentration, meditation
and the three deliverances; death; and the
purification of all imperfections .
DASHABHUMI. " Ten lands", “ten paradises”.
This refers to the “ten stages” of the Buddha
Shakyamuni.
DASHAGUNA. “Ten, primordial property”.
Sanskrit name for the decimal base. This word
can be found in such works as the *Trishatika
by Shrldharacharya [see TsT, R. 2 - 31 and in
the *Lilavdti by Bhaskaracharya [see Ul, p. 2],
DASHAGUNASAMJNA. “Words representing
powers of ten”. Term which applies to names of
numbers of the Sanskrit numeration, distrib-
uted according to a decimal scale (base 10). See
Dashaguna, Names of numbers and High
numbers. This word can be found in such
works as the *Trishatika by Shrldharacharya
[see TsT, R. 2 - 3],
DASHAGUNOTTARASAMJNA. “Words rep-
resenting powers of ten”. Term which applies
to names of numbers of the Sanskrit numera-
tion, distributed according to a decimal scale
(base 10). The contrast is made here with the
word shatottarasmjna which applies to the
centesimal scale (base 100). See Dashaguna,
Names of numbers. High numbers and
Shatottarasamjha.
DASHAHARA. Name of the Feast of the tenth
day. See Dasha and Durga.
DASHAKOT1. Literally “ten *kotis". Name
given to the number ten to the power eight ( =
a hundred million). See Names of numbers
and High numbers.
Source: * Gamtasarasamgraha by Mahaviracharya
(850 CE).
DASHALAKSHA. Literally “ten *lakshas”.
This is the name given to the number ten to tile
power six (one million). See Names of num-
bers and High numbers.
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
DASHAN. Ordinary Sanskrit name for the
number ten. See Dasha.
DASHASAHASRA. Literally “ten *sahastras".
Name given to the number ten to the power
four (ten thousand). See Names of numbers
and High numbers.
Source: *(ianitasdrasamgruha by Mahaviracharya
(850 CE).
DASHAVATARA. Name of the “ten major
incarnations” of *Vishnu, which are as follows:
Matsya (incarnation as a fish); Kurma (incarna-
tion as a tortoise); Varaha (as a boar);
Narasimha (as a lion-man); Vamana (as a
dwarf); Parashu-Rdma (as Rama of the axe);
*Rama (the hero of Rdmayana)\ * Krishna (the
god); Budha (the god); and Kalki. See Avatara.
DASRA. (S). Value = 2. Name of one of the two
twin gods Saranyu and Vivashvant of the Hindu
pantheon (also called Dasra and Nasatya).
Symbolism through association of ideas with the
“Horsemen”. See Ashvin and Two.
DAY. [S). Value = 15. See Tithi, Ahar and
Fifteen.
DAY OF BRAHMA (Arithmo-cosmogonical
speculations about the). According to Brahman
cosmogony, the lifespan of the material universe
is limited, and it manifests itself by *kalpa
cycles: “All the planets of the universe, from the
most evolved to the most base, are places of suf-
fering, where birth and death take place. But for
the soul that reaches my Kingdom, O son of
Kunti, there is no more reincarnation. One day
of *Brahma is worth a thousand of the ages
1 *yuga] known to humankind; as is each night”
(*Bhagavad Gita, VIII, lines 16 and 17). Thus
each kalpa is worth one day in the life of
Brahma, the god of creation. In other words, the
four ages of a *mahdyuga must be repeated a
thousand times to make one “day of Brahma”,
a unit of time which is the equivalent of
4,320,000,000 human years.
According to this cosmogony, this is the
total length of one created universe. The kalpa
or “day of Brahma” is meant to correspond to
the appearance, evolution and disappearance
of a world, and this cycle is followed by a
period of “cosmic repose” of equal length,
w'hich is followed by a new kalpa, and so on
indefinitely. In other words, each kalpa ends
with the total destruction (pralaya) of the uni-
verse w'hich is followed by a period of
reabsorption which is equivalent to a “night of
Brahma”, of equal length to the corresponding
“day”, before life is breathed into a new uni-
verse. It is precisely during this period of
non-creation that *Vishnu, lying on *Ananta,
the serpent of Infinity and Eternity, rests w'hile
he waits for Brahma to accomplish his work of
Creation. This philosophy was developed as far
as to speculate on the “length of the life of the
god Brahma”. A Commentary on the
*Bhagavad Gita says: “. . . nothing in the mater-
ial universe, not even Brahma can escape birth,
ageing and death ... the Causal Ocean con-
tains countless Brahmas, who, being in a con-
stant state of flux, appear and disappear like
bubbles of air”.
Here are some calculations relating to this
this subject. Given that one whole “twenty-four
hour day” in this god’s life is the sum of one of
his “days” and one of his “nights", “twenty-four
hours in the life of Brahma” corresponds to:
4.320.000. 000 + 4,320,000,000 = 8,640,000,000
(= eight thousand, six hundred and forty mil-
lion) human years. One “year of Brahma”
is made up of 360 of these “days”. Thus it
corresponds to 8,640,000,000 x 360 =
3.110.400.000. 000 (= three billion, one hun-
dred and ten thousand, four hundred million)
human years. As this god is said to live for one
hundred of his “years”, the total length of his
existence is equal to: 3,110,400,000,000 x 100
= 311,040,000,000,000 ( = three hundred and
eleven billion, forty thousand million) human
years. According to certain traditions reported
notably by al-Biruni, the “day of Brahma” does
not correspond to a simple kalpa, but to a
*parardha of kalpas, which is the length of a
kalpa multiplied by ten to the power seventeen.
Thus: 1 “day of Brahma” = 100,000,000,
000, 000, 000 (= one hundred trillion) kalpas.
As one kalpa is 4,320,000,000 years long, one
“day” of this god corresponds to: 432,000,000,
000,000,000,000,000,000 (= four hundred and
thirty-two sextillions) human years. Thus the
complete “day” = 864,000,000,000,000,000,
000,000,000 (= eight hundred and sixty-four
sextillions) human years. If we multiply this
number by 36,000, the "life of Brahma" lasts
thirty-one octillion and one hundred and four
septillion human years. Childish at first sight,
such speculations are very revealing of the
Indian tendency towards metaphysical abstrac-
tion and of the high conceptual level achieved
at an early stage by this civilisation. See
Ananta, Asamkhyeya, Calculation, High num-
bers, Infinity, Speculative arithmetic,
Sanskrit, Sheshashirsha, Indian mathematics
(The history of) and Yuga (Cosmogonical
speculations on).
DAY OF BRAHMA. Cosmic period corre-
sponding to the total length of one creation of
the universe. According to Brahman cos-
mogony, this “day” is equal to 12,000,000
divine years (*divyavarsha); and as one divine
year is equal to 360 human years, the “day of
Brahma” is equal to 4,320,000,000 human
years. See Divyavarsha, Mahay uga and Yuga.
DAY OF THE WEEK. [S]. Value = 7. See Vara
and Seven.
DECIMAL NUMERATION. See Dashaguna
and Dashagundsamjha.
DELECTATION. [S]. Value = 6. See Rasa and
Six.
DEMONSTRATION. [SI- Value = 6. See
Darshana and Six.
DESCENT. IS]. Value = 10. See Avatara and
Ten.
DEVA. [S]. Value = 33. “Gods”. This is the gen-
eral name given to all the divinities of the
Hindu, Brahmanic, Vedic and Buddhist pan-
theons. These are the inhabitants of Mount
Meru (mythical mountain, situated at the axis
of the universe), who are ruled by a god. Unlike
the great divinities ( Mahddeva ) such as
*Brahma, *Vishnu and *Shiva, these divinities
have neither strength nor creative power.
Theoretically numbering thirty-three million,
they are reduced to thirty-three in Hindu cos-
mogony, w'hich also gives their group the name
*Traiyastrimsha (“thirty-three”). See Thirty-
three. See also Mount Meru.
DEVANAGARI NUMERALS. See Nagari
numerals.
DEVAPARVATA. “Mountain of the gods”. One
of the names of Mount Meru, the home of the
gods in Brahmanic mythology and Hindu cos-
mology. See Mount Meru, Adri, Dvipa, Puma,
Patdla, Sdgara, Pushkara, Pavana and Vayu.
DHARA. (SJ. Value = 1. “Earth”. See Prithivi
and One.
DHARANI. [S]. Value = 1. Literally “Bearer”.
This is synonymous here with the “earth", in
the sense of “the bearer”. See Prithivi and One.
DHARMA. In Indian philosophies, the
Dharma is the general law, the Duty, the thing
which is permanently fixed, the ensemble of
rules and natural phenomena which rule the
order of things and of men. In Buddhist philos-
ophy in particular, the dharma is considered to
be one of the three Treasures ( *Triratna ) and
one of the three refuges of the faithful. It is thus
the social duty of the disciple. It represents the
ultimate Only Reality, Virtue, Natural Order of
all that exists, the Doctrine of Buddha as well
as all the perceptions (ideas) hidden in
the Manas [see Frederic, Dictionnaire ]. See
Shunyata.
DHARTARASHTRA. [S]. Value = 100. There is
a legend related in the Mahabharata about the
blind king Dhritarashtra, son of Ambika and
the king Vichitravirya, who married Gandhari,
with whom he had a hundred sons, called
Dhartarashtra. During the Great Battle against
the sons of Pandu, the latter were all killed and
became demons [see Frederic, Dictionnaire].
See Pandava and Hundred.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
456
DHATRI. [SJ. Value = 1. “Earth”. See Prithivi
and One.
DHRITI. IS]. Value = 18. This refers to the
metre of four times eighteen syllables per verse
in Sanskrit poetry. See Indian metric.
DHRUVA. [S]. Value = 1. In Hindu mythology,
this was the son of a king called Uttanapada
and his queen Suniti, who, through the power
of his will, became the Sudrishti, the “divinity
who never moves”: the Pole star, whose unique-
ness and fixedness are doubtless at the root of
this symbolism. See One.
DHULiKARMA. Literally “work on dust”
(from Dhuli, "sand”, "dust”, and karma, "act”).
Term used in ancient Sanskrit literature to
denote the “act of carrying out mathematical
operations”, in allusion to the ancient Indian
practice of carrying out calculations on a board
covered in sand. Today, the word is only used
in the abstract sense of "superior mathemat-
ics”. See Calculation (Methods of).
DHVAJAGRANISHAMANI. Name given to
the number ten to the power 145. See Names
of numbers and High numbers.
Source: *I.aIitavistara Sutra (before 308 CE).
DHVAJAGRAVATI. Name given to the number
ten to the power ninety-nine. See Names of
numbers and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
DIAMOND. A representation of the number
ten to the power thirteen. See Shanku.
DIGGAJA. [S]. Value = 8. In Hindu cos-
mogony, the collective name given to the
Ashtadiggaja, the “eight Elephants”, who are
said to guard the eight horizons of space. See
Dish and Eight.
DIGITAL CALCULATION. See Mudrd.
DIKPALA. [SI. Value = 8. "Guardian of the
points of the compass". In Hindu cosmogony,
this is the collective name given to the eight
divinities considered to be the guardians of the
horizons and the points of the compass
(*Indra in the east, *Agni in the southeast,
*Yama in the south, Nirriti in the southwest,
*Varuna in the west, Kuvera in the north,
*Vayu in the northwest and Ishana in
the northeast). See Diggaja, Dish, Lokapala
and Eight.
DISH. [S]. Value = 4. "Horizon". The four
cardinal points (north, south, east and west).
See Four.
DISH. [S]. Value = 8. "Horizon”. The horizons
corresponding to the eight points of the com-
pass: the north, the northwest, the west, the
southwest, the southeast, the south, the east
and the northeast. See Eight.
DISH. [S]. Value = 10. "Horizon”. The ten hori-
zons of space: the eight normal horizons, plus
the nadir and the zenith. See Ten.
DISHA. [S). Value = 4. “Horizon”. See Dish
and Four.
DISHA. [S]. Value = 10. “Horizon”. See Dish
and Ten.
DIVAKARA. (S). Value = 12. “Sun". See Surya
and Twelve.
DIVIDEND. [Arithmetic]. See Bhdjya.
DIVINATION. See Numeral alphabet, magic,
mysticism and divination, Indian astrology,
and Indian astronomy (The history of).
DIVINE MOTHER. [S]. Value = 7. See Mdtrikd
and Seven.
DIVINE PERFECTION. As a symbol for a
large quantity. See High numbers (Symbolic
meaning of).
DIVINE YEAR. See Divyavarsha.
DIVISION. [Arithmetic]. See Chhedana ,
Bhagahara. Labdha, Shesha and Bhdjya.
DIVISOR. [Arithmetic]. See Bhagahara.
DIVYAVARSHA. “Celestial or divine year”. To
convert a number of divine years into human
years, it must be multiplied by 360.
DOGRI NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Gupta
and Sharada numerals, and constituting a vari-
ation of Takari numerals. These are currently
used in the Indian part of Jammu (in the south-
west of Kashmir). The corresponding system
uses the place-value system and possesses zero
(in the form of a little circle). See Indian writ-
ten numeral systems (Classification of). See
Fig. 24.13, 52 and 24.61 to 69.
DOT. A graphical sign representing zero, see
Numeral 0, Bindu, Shunya-bindu, Zero.
DOT. A name for ten to the power forty-nine.
See Bindu, High numbers, High numbers
(Symbolic meaning of).
DOT. [S]. Value = 0. See Bindu, Indian atom-
ism and Zero.
DRAVIDIAN NUMERALS. Numerals used in
the southern regions of India, namely Tamil
Nadu, Karnataka, Andhra Pradesh and Kerala,
where the people are referred to as “Dravidian",
and who, unlike the people from northern and
central India, do not speak Indo-European lan-
guages. These signs are derived from *Brahmi
numerals, through the intermediary of Shunga,
Shaka. Kushana, Andhra, Pallava, Chalukya,
Ganga, Valabhi and Bhattiprolu numerals. The
corresponding system has not always used the
place-value system or possessed zero. See Tamil
numerals, Malayalam numerals, Telugu
numerals. Kannara numerals and Indian writ-
ten numeral systems (Classification of).
DRAVYA [SI. Value = 6. "Substances”. The six
“bodies”, or "substances" which make up exis-
tence according to *Jaina philosophy (these
are: dharmashlikaya, which constitutes the
means of movement; adharmashtikaya, which
allows the animate to become inanimate;
akshatikdya, which creates the space in which
the animate and the inanimate live;
pudgaJashtikaya, which enables the very exis-
tance of matter; jivashtikdya, which allows the
mind to exist through inferences; and kala,
which is nothing other than time [see Frederic
(1987)). See Six. This symbol is found in
*Ganitasdrasamgraha by Mahaviracharya [see
Datta and Singh (1938), p. 55).
DRIGG ANITA. See Parameshvara.
DRISHTI. [S]. Value = 2. This term is generally
used in the sense of "vision", “contemplation",
“revelation”, “conception of the world” and
“theory”. Its primary sense, however, is “eye”;
hence drishti = 2. See Netra and Two.
DROP. [SI. Value = 1. See Indu and One.
DUALITY. See Dvaita.
DURGA. [S]. Value = 9. “Inaccessible”. This is
the name of a Hindu divinity, the “Divine
Mother”, wife of Shiva, who is worshipped
during the “Feast of the nine days” ( navardtri ),
which is celebrated at the end of the rain
season in the month of Ashvina (September -
October). The association of ideas which led to
Durga becoming the numerical symbol equiva-
lent to nine is obvious, but the choice of this
value for the number of days of the feast is diffi-
cult to explain. This divinity, who is said to
possess great powers, is often represented as
having ten arms; moreover, she is depicted
standing on a lion, which symbolises her
power. The “Feast of nine days”, which marks
the end of the monsoon, ends on the tenth day
with the grand feast of the dashahara (from
dasha, “ten”), which is dedicated to Durga. The
Hindus commemorate the victory of their
divinities over the forces of Evil.
In this double religious symbolism, it is
possible that, in accordance with the character-
istic Indian way of thinking, these nine days
were associated with the nine numerals of the
place-value system, with which it is possible to
write all numbers. The tenth day might then be
associated with the tenth sign in this system:
the zero, which corresponds to the most elu-
sive, “inaccessible" and abstract concept; a
concept whose invention is attributed to
*Brahma, and which certainly constituted a
great victory over the difficulties presented by
numerical calculation. As for the tenth whole
number, which in this system is written using a
1 and a 0, this would have corresponded, in the
Indian symbolic mind, to an achievement, fol-
lowed by the return to the unit at the end of the
development of the cycle of the first nine num-
bers. However, this is mere conjecture for
w'hich there is no proof or foundation, it is
simply based on one of the possible attitudes
which characterise Indianity so well. See
Shunya, Shunyatd, Zero and Nine.
DUST BOARD. See Pati, Patiganita.
DVA (or DVE, DVI). Ordinary Sanskrit name
for the number two, w'hich forms a component
of many words which have a direct relationship
with the concept of duality, opposition, com-
plementarity, etc. Examples: *Dvaipdyanayuga ;
*Dvaita, *Dvandva, *Dvandvamoha, *Dvapar-
ayuga, *Dvaya, *Dvija, *Dvivachana.
For words which have a more symbolic link
with this number, see Two and Symbolism of
numbers.
DVADASHA. Literally "twelve”. This term is
used symbolically in the Rigveda to mean “year”,
in allusion to the twelve months of the year.
Ref: Rigveda, VII, 103, 1; Datta and Singh
(1938), p. 57.
DVADASHA. Ordinary Sanskrit name for the
number twelve. For words which have a sym-
bolic link with this number, see Symbolism
of numbers.
DVADASHADVARASHASTRA. “Tract of the
twelve doors”. Title of a work by Nagarjuna,
one of the principal Buddhist philosophers,
founder of the school of *Madhyamika. See
Dvddasha and Shunyatd.
DVAIPAYANAYUGA. Synonym of *dvd-
parayuga.
DVAITA. "Duality”. Term applied to a dualist
philosophy, according to which a human crea-
ture is different from the *Brahman, its creator.
This philosophy opposes the pure doctrine of
the Vedantas, which is monistic ( Advaitav -
eddnta, “non dualist Vedanta”).
457
DVANDVA
DVANDVA [S]. Value = 2. "Couple, contrast”.
The symbolism is self-explanatory. See Two.
DVANDAMOHA. From dvandva, “couple, con-
trast”, and moha, “illusion”. This is the name
given by the Hindus to what they consider to
be the illusory impression that couples com-
posed of opposites exist, such as shadow and
light, joy and pain, etc.
DVAPARAYUGA (or DVAIPAYANAYUGA).
Name of the third of the four cosmic ages
which make up a *mahdyuga. This cycle, which
is meant to be the equivalent of 864,000
human years, is regarded as the age during
which humans have only lived for half of their
lives, and where the forces of good have bal-
anced out those of evil. See Yuga (Definition).
DVATRIMSHADVARALAKSH ANA. “Thir ty-
two distinctive signs of perfection”. According
to Buddhism, these are the signs which allow
Buddha to differentiate between ordinary
humans from a moral, physical or spiritual per-
spective. See Dvatrimshati.
DVATRIMSHATI (or DVITRIMSHATI).
Ordinary Sanskrit name for the number thirty-
two. For words which have a symbolic
connection with this number, see Thirty-two
and Symbolism of numbers.
DVAVIMSHATI (or DVIVIMSHATI).
Ordinary Sanskrit name for the number
twenty-two. For words which have a symbolic
link with this number, see Twenty-two and
Symbolism of numbers.
DVAYA. [S], Value = 2. Word meaning “pair”.
The symbolism is self-explanatory. See Two.
DVE. Ordinary Sanskrit name for the number
two. See Dva.
DVI. Ordinary Sanskrit name for the number
two. See Dva.
DVIJA. [S], Value = 2. “Twice born”. Epithet
given to people belonging to the first three
Brahmanic casts having the right to wear the
sacred sash and who, during the ceremony of the
handing over of the sash, are considered to be
beginning a second life, this time of a spiritual
nature [see Frederic (1987)]. See Dva and Two.
DVIPA. [S], Value = 7. “Island -continent”.
Allusion to the seven island-continents which,
in Hindu cosmology, are meant to radiate out
from *Mount Meru. See Adri and Seven. See
also Sapta Dvipa.
DVIPA. [S]. Value = 8. “Elephant”. See Diggaja
and Eight.
DVITRIMSHATI. Synonym of * dvatrimshati.
DVIVACHANA. Name of the dual of Sanskrit
verbs.
DVIVIMSHATI. Synonym of *dvavimshati.
DYUMANI. [Si. Value = 12. “Sun”. See Surya
and Twelve.
E
EARTH. As a mystical symbol for the number
four. See Naga, Jala, Ocean, Serpent
(Symbolism of the).
EARTH. As a name for the number ten to the
power sixteen, ten to the power seventeen, ten
to the power twenty, ten to the power twenty-
one. See Kshiti , Kshoni, Mahakshiti,
Mahdkshoni and High numbers.
EARTH. [SI. Value = I. See Avani, Bhu, Bhumi,
Dhara, Dharani, Dhdtri, Go, Jagati, Kshauni,
Kshemd, Kshiti, Kshoni, Ku, Mahi, Prithivi,
Vasudhd, Vasundhara and One.
EARTH. [S]. Value = 9. See Go and Nine.
EASTERN ARABIC NUMERALS. Signs
derived from *Brahmi numerals, through
the intermediary of Shunga, Shaka,
Kushana, Andhra, Gupta and Nagari numer-
als. Currently in use in Near and Middle East
and in Muslim India, Malaysia and Indonesia.
The corresponding system functions according
to the place-value system and possesses zero
(formerly either in the form of a little circle or
dot but today exclusively represented by a dot).
See Indian written numeral systems
(Classification of). See Fig. 24.2, 24.52 and
24.61 to 24.69.
EIGHT. Ordinary Sanskrit names for the
number eight: *ashta, * ash tan. Here is a list of
the corresponding numerical symbols: *Ahi,
Anika, *Anushtubh, Bhuti, *Dantin, * Diggaja ,
Dik, *Dikpala, *Dish, Durita, * Dvipa, Dvirada,
*Gaja, *Hastin, Ibha, Karman, *Kuhjara,
*lokapala, Mada, *Mar\gala, *Matanga,
*Murti, *Naga, Pushkarin, *Sarpa, *Siddhi,
Sindhura, *Takshan, *Tanu, *Vasu and Varna.
These words can either be translated by the
following words or have a symbolic relationship
with them: 1. The serpent (Ahi, Naga, Sarpa). 2.
The serpent of the deep (Ahi). 3. The elephant
(Dantin, Dvipa, Diggaja). 4. The eight elephants
(Diggaja). 5. That which augurs well (Mangala).
6. The jewel (Mangala). 7. The shapes, or forms
(Marti). 8. The horizons (Dish). 9. The guardians
of the horizons and of the points of the compass
(Lokapala). 10. The guardians of time (Dikpala).
11. Supernatural powers (Siddhi). 12. Certain
groups of lines of Vedic poetry (Anushtubh). 13. A
group of eight divinities (Vasu). 14. The spheres
of existence of Adibhautika (Vasu). 15. The “acts”
(Karman) (only in *Jaina philosophy). 16. The
“body” (Tanu).
See Numerical symbols.
EIGHTEEN. Ordinary Sanskrit name: *ash-
tadasha. The corresponding numerical symbol
is *Dhriti.
EIGHTY. See Ashiti.
EKA. Ordinary Sanskrit word for the number
one, which appears in the composition of
many words which have a direct relationship
with the concept of unity. Examples:
*Ekachakra, *Ekadanta, *Ekdgrata, Ekakshara,
*Ekdntika, *Ekatva.
For words which have a symbolic connec-
tion with the concept of this number, see One
and Symbolism of numbers.
EKACHAKRA. “Who has only one wheel”.
Attribute of *Surya (the Sun-god).
EKADANTA. “Who has only one tooth”.
Attribute of *Ganesha, son of *Shiva and
Parvati, who is represented as having the body
of a man and the head of an elephant, endowed
with a unique defence. He is the Hindu divinity
of wisdom, guaranteeing success in terrestrial
existence and spiritual life.
EKADASHA. The ordinary Sanskrit name for
the number eleven. For words which have a
symbolic link with this number, see Eleven and
Symbolism of numbers.
EKADASHARASHIKA. [Arithmetic]. Sanskrit
name for the Rule of Eleven.
EKADASHI. Name of the eleventh day after
the new moon, which orthodox Hindus spend
fasting and meditating.
EKAGRATA. In Hindu philosophy, a term
which denotes a particular type of esoteric
yoga, consisting in concentrating all of
one’s attention on a single point or object,
which allows one to achieve dhyana or “active
contemplation”.
EKAKSHARA. “Unique and indestructible".
Name of the sacred syllable of the Hindus
(*AUM).
EKANNACHATVARIMSHATI. “One away
from forty”. The ancient form of the Sanskrit
name for the number thirty-nine (in Vedic
times). See Names of numbers.
EKANNATRIMSHATI. “One away from
thirty”. The ancient form of the Sanskrit name
for the number twenty-nine (in Vedic times).
See Names of numbers.
EKANNAVIMSHATI. “One away from
twenty”. The ancient from of the Sanskrit
name for the number *nineteen (during Vedic
times). See Names of numbers.
EKANTIKA. Name of the monotheistic doc-
trine of the Vishnuite tradition.
EKATVA. In Hindu philosophical systems, a
term denoting Unity, the contemplation of
Everything. This is the ability to see the Self or
the Divine in everything, and everything in the
Self or the Divine.
EKAVIMSHATI. Ordinary Sanskrit name for
the number twenty-one. For words which have
a symbolic connection with this number, see
Twenty-one and Symbolism of numbers.
ELEMENT. [S]. Value = 5. See Bhuta, Five and
Pahchabhuta.
ELEMENTS OF THE REVELATION. See
Bhuta, Pahchabhuta, Jala, Five, Numeral
alphabet, magic, mysticism and divination
and Ocean.
ELEPHANT. A symbol for ten to the power
twenty-one, ten to the power twenty-seven,
ten to the power 105 or ten to the power
112. See Kumud, Kumuda, Pundarika and
High numbers.
ELEPHANT. [S]. Value = 8. See Dantin,
Diggaja, Gaja, Hast in, Kuhjar, Eight and
Ashtadiggaja.
ELEVEN. Ordinary Sanskrit name: *ekadasha.
Here is a list of corresponding numerical sym-
bols: Akshauhini, *Bharga, *Bhava, *Hara,
*lsha, *lshvara, Labha, *Mahadeva, *Rudra,
* Shiva, * Shuiin, Trishtubh.
These words have the following translation or
symbolic meaning: 1. A name or attribute of
Rudra-Shiva (Bharga, Bhava, Hara, Isha, Ishvara,
Mahade\>a, Rudra, Shiva, Shuiin). 2. The “Supreme
Divinity” (Ishvara). 3. The “Lord of the Universe”
(Ishvara). 4. The “Great God” (Mahadeva).
5. “Grumbling” (Rudra). 6. The “Lord of tears”
(Rudra). 7. “Violent” (Rudra). 8. The “Master of
the animals" (Shuiin), See Numerical symbols.
ENERGY (feminine). [SJ. Value = 3. See Shakti
and Three.
EQUATION. [Algebra]. See Ghana, Varga,
Vargavarga, Samikarana, Vyavahara,
Ydvattdvat and Indian mathematics (The
history of).
ERAS (of Southeast Asia). See Shaka,
Buddhashakardja and Indian calendars.
ESOTERICISM. See Akshara, Numeral alpha-
bet and secret writing, Numeral alphabet,
magic, mysticism and divination, Atman,
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
4 58
AVM, Bija. Ekagratd, Ekakshara, Kavacha,
Mantra, Trivarna, Vachana and Serpent.
ETERNITY. See Atlanta and Infinity.
ETHER. [S]. Value = 0. See Akasha, Shunya
and Zero.
EUROPEAN NUMERALS (Algorisms).
Numerals used after the twelfth century by
European mathematicians (written calcula-
tion). The corresponding system functioned
according to the place-value system and pos-
sessed a zero (in the form of a little circle).
These signs derived from *Brahmi numerals,
firstly through the intermediary of types of
Indian numerals such as Shunga, Shaka,
Kushana, Andhra, Gupta and Nagari, and then
via the numerals used by the Arabs. The
appearance of the numerals varied greatly from
one school to another. Some styles derived
from “Hindi” numerals, but most came from
Arabic numerals. One such style, standardised
due to the requirements of typography, became
the origin of the numerals we use today: 12 3 4
5 6 7 8 9 0. See Indian written numeral sys-
tems (Classification of). See also Fig. 24.52
and 24.61 to 69.
EUROPEAN NUMERALS (Apices of the
Middle Ages). Numerals used by European
mathematicians in the Middle Ages (who car-
ried out their calculations on an abacus). They
derive from *Brahmi numerals, first through
the intermediary of types of Indian numerals
such as Shunga, Shaka, Kushana, Andhra,
Gupta and Nagari, and then via Ghubar
numerals of North African Arabs. The appear-
ance of the numerals varied greatly from one
school to another. The corresponding system
did not possess zero because calculations were
carried out on the abacus. See Indian written
numeral systems (Classification of). See also
Fig. 24.52 and 24.61 to 69.
EYE. [SJ. Value = 2. See Netra, Drishti and
Two.
EYE. [S]. Value 3. See Netra and Three.
EYE OF SHUKRA. [SJ. Value = 1. See
Shukranetra and One.
EYES. [SI. Value = 2. See Lochana and Two.
EYES OF INDRA. [S[. Value = 1,000. See
Indradrishti and Thousand.
EYES OF SENANL [S[. Value = 12. See
Senaninetra and Twelve.
EYES OF SHIVA. [S[. Value = 3. See Haranetra
and Three.
F
FACE. [SJ. Value = 4. See Mukha and Four.
FACES OF BRAHMA. [S[. Value = 4. See
Brahmasya and Four.
FACES OF KARTTIKEYA. [S[. Value = 6. See
Karttikeyasya and Six.
FACES OF KUMARA. [S[. Value = 6. See
Kumaravadana and Six.
FACES OF RUDRA. [S[. Value = 5. See
Rudrasya and Five.
FACULTY. [SJ. Value = 5. See Indriya and Five.
FIFTEEN. Ordinary Sanskrit name:
‘pahchadasha. Here is a list of corresponding
numerical symbols: *Ahar, Dina, Ghasra,
‘Paksha, ‘Tithi. These words have the follow-
ing translation or symbolic meaning: 1.
"Wing”, in allusion to the number of days in
one of the two “wings” of the month ( Paksha ).
2. “Day", in allusion to the number of days in
one of the two “wings” of the month (Ahar,
Tithi). See Numerical symbols.
FIFTY. See Pahchashat and Names of
numbers.
FINGER. [S[. Value = 10. See Anguli and Ten.
FINGER (or Digit). [S], Value = 20. See Anguli
and Twenty.
FINITE (Number). See Infinity and Indian
mathematics (The history of).
FIRE. [S[. Value = 3. See Agni, Anala, Dahana,
Hotri, Hutashana, Jvalana , Krishanu, Pdvaka,
Shikhin, Tapana, Udarchis. Vahni, Vaishvanara
and Three.
FIRE [S[. Value = 12. See Tapana. Twelve.
FIRMAMENT. [S[. Value = 0. See Shunya,
Zero and Infinity.
FIRST FATHER. [S], Value = 1. See Pitamaha
and One.
FIVE. Ordinary Sanskrit name: ‘pahcha. Here is
a list of corresponding numerical symbols:
Artha, ‘Bana, Bhava, ‘ Bhuta , * Gavyd , * Indriya ,
‘Ishu, * Kalamba , ‘Karaniya, Kshara, Lavana,
* Mahabhuta , * Mahapapa , ‘Mahayjha,
‘ Morgana , Pallava, ‘Pandava, Parva, Parvan,
‘ Pataka , ‘ Pavana , ‘Prana, ‘ Puranalakshana ,
‘ Putra , * Patna, ‘ Rudrasya , * Say aka, ‘ Shara ,
Shastra, * Suta , Tanmdtra, Tata, ‘Tattva,
‘Tryakshamukha, ‘ Vishaya , ‘Vishikha.
The translation, or symbolic meaning of
these words is as follows: 1. Arrows ( ‘Bana ,
* Ishu , *Kalamha, ‘ Mdrgana , ‘Sdyaka, ‘ Shara ,
‘ Vishikha ). 2. Statistics (* Puranalakshana).
3. “That which must be done” (‘Karaniya).
4. Purification (* Pavana). 5. The gifts of the
Cow (* Gavya). 6. The elements, in allusion to
the five elements of the revelation (* Bhuta).
7. The Great Elements, in allusion to the five
elements of the revelation ( ‘Mahabhuta ).
8. The faculties (* Indriya). 9. The worst sins
( "Mahapapa ). 10. The great sacrifices
(* Mahdyajhc). 11. The main observances
( ‘Karaniya ). 12. The fundamental principles,
realities, truths, the “true natures” ( Tattva ).
13. The Jewels ( ‘Ratna ). 14. The breaths
(‘Prana). 15. The senses, or the sense organs
(* Vishaya ). 16. The Sons of Pandu ( ‘Pandava ).
17. The Sons, in allusion to the sons of Pandu
(* Putra). 18. The faces of Rudra (* Rudrasya,
‘Tryakshamukha). See Numerical symbols.
FIVE ELEMENTS (philosophy of the). See
Bhuta, Pahchabhuta, Jala, Five, Numeral
alphabet. Magic, Mysticism and Divination.
FIVE SUPERNATURAL POWERS. See
Pahchabhijhd.
FIVE VISIONS OF BUDDHA. See
Pahchachakshus.
FORM. [S[. Value - 1. See Rupa and One.
FORM. [S[. Value = 3. See Murti, Trimurti and
Three.
FORM. [S[. Value = 8. See Murti and Eight.
FORTY. Ordinary Sanskrit name:
‘chatvdrimshati. Corresponding numerical
symbol: Naraka.
FORTY-EIGHT. Ordinary Sanskrit name:
* ashtachatvarimshati . Corresponding numeri-
cal symbol: ‘Jagati.
FORTY-NINE. Ordinary Sanskrit name:
‘navachatvdrimshati. Corresponding numeri-
cal symbols: * Tana and * Vdyu.
FOUR. Ordinary Sanskrit name for this
number: ‘chatur. Here is a list of the correspond-
ing numerical symbols: ‘Abdhi, ‘Ambhodha,
Ambhodhi, * Ambhonidhi , Ambudhi, ‘Amburdshi,
‘ Arnava , Ashrama, Aya, Aya, Bandhu,
* Brahmasya , *Chaturananavadana, Dadhi,
* Dish , *Disha, *Gati, Gostana, * Haribahu , *lrya,
*Jaia, *Ja!adhi, *Jalanidhi,JaIashaya,Kashdya,
Kendra, Khatvapada, Koshtha, *Krita, *Mukha,
Payodhi, Payonidhi, Purushartha, * Sagara ,
Salilakara, *Samudra, Senanga, * Shruti , * Sindhu ,
*Turiya, *Udadhi, Vanadhi, *Varidhi, *Varimdhi,
‘Veda, Vishanidhi, Vyiiba, ‘Toni, ‘Yuga.
These words have the following translation
or symbolic meaning: 1. Water (Jala). 2. Sea or
ocean (* Abdhi , ‘ Ambhonidhi , ‘Ambudhi,
* Amburashi , ‘Arnava, ‘ Jaladhi , ‘ Jalanidhi ,
‘Jalashaya, ‘Sagara, ‘Samudra, ‘Sindhu,
‘Udadhi, ‘Varidhi, ‘Varinidhi). 3. The four
oceans (Chaturdnanavadana). 4. The “horizons”,
in the sense of the cardinal points (Dish, Disha).
5. The conditions of existence (Gati). 6. The
“Fourth” as an epithet of the Brahman ( Turiya ).
7. The “revelations” (Shruti). 8. The "positions"
(iryd). 9. The arms of Vishnu (Haribahu). 10. The
births (Gati, Yoni). 11. The vulva ( Yoni ). 12. The
Vedas (Veda). 13. The faces of Brahma
( Brahmasya ). 14. The “faces” (Mukha). 15. The
four ages of a mahdyuga ( Yuga). 16. The last of the
four ages of a mahdyuga ( Krita ). See Numerical
symbols. See also Ocean.
FOUR CARDINAL POINTS. [S[. Value = 4.
See Dish and Four.
FOUR ISLAND-CONTINENTS. See
Chaturdvipa and Ocean.
FOUR OCEANS (or FOUR SEAS). See
Chatursagara, Sagara (= 4) and Ocean.
FOUR STAGES. See Chaturashrama.
FOURTEEN. Ordinary Sanskrit name:
* chaturdasha . Here is a list of corresponding
numerical symbols: Bhuvana, ‘lndra, ‘Jagat,
‘Loka, ‘ Manu , Purva, ‘Ratna, ‘Shakra, ‘Vidya.
These words have the following translation
or symbolic meaning: 1. The god lndra (lndra).
2. “Courage”, “strength”, “power” (lndra).
3. Powerful (Shakra). 4. “Human", in the sense
of progenitor of the human race (Manu). 5. The
worlds (Bhuvana, Jagat, Loka). 6. The Jewels
(Ratna). See Numerical symbols.
FOURTH (The). [S[. Value = 4. Word used as
an epithet for *Brahma. See Turiya and Four.
FRACTIONS. [Arithmetic]. See Bhinna,
Kalavarna, Pahcha Jdti.
FUNDAMENTAL PRINCIPLE. [SJ. Value = 1.
See Adi and One.
FUNDAMENTAL PRINCIPLE. [S[. Value = 5.
See Tattva and Five.
FUNDAMENTAL PRINCIPLE. [SJ. Value = 7.
See Tattva and Seven.
FUNDAMENTAL PRINCIPLE. [SJ. Value =
25. See Tattva and Twenty-five.
G
GAGANA. [S]. Value = 0. Word meaning "the
canopy of heaven”, “firmament". This symbol-
ism is explained by the fact that the sky is
nothing but a “void”. See Zero and Shunya.
459
G AJ A
GAJA. [S]. Value = 8. “Elephant". See Diggaja
and Eight.
GAME OF CHESS. See Chaturanga.
GANANA. Word meaning “arithmetic” in
ancient Buddhist literature. More commonly,
however, it has been used in the sense of
“mental arithmetic” (which was and still is
particularly developed in the art of Indian
calculation).
GANANAGATI. From *ganana, “arithmetic”,
and *gati, “condition of existence". Name given
to the number ten to the power thirty-nine. See
Names of numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Source: *L,alitavistara Sutra (before 308 CE).
GANESHA. Hindu divinity of wisdom, also
called *Ekadanta. See Eka.
GANESHA. Indian mathematician who lived
around the middle of the sixteenth century.
Notably his works include a work entitled
Ganitamanjari.
GANGA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
ofShunga, Shaka, Kushana, Andhra, Pallava
and Chalukya numerals. These were contempo-
raries of the beginnings of the dynasty of the
Gangas of Mysore (sixth to eighth century CE)
The corresponding system did not use the
place-value system or zero. See Indian written
numeral systems (Classification of). See Fig.
24.46, 52 and 24.61 to 69.
GANITA. Sanskrit name for mathematics. In
Vedic literature, this word is used to mean “the
science of calculation”, which is no doubt its
original meaning. By extension, this word later
acquired the meaning “science of measuring”.
In ancient Buddhist literature, there are three
types of ganita: *mudrd or “manual arith-
metic”; *ganand or “mental arithmetic”; and
*samkhyana or “high arithmetic". Note that the
word ganita was often used in ancient times to
mean astronomy and even geometry (kshetra-
ganita). See Arithmetic, Calculation and
Indian mathematics (The history of).
GANITAKAUMUDI. See Narayana.
GANITANUYOGA. Word meaning ‘‘explana-
tion of mathematical principles". Term used
mainly in *Jaina texts.
GANITASARASAMGRAHA. See
Mahaviracharya.
GATI. [S]. Value = 4. Literally ‘‘condition of
existence". This word denotes the different
forms of existence that reincarnation can
assume ( *samsara ). The word became the
numerical symbol for 4, synonymous with
*yoni, ‘‘birth’’ [see Frederic (1994)1. See
Chaturyoni and Four.
GAUTAMA SIDDHANTA. (Not to be con-
fused with Gautama Siddhartha, the Buddha).
Chinese Buddhist astronomer of Indian origin,
author of a work on astronomy and astrology
entitled Kai yuan zhan jing (718 - 729 CE),
where he describes zero, the place-value system
and Indian methods of calculation. See Place-
value system, and Zero.
GAVYA. [S], Value = 5. "Gifts of the Cow".
These are the *Pahchagavya, the "five gifts of
the Cow" (namely: milk, curds, dung, ghi and
urine), which make up the sacred drink gavya,
used by certain samnyasin ascetics for its sup-
posedly curative and purifying properties [see
Frederic (1994)]. See Five.
GAYATRL [S J. Value = 24. In expressive
Sanskrit poetry, this is a stanza composed of
three times eight syllables. See Indian metric.
GEOMETRY. See Kshetraganita and Indian
mathematics (The history of).
GHANA. “Cube”. Sanskrit term used in arith-
metic and algebra to denote the operation of
cubing a number.
GHANA. Word used in algebra to denote
“cube", in allusion to the third degree of
equations of this order. See Varga, Varga-
Varga and Ydvattai at
GHANAMULA. Sanskrit term used in arith-
metic and algebra to denote the operation of
the extraction of the cubic root.
GHUBAR NUMERALS. Signs derived from
‘Brahmi numerals, through the intermediary
ofShunga, Shaka, Kushana, Andhra, Gupta
and Nagari numerals. Formerly used by the
Arabic mathematicians of North Africa (for
calculations carried out on the “dust” abacus).
The corresponding system did not always pos-
sess zero. See Indian written numeral
systems (Classification of). See Fig. 24.52
and 24.61 to 69.
GIFTS OF THE COW. [S[. Value = 5. See
Gavya and Five.
GIRL [S] . Value = 7. “Mountain, hill”. See Adri
and Seven.
GO. [S]. Value = 1. “Cow”, “Earth". This is the
name of the sacred cow worshipped by the
Hindus. This cow is said to have been created
by *Brahma on the first day of the month of
Vaishakha (April-May). The word forms part
of the composition of the name Govinda
(“Cowherd") attributed to *Vishnu as “Saviour
of the earth”. This is also an allusion to the fact
that the earth ( *Prithivi ) is often symbolically
associated with a cow named Prishni. This rela-
tionship (which also explains the veneration of
the cow in Hindu religion) stems from the fact
that the cow, like the earth, gives life [see
Frederic (1994)). See One.
GO. [S] . Value = 9. “Cow, Earth”. Another
meaning of this word is “radiance”, and by
extension “star”. This is why the word became
synonymous with *graha, “planets” (in the
sense of *navagraha, the “nine planets of the
Hindu cosmological system”). Thus Go = 9.
See Nine.
GOAL (The three). See Trivarga.
GOAL (The four). See Chaturvarga.
GOD OF CARNAL LOVE. [S], Value = 13. See
Kama and Thirteen.
GOD OF COSMIC DESIRE. [S], Value = 13.
See Kama and Thirteen.
GOD OF SACRIFICIAL FIRES. [S], Value = 3.
See Agni and Three.
GOD OF WATER AND OCEANS. See Varuna.
GODS. [Sj. Value = 33. See Deva and Thirty-
three.
GOOGOL. This term is of English origin. It
was invented by the American mathematician
Edward Kastner in the 1940s. It denotes the
number ten to the power 100. This number,
which no longer represents anything palpable,
surpasses all that is possible to count or mea-
sure in the physical world. See Infinity and
High numbers.
GOVINDASVAMIN. Indian astronomer
c. 830 CE. Notably, his works include
Bhdskariyabhasya. in which there are many
examples of the use of the place-value system
using Sanskrit numerical symbols [see Billard
(1971), p. 8], See Numerical symbols, and
Numeration of numerical symbols.
GRAHA. [S], Value = 9, “Planet”. This alludes
to the *navagrahas, the “nine planets" of the
Hindu cosmological system (namely: *Surya,
the Sun; *Chandra, the Moon; Angdraka,
Mars; Budha, Mercury; Brihaspati, Jupiter;
Shukra, Venus; Shani, Saturn; and the two
demons of the eclipses *Rahu and Ketu. See
Paksha and Nine.
GRAHA. “Planet”. See previous entry,
Saptagraha and Navagraha.
GRAHACHARAN1BANDHANA. See
Haridatta.
GRAHADHARA. “Axis of the planets”. Name
given to the Pole star. See Dhruva and Sudrishti.
GRAHAGANITA. Name given to astronomy
by Brahmagupta (628 CE). Literally: “calcula-
tion of the planets", and, by extension,
“mathematics of the stars”. See Indian astron-
omy (The history of) and Ganita.
GRAHAPATI. “Master of the planets". Name
sometimes given to *Surya, the Sun-god. See
Graha.
GRAHARAJA. “King of the planets". Name
sometimes given to *Surya, the Sun-god. See
Graha.
GRANTHA NUMERALS. Symbols derived
from *Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi and Bhattiprolu
numerals. Formerly used by the Dravidian peo-
ples of Kerala and Tamil Nadu. The symbols
corresponded to a mathematical system that
was not based on place-values and therefore
did not possess a zero. See: Indian written
numeral systems (Classification of). See also
Fig. 24.52 and 24.61 to 69.
GREAT ANCESTOR. [S[. Value = 1. See
Pitdmaha and One.
GREAT ELEMENT. See Mahdbhuta. Value
= 5.
GREAT GOD. [S]. Value = 11. See Mahddeva.
Eleven and Rudra-Shiva.
GREAT KINGS (The four). See
Chaturmaharaja.
GREAT SACRIFICE. [S], Value = 5. See
Mahayajha and Five.
GREAT SIN. [SI- Value = 5. See Mahapapa and
Five.
GREAT YEAR OF BEROSSUS. Cosmic period
mentioned in the w'ork of the Babylonian
astronomer Berossus (fourth - third century
BCE), 432,000 years long. There is an “arith-
metical" relationship between this “Great year”
and the Indian cosmic cycles called *yugas,
because it corresponds: to a *kaliyuga, to 1/10
of a *mahdyuga, and to 2/5 of a *yugapada.
However, it is not known if there is a historical
link between this “year” and the Indian *yugas.
See Great year of Heraclitus and Yaga
(Astronomical speculations).
GREAT YEAR OF HERACLITUS. Cosmic
period of the ancient Mediterranean world
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
460
which, according to Censorinus, is 10,800 years
long. There is a mathematical relationship
between this “Great year" and the Indian
cosmic cycles known as *yugas, because it cor-
responds: to 1/40 of a *kaliyuga, to 1/100 of a
*yugapada and to 1/400 of a *mahayuga.
However, it is not known if there is a historical
link between this “year” and the Indian *yugas.
See Great year of Berossus.
GUARDIAN OF THE HORIZONS. [SJ. Value
= 8. See Lokapdla and Eight.
GUARDIAN OF THE POINTS OF THE COM-
PASS. [S] . Value = 8. See Lokapdla, Dikpala
and Eight.
GUJARATI NUMERALS. Signs derived from
‘Brahmi numerals, through the intermediary
ofShunga, Shaka, Kushana, Andhra, Gupta,
Nagari and Kutila numerals. Currently in use in
Gujarat State, on the Indian Ocean, between
Bombay and the border of Pakistan. The corre-
sponding system functions according to the
place-value system and possesses zero (in the
form of a little circle). See Indian written
numeral systems (Classification of). See also
Fig. 24.8, 52 and 24.61 to 69.
GULPHA. [S]. Value = 2. “Ankle”. This symbol-
ism is due to the symmetry of this part of the
body. See Two.
GUNA. [S]. Value = 3. “Merit”, “Quality", “pri-
mordial property”. Philosophically, the gunas
are the qualities or conditions of existence
which make up Nature. They are in a state of
rest when the qualities are in perfect equilib-
rium, and in a state of evolution when one or
more of them prevail over the others. According
to the philosophy of the *Samkhya, these quali-
ties are composed of three natural “materials”:
Sattva (representing kindness, the pure essence
of things). Rajas (active energy, passion), and
Tamas (passivity, apathy). Here the word is syn-
onymous with Triguna, “three qualities”, “three
primordial properties" [see Frederic,
Dictionnaire (1987)]. See Triguna and Three.
GUNA. IS]. Value = 6. “Merit”, “quality”, “pri-
mordial property”. The allusion here is to
*shadayatana, the “six^wwrts” of Buddhist phi-
losophy. This value was only acquired relatively
recently. See Shadayatana and Six.
GUNANA. Term used in arithmetic to mean
multiplication. Other synonyms: banana ,
vadha, kshaya, etc. (which literally mean:
“destroy”, “kill”, etc., in allusion to the succes-
sive erasing of the results of the partial
products whilst carrying out calculations on
sand or using chalk on a board). See
Calculation, Patiganita, and Indian methods
of calculation. See also Chapter 25.
GUNDHIKA. Name given to the number ten
to the power twenty-three. See Names of num-
bers and High numbers.
Source: * /. alitavistara Sutra (before 308 CF.).
GUPTA (Calendar). A calendar (with normal
years) established by Chandragupta I begin-
ning in 320 CE. To find the date in the
universal calendar which corresponds to one
expressed in Gupta years, add 320 to the Gupta
date. Sometimes the first year of this calendar
is given as 318 or 319. It was used during the
Gupta dynasty. In Central India and Nepal, it
persisted until the thirteenth century. See
Indian calendars.
GUPTA NUMERALS. Signs derived from
‘Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana and Andhra
numerals. Contemporaries of the Gupta
dynasty (inscriptions of Parivrajaka and
Uchchakalpa). The corresponding system does
not use the place-value system or zero. These
numerals were the ancestors of Nagari,
Sharada and Siddham notations. See Fig. 24.38
and 24.70. For notations derived from Gupta
numerals, see Fig. 24.52. For their graphical
evolution, see Fig. 24.61 to 69. See Indian writ-
ten numeral systems (Classification of).
GURKHALI NUMERALS. See Nepali
numerals.
GURUMUKHI NUMERALS. Signs derived
from ‘Brahmi numerals, through the interme-
diary of Shunga, Shaka, Kushana, Andhra.
Gupta and Sharada numerals, and constituting
a sort of mixture ofSindhi and Punjabi numer-
als. Once used by the merchants of Shikarpur
and Sukkur. (These merchants also used Sindhi
or Punjabi numerals, as well as the eastern
Arabic “Hindi” numerals.) The corresponding
system functions according to the place-value
system and possesses zero (in the form of a
little circle). See Fig. 24.7. See also Indian writ-
ten numeral systems (Classification of) and
Fig. 24.52 and 24.61 to 69.
H
HALF OF THE BEYOND. As a representation
of the numbers ten to the power twelve, ten to
the power seventeen and ten to the power eigh-
teen. See Parardha.
HALF OF THE MONTH. [S]. Value = 2. See
Paksha.
HALF OF THE MONTH. [SJ. Value = 15. See
Paksha.
HAND. [S]. Value = 2. See Kara and Two.
HARA. IS]. Value = 11. One of the names of
‘Shiva who is an emanation of ‘Rudra, the
symbolic value of which is eleven. See Rudra-
Shiva and Eleven.
HARANAYANA. [S]. Value = 3. The “eyes of
*Hara”. See Haranetra.
HARANETRA. IS). Value = 3. “Eyes of *Hara”.
‘Shiva, who has a multitude of names and
attributes, one of which is *Hara, often repre-
sented with a third eye in his forehead, which is
meant to symbolise perfect knowledge. From
which: Haranetra = 3. See Three.
HARIBAHU. (SI. Value = 4. “Arms of Hari”.
Mari (literally “he who removes sin”) is one of
the names for *Vishnu, who is always repre-
sented as having four arms.
HARIDATTA. Indian astronomer
c. 850 CE. Notably, his works include
Grahacharanibandhana, in which he tells of
the fruit of his invention: a system of numeri-
cal notation which uses the letters of the
Indian alphabet. This is based on the place-
value system and a zero (always expressed by
one of two letters). This system is called kata-
payddi : the first ever alphabetical positional
number system [see Sarma (1954)]. See
Katapayadi numeration, and Indian
Mathematics (The history of).
HARSHAKALA (Calendar). Calendar begin-
ning in the year 606 CE, created by
Harshavardhana, King of Kanauj and
Thaneshvar. To find the date in the universal
calendar which corresponds to one expressed
in Harshakala years, add 606 to the Harshakala
date. This calendar was only used during the
reign of Harshavardhana and for a short time
afterwards in Nepal. See Indian calendars.
HARYA. “Dividend" (in the mathematical
sense). See Bhdjya.
HASTIN. IS]. Value = 8. “Elephant”. See
Diggaja and Eight.
HEADS OF RAVANA. IS]. Value = 10. See
Ravanashiras and Twenty.
HEADS OF RUDRA. See Rudrdsya.
HEGIRA (Calendar of the). See Hijra.
HELL. Value = 7. See Patala.
HEMADRi. One of the names of ‘Mount
Meru.
HETUHILA. Name given to the number ten to
the power thirty-one. See Names of numbers
and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
HETVINDRIYA. Name given to the number
ten to the power thirty-five. See Names of
numbers and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
HIGH NUMBERS. Early in Indian civilisation,
there was a sort of “craze” for high numbers.
‘Sanskrit numeration lent itself admirably to
the expression of high numbers because it pos-
sessed a specific name for each power of ten.
There are numerous examples to be found, not
only in works on mathematics, but also in
those concerning astronomy, cosmology,
grammar, religion, legends and mythology.
This proves that these names were not in
everyday use in India, but rather they were
familiar in learned circles, at least as early as
the beginning of the Common Era. See Names
ofnumbers.
In the naming of high numbers, these texts
generally reached the highest numbers that
were used in calculations. Thus each of the
ascending powers of ten up to a ‘billion (ten to
the power twelve), or even up to ‘quadrillion
(ten to the power 18) were named. In cosmo-
logical texts, however (especially those
developed by members of the religious cult of
‘Jaina, such as the Anuyogadvdra Sutra), this
limit was pushed much further, bearing wit-
ness to the extraordinary fertility of Indian
imagination. The Jainas attempted to define
their vision of an eternal and infinite universe;
thus they undertook impressive arithmetical
speculations, which always involve extremely
high numbers, equal to or higher than num-
bers such as ten to the power 190 or ten to the
pow-er 250.
This obsession with high numbers is also
found in *Vydkarana, a famous Pali grammar
of Kachchayana, and in the legend of Buddha,
related in the * Lalitavistara Sutra, which
juggles with numbers as high as ten to the
power 421. At first glance childish, this pas-
sion for high numbers can tell us something
about the high conceptual level achieved early
on by Indian arithmeticians. It led the Indians
not only to expand the limits of the “calcula-
ble”, physical world, but also and above all to
conceive of the notion of infinity, long before
the Western world. See Googol and all other
entries entitled High numbers as well as those
entitled Infinity.
461
HIGH NUMBERS
HIGH NUMBERS. Here is a (non-exhaustive)
alphabetical list of Sanskrit words which repre-
sent high numbers:MZwi> (= 10 17 ), * Ababa
(= 10 77 ), *Abbuda (= 10 56 ), *Abja (= 10 9 ),
* Ababa (= 10 7 °), *Akkhobhini (= 10 42 ), *Akshiti
(= 10 15 ), *Ananta (= 10 13 ), *Anta (= 10 11 ),
*Antya (= 10 12 ), *Antya (= 10 15 ), *Antya
(= 10 16 ), *Arbuda (= 10 7 ), *Arbuda{- 10 s ),
*Arbuda (= 10 10 ), * Asankhyeya (= 10 140 ), *Atata
(= 10 84 ), *Attata (= 10 19 ), *Ayuta (= 10 4 ), *Ayuta
(- 10 9 ), *Bahula (= 10 23 ), *Bindu (= 10 49 ),
*Dashakoti (= 10 8 ), *Dashalaksha (= 10 6 ),
*Dashasahasra (= 10 4 ), * Dhvajagravati (= 10"),
*Dhvajdgranshamani (= 10 145 ), *Gananagati
(= 10 39 ), *Gundhika (= 10 23 ), *Hetuhila (= 10 31 ),
*Hetvindriya (= 10 35 ), *Jaladhi (= 10 14 ),
*Kankara {= 10 13 ), *Karahu (= 10 33 ), *Kathana
(= 10 U9 ), *Khamba (= 10 13 ), *Kharva (= 10*°),
*Kharva (=10 12 ), *Kharva (= 10 39 ), *Koti
(= 10 7 ), * Kotippakoti (= 10 21 ), *Kshiti (= 10 2 °),
*Kshobha (= 10 22 ) *Kshobhya (= 10 17 ), *Kshoni
(= 10 16 ), *Kumud (=10 21 ), *Kumuda (= 10 105 ),
*Lakh (= 10 s ), *Lakkha (= 10 5 ), *Laksha (= 10 s ),
*Madhya (= 10 10 ), * Madhya (= 10 u ), * Madhya
(= 10 15 ), * Madhya (= 10 16 ), *Mahabja (= 10 12 ),
*Mahdkathana (= 10 126 ), *Mahdkharva
(= 10 13 ), *Mahdkshiti (= 10 21 ), *Mahakshobha
(= 10 23 ), *Mahakshoni (= 10 17 ), *Mahapadma
(= 10 12 ), *Mahdpadma (=10 1S ), *Mahapadma
(= 10 34 ), *Mahasaroja (= 10 12 ), *Mahashankha
(= 10 19 ), * Mahavrindd (= 10 22 ), * Mudrabala
(= 10 43 ), *Nagabala (= 10 25 ), *Nahut (= 10 9 ),
*Nahuta (= 10 28 ), *Nikharva (= 10 9 ), *Nikharva
(= 10 u ), *Nikharva (= 10 13 ), *Ninnahut (= JLO u ),
*Ninnahuta (= 10 35 ), *Nirabbuda (= 10 63 ),
*Niravadya (= 10 41 ), *Niyuta (= 10 5 ), *Niyuta
(= 10 6 ), *Niyuta (= 10 u ), *Nyarbuda (10 s ),
*Nyarbuda (= 10 u ), *Padma (= 10 9 ), *Padma
(= 10 14 ), *Padma (= 10 29 ), *Paduma (= 10 29 ),
*Paduma (= 10 u9 ), *Pakoti (= 10 14 ), *Pardrdha
(= 10 12 ), *Parardha {= 10 17 ), *Paravdra (= 10 14 ),
*Prayuta (= 10 s ), *Prayuta (= 10 6 ), *Pundarika
(= 10 27 ), *Pundarika (= 10 u2 ), *Salila (= 10 u ),
* Samaptalambha (= 10 37 ), *Samudra (= 10 9 ),
*Samudra (= 10 10 ), *Samudra (= 10 14 ),
*Saritapati (= 10 14 ). *Saroja (= 10 9 ), *Sarvabala
(= 10 45 ), *Sarvajna (= 10 49 ), *Shankha (= 10 12 ),
* Shankha (= 10 13 ), *Shankha (= 10 1B ), *Shanku
(= 10 13 ), *Shatakoti (= 10 9 ), *Sogandhika
(= 10 9i ), *Tallakshana (= 10 53 ), *Titilambha
(= 10 27 ), *Uppala (= 10 98 ), * Utpala (= 10 25 ),
*Utsanga (= 10 21 ), * Vadava (= 10 9 ), *Vadava
(= 10 14 ), *Vibhutangama (= 10 5i ), *Visamjhagati
(= 10 47 ), * Viskhamba (= 10 15 ), *Vivaha (= 10 19 ),
*Vivara (= 10 15 ), *Vrinda (= 10 9 ), *Vrinda (=
10 17 ), *Vyarbuda (= 10 8 ), * Vyavasthanaprajha-
pati (= 1029).
HIGH NUMBERS (SYMBOLIC MEANING
OF). The preceding list is enough to give the
reader some idea of the arithmetical genius of
the Indian scholars. However, it only gives the
mathematical value of the words in question,
and neither their literal nor their symbolic
meaning. The following (summary) explana-
tions should give the reader a precise idea of
the associations of ideas and the symbolism
which is implied in this unique terminology.
Firstly, the word *padma (which represents
the number ten to the power nine, ten to the
power fourteen or ten to the power twenty-
nine) literally means “*lotus”. However, there
is another word *paduma (which can represent
ten to the power twenty-nine as well as ten to
the power 119), as well as the terms * utpala
(ten to the power twenty-five), *uppala (ten to
the power ninety-eight), *pundarika (ten to the
power twenty-seven or ten to the power 112),
*kumud (ten to the power twenty-one) and
*kumuda (ten to the power 105), which also
mean “lotus”. The reasoning behind this
metaphor lies in the fact that the lotus flower is
the best-known and most widespread symbol
in the whole of Asia. “Bom of miry waters, this
flower maintains a flat and immaculate purity
above the water in all its splendour. Thus it
became the symbol of a pure spirit leaving the
impure matter of the body. Nearly all Indian
philosophies and religions adopted the flower
as their symbol, and its diffusion throughout
Asia took place due to the spread of Buddhism,
even though it is almost certain that the lotus
flower was already used as a symbol by many
peoples before the advent of Buddhism. Indian
philosophers saw it as the very image of divin-
ity, which remains intact and is never soiled by
the troubled waters of this world. The closed
flower of the lotus, in the shape of an egg, rep-
resents the seed of creation which rose out of
the primordial waters, and as it opens all the
latent possibilities contained within the seed
develop in the light. This is why, in Hindu
iconography, * Brahma is seen to be born from
a lotus flower growing out of the navel of
‘Vishnu who lies upon the serpent *Ananta
who is coiled up on the primordial waters
which represent infinity [see Fig. D. 1, p. 446,
of the entry entitled Ananta]. Similarly, this
flower is the ‘throne’ of Buddha and most of his
manifestations: here the lotus represents the
bodhi, the ‘nature of Buddha’ which remains
pure when it leaves the *samsara, the cycle of
rebirth of this world. A whole symbolic system
developed around the lotus flower, which takes
into account its colour, the number of petals it
possesses, and whether it is open, half-closed
or in bud. In the Kundalini Yoga, it is the stem
of the lotus which forms the spiritual axis of
the world and upon which the iotus flowers
become steadily more fully open and the
number of petals becomes greater and greater
up until the highest illumination where the
corolla, which has become divine and of
unequalled brilliance, possesses a thousand
petals,” [see Frederic, Le Lotus, (1987)].
Indian art also seized upon this flower and it
has been widely represented in painting as well
as in sculpture. We can appreciate why Indian
mathematicians, with their perfect command of
symbolism, also adopted the lotus flower and all
its corresponding mysticism in order to express
in Sanskrit gigantic quantities. The padma (or
paduma ), is the pink lotus. As well as the purity
which it represents, this flower, to the Indian
mind, symbolises the highest divinity, as well as
innate reason. Written Padma (with a long a),
the pink lotus flower figures amongst the names
of Lakshml as feminine energy ( *shakti ) of
*Vishnu. In the word *sahasrapadma (the
“thousand-petalled lotus”), it represents the
“third eye”, that of perfect Knowledge; it also
represents the superior illumination and the
divine corolla, of unequalled brilliance, flower-
ing on the axis of the spirituality of the world as
a thousand-petalled pink lotus [see Frederic, Le
Lotus {mi).
It is probably the idea of absolute and
divine perfection which gave padma a value as
elevated as ten to the power 119. However, it
did not initially represent such a quantity.
Initially, as the Indian mathematicians were
gradually becoming more accustomed to deal-
ing with large quantities, and with the idea of
perfection and absolution, they probably gave
the lotus the value often to the power nine. Its
value gradually increased as it was successively
attributed the values of ten to the power four-
teen, then ten to the power twenty-nine and
finally ten to the power 119. The flower in ques-
tion here possesses a thousand petals.
Padma is also the name of one of the
branches of the Ganges at its mouth. It is inter-
esting to note that this swampy delta with
branches radiating from it, like the petals of a
lotus flower, is often referred to as *jahnavivak-
tra, literally the “mouths of Jahnavi (the
Ganges)". The name, as an ordinary numerical
symbol, corresponds to the number one thou-
sand, precisely because of the hundreds of
branches which characterise it. Moreover
*Vishnu is associated with the thousand-
petalled lotus and has a thousand attributes,
amongst which are: *Sahasranama, “the thou-
sand names (of Vishnu)”. What is more, in
Hindu mythology, it is from the “feet of
*Vishnu” (*Vishnupada) that the celestial
Ganga (the Ganges) sprang, considered to be
the “divine mother of India”. Thus this flower
was associated with both the name and the
concept of thousand {*sahasra). However, the
terminology which was used had recourse to a
secondary symbolism: ‘thousand was no
longer really a numerical concept; its figurative
sense was the idea of plurality, of “vast
number”. Like the word padma which initially
only meant ten to the power nine, the name of
this vast number grew to have the value of ten
to the power twelve; which then gave ten to the
power fifteen the name *mahapadma, which
means “great lotus”. Through a similar associa-
tion of ideas, the word * shankha, which means
“sea conch”, was assigned to the numbers ten
to the power thirteen and ten to the power
eighteen. This symbolises certain Buddhist or
Hindu divinities (such as ‘Vishnu or ‘Varuna
for example). In India, the conch represents
riches, good fortune and beauty. This can be
associated with the image of a diamond which
is pure and beautiful in equal measures. As the
diamond is everlasting and shines with a thou-
sand fires, the beauty represented by the conch
can be compared to this precious stone. Thus,
for some Indian arithmeticians, *shanku (“dia-
mond”) is equal to ten to the power thirteen.
Returning to the conch, one of the attrib-
utes of Vishnu is expressed by the Sanskrit
word for conch {shankha), which symbolises
the conservative principle of the revelation,
due to the fact that the sound and the pearl are
conserved within the shell. The conch is also
the symbol of abundance, fertility and fecun-
dity, which are precisely the characteristics of
the sea from which the shell comes. The shell is
also related to the water. This explains the con-
nection with ‘Varuna, the lord of the Waters.
Here there is also the connection with the
lotus, which also symbolises not only abun-
dance, but also and above all, in the eyes of
humans, a limitless expanse. This is why the
word *samudra, which means “ocean”, came to
mean, in this symbolic terminology, the
number ten to the power nine, ten to the power
ten or ten to the power fourteen. This is the
reason why ‘Bhaskaracharya used the word
*jaladhi, which also means ‘ocean, to denote
the number ten to the power fourteen. The
mathematician must have chosen this word
because he was writing in verse and in Sanskrit
and he chose his words in order to achieve the
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
462
desired effect, using an almost limitless choice
of synonyms, following the exacting rules of
Sanskrit versification. See Poetry and writing
of numbers.
The Indians also see *samudra as the waves
of superior consciousness which bring immor-
tality to mere mortals; eternal existence and
infinity. This explains the connection with
*soma, which is the *amrita, the “drink of
immortality”. Soma can also mean *moon,
which became a metaphor for a goblet full of
the heady brew. Thus it was quite natural that
this star should also be associated with incalcu-
lably vast quantity. So *abja ("moon”) and
*mahabja (“great moon”), came to represent
numbers such as ten to the power nine or a bil-
lion (ten to the power twelve). As well as being
connected to water, the conch is symbolically
related to the moon, as it is white, the colour of
the full moon. This gives double justification to
this association of ideas. The apparently limit-
less expanse of the sea is the most immense
thing in the “terrestial world”. As the *earth is
called *kshiti or *kshoni, (also referred to as
*mahdkshiti and *mahakshoni, meaning “great
earth”) w r e can see how these words came to
represent such immense values as ten to the
power sixteen, ten to the power seventeen, ten
to the power twenty or ten to the power
twenty-one.
The Sanskrit word *abhabagamana means
“the unachievable”. The term *ababa is used in
the * Lalitavistara Sutra (before 308 CE), to
express the quantity ten to the power seventy,
and it is possible that this is an abbreviation of
*abhabdgamana. The word *ahaha, used in the
same text to express ten to the power seventy-
seven, is almost definitely an abbreviation of
the word abaharaka, which means extravagant
and is similar to our word “abracadabra”.
*Pundarika means white lotus with eight
petals and is the symbol of spiritual and
mental perfection. The term is generally
reserved for esoteric divinities, and was dedi-
cated to Shikhin. the second Buddha of the
past. This lotus has the same number of petals
as the eight directions of space, the eight points
of the compass and the eight elephants ( *dig -
gaja ) of Hindu cosmogony. Amongst these
elephants figures *Pundarika who guards the
“southeastern horizon” of the universe for the
god of fire *Agni. The “southwestern horizon”
is guarded by the elephant *Kumuda , whose
real name also means “lotus”, but this time
refers to the white-pink flower. The sun is not
far from this lotus, as it is situated at the axis of
the eight horizons. The elephant Kumuda also
symbolises the Sun-god *Surya, who is often
denoted by names which evoke the idea of a
thousand or the lotus flower: *Sahasramshu
(“Thousand of the Shining", in allusion to its
rays), *Sahasrakirana (“Thousand rays”),
*Sahasrabhuja (“Thousand arms") and
Padmapani (“Lotus carrier”). Thus the Indians
expressed the omniscience of this god and his
incalculable powers.
If the sun is a source of light, warmth and
life, then like the petals of a lotus, its rays must
also contain the spiritual influences received by
all things on earth. This is why the names of
*Pundarika, *Kumud and * Kumuda came to rep-
resent such vast quantities as ten to the power
twenty-one, ten to the power 105 and ten to the
power 112. Indian mathematicians soon took
the step from the Sun to the canopy of
heaven. * Parardha, one of the names attributed
to ten to the power twelve or ten to the power
seventeen, comes from para, “beyond”, and
from ardha, “half of beyond”. Due to a similar
association of ideas, * Madhya, “middle” (repre-
senting the “middle of the beyond”) was used to
express such numbers as ten to the power ten,
ten to the power eleven, ten to the power fifteen
or ten to the power sixteen.
According to the Indians, parardha is the
spiritual half of the path which leads to death.
It is the same as devaydna, the “path of the
gods”, which, according to the * Vedas, was one
of the two possibilities offered to human souls
after death, parardha leading to deliverance
from *samsara or cycle of reincarnation. On
reaching the sky, one cannot fail to achieve
divine transcendence, power, durability and
sanctity, thus touching upon the incalculable
in terms of mental and physical perfection,
represented by the word *pundarika.
Intelligence, wisdom and the triumph of the
mind over the senses is represented by *utpala,
the blue, half open lotus. This is why these
words came to be worth such quantities as ten
to the power twenty-five, ten to the power
twenty-seven, ten to the power ninety-eight or
even ten to the power 112.
No living being can attain divine transcen-
dence, which is conveyed by the Inaccessible,
the Absolute, and the Ineffable. This is similar
to the “incalculable”, * Asamkhyeya (or
* Asankhye}>a), “that which cannot be counted”.
According to the * Lalitavistara Sutra, this word
means “the sum of all the drops of rain which,
in ten thousand years, would fall each day on
all the worlds”. In other words, this is the
“highest number imaginable". * Asankhyeya is
the term used to express the number ten to the
power 140. The terminology used here deals
symbolically with the notion of eternity. This is
explained by al-Biruni in the thirty-third chap-
ter of his work on India, where he gives this
word the value often to the power seventeen
(Fig. 24. 81):
“The name of the eighteenth order is
* parardha, which means half of the sky, or
more precisely, half of what is above. The
reason for this is that if a period of time is made
up of *kalpas (cycles of 4,320,000,000 years), a
unit of this order is a day of the purusha (= one
day of the Supreme Being, namely *Brahma).
As there is nothing beyond the sky, this is the
largest body. Half of the biggest nychthemer (=
the longest possible day) is similar to the other;
in doubling it, we obtain a “night" with a “day",
and thus complete the biggest nychtemer. It is
certain that parardha is from para, which
means the whole sky”. Ref: Kitab fi tahqiq i ma
li’l hind (1030 CE).
Al-Biruni also tells that “according to some,
the day of purusha (the day of Brahma) is made
up of a parardha and a kalpa ”. As a kalpa is
4.320.000. 000.000 years, this “day” corresponds
to: 432,000,000,000,000,000,000,000,000 (four
hundred and thiry-two sextillion) years. See
Day of Brahma.
Traditional brahmanic cosmogony more
modestly attributes the length of
4.320.000. 000 human years to the “*day of
Brahma”. This is also what it refers to as
*asankhyeya , the “incalculable”. The Bhagavad
Gita assigns 311,040,000,000,000 years to this
word. In a commentary on the text is written:
"This formidable longevity, to us infinite, only
represents a zero in the tide of eternity.” The
word Padmabhuta, “born from the lotus (with a
thousand petals)" is an attribute of *Brahma.
Brahma is said to have been born from the
lotus w'hich grew out of Vishnu’s navel as he lay
on the serpent *Ananta floating on the primor-
dial waters (see Fig. D. 1 and Ananta). Another
attribute of Brahma is Padmandbha, w'hich
means “the one w'hose navel is the lotus
(Vishnu)”. This is why the word ananta, which
means “infinity” and “eternity”, has sometimes
been used to express the number ten to the
pow'er thirteen, in memory of distant times
when the Sanskrit names for numbers went no
further than ten to the power twelve. See Antva
(first entry, note in the reference).
Ananta is another name for *Shesha, the
king of the *ndga who resides in the lower
regions of the *Pdtdla. It is an immense serpent
with a thousand heads, who serves as a seat for
*Vishnu as he rests amongst creatures between
two periods of creation. At the end of each
*kalpa, he spits the fire which destroys cre-
ation. Considered as the “Remainder”
( *Shesha ), the “Vestige” of destroyed universes
and as the seed of all future creations, he repre-
sents immensity and space, and infinity and
eternity all at once. See Serpent (Symbolism
of the).
The words eternal and infinity mean “that
which has no end, that which never ends, that
which can never be reached”. This leads to
ideas of absoluteness and totality, in the
strongest sense of the terms. Words such as
*Sarvabala and *Sarvajna, formed with the
Sanskrit adjective sarva, meaning “everything”
or “totality”, have respectively been associated
with numbers as high as ten to the power forty-
five or ten to the power forty-nine. Moreover,
Sarvajnata expresses omniscience in
Buddhism, the knowledge of Buddha, one of
his fundamental attributes. In the Buddhism of
Mahayana, this word has even acquired the
meaning “the knowledge of all the *dharmas
and of their true nature”; a nature which, in
essence, is *shunyatd, vacuity. According to the
Indians, vacuity materialises in the centre of
the *vajra, the “diamond", symbol of what
remains once appearances have disappeared.
The vajra is also the projectile “of a thousand
points”, reputed to never miss its mark, and
made of bronze by Tvashtri for *lndra; but this
is above all a religious instrument, symbol of
the linga and divine power, indicating the sta-
bility and resoluteness of mind as well as its
indestructible character. And as vacuity also
means the void for Indians (also caused as
much by nothingness, absence or insignifi-
cance as by the unthought, immateriality,
insubstantiality and non-being), this explains
why the *bindu, the “point” (destined to
become a numerical symbol and a graphical
representation for zero), represented, for
Indian arithmetician-grammarians, a number
as high as ten to the power forty-nine.
Before the discovery of infinity or zero,
the bindu (the “point”), was the Indian
symbol for the universe in its non-manifest
form, thus that of a universe before its trans-
formation into riipadhdtu, the “world of
appearances”. For Indian scholars, “nothing”
could be united with “everything”, even
before mathematics made these two concepts
inverse notions of one another. See Zero,
Infinity and Indian Mathematics (The his-
tory of)- See Names of numbers, Numerical
symbols, Arithmetical speculations,
Cosmogonical speculations, Sheshashirsha,
463
HIJRA
Shunya, Shunyata, Indian atomism. See also
Serpent (Symbolism of the).
HIJRA (Calendar). Arabic name for the
Islamic calendar, which, according to tradition,
begins on the 16 July 622 CE, day of the
“Escape” or "Flight” (hijra, “Hegira”) of the
prophet Mohammed of Mecca to Medina. As
the Muslim year has twelve lunar months each
twenty-nine or thirty days long, making up a
year of 354 days, this calendar must be recti-
fied by the addition of eleven intercalary years
of 355 days every thirty years to catch up with
normal solar years. To obtain a date in the uni-
versal calendar from one in the Hegira,
multiply the latter by 0.97 and add 625.5 to the
result. For example: the start of the year of the
Hegira 1130 corresponds to July 1677:
1130 Hegira = (1130 X 0.97) + 625.5 =
1721.6. Inversely, to find a date of the Hegira
from a date in the universal calendar, subtract
625.5 from the latter, then multiply the result
by 1 .0307 and add 0.46. If there are decimals
remaining, add a unit.
For example, to convert the year 1982 into
the Hegira calendar, proceed as follows: 1st
stage: 1982-625.5 = 1356.5.
2nd stage: 1356.5 X 1.0307 = 1398.14.
3rd stage: 1398.14 + 0.46 = 1398.6.
In rounding off this result, the year of the
Hegira 1399 is obtained [see Frederic,
Dictionnaire (1987)]. See Indian calendars.
HINDI NUMERALS. See Eastern Arabic
numerals.
HINDU RELIGION. See Indian religions and
philosophies.
HINDUISM. See Indian religions and
philosophies.
HOLE. [S], Value = 0. See Randhra.
HOLE. [S], Value = 9. See Chhidra, Randhra
and Nine.
HORIZON. [S], Value = 4. See Dish and Four.
HORIZON. [S). Value = 8. See Dish and Eight.
HORIZON. [S], Value = 10. See Dish and Ten.
HORSE. [S], Value = 7. See Ashva and Seven.
HORSEMEN. [SJ. Value = 2. See Ashvin and
Two.
HOTRI. [S], Value = 3. “Fire”. See Agni and
Three.
HUMAN. [S], Value = 14. In the sense of
progenitor of the human race. See Manu
and Fourteen.
HUNDRED. Ordinary Sanskrit name: *shata.
Corresponding numerical symbols: Abjadala,
*Dhdrtardshtra, Purushayus and Shakrayajha.
See Numerical Symbols.
HUNDRED BILLION (= ten to the power
fourteen). See Jaladhi, Padma, Pakoti,
Paravara, Samudra, Saritapali, Vadava. See
also Names of numbers.
HUNDRED MILLION (= ten to the power
eight). See Arbuda, Dashakoti, Nyarbuda,
Vyarbuda. See also Names of numbers.
HUNDRED THOUSAND BILLION (UK) (=
ten to the power seventeen). See Abab.
Kshobhya, Mahakshoni, Parardha, Vrinda. See
also Names of numbers.
HUNDRED THOUSAND MILLION (UK) (=
ten to the power eleven). See Anta, Madhya,
Nikharva, Ninnahut, Niyuta, Nyarbuda, Salila.
See also Names of numbers.
HUNDRED THOUSAND TRILLION (UK)
(ten to the power twenty-three). See Bahula,
Gundhika, Mahakshobha. See also Names
of numbers.
HUNDRED TRILLION (UK) (= ten to the
power twenty). See Kshiti. See also Names
of numbers.
HUTASHANA. [S], Value = 3. “Fire”. See Agni
and Three.
I
IMMENSE. [SJ. Value = 1. See Prithivi
and One.
IMPURITIES (The five). See Pahchaklesha.
IMRAjt (Calendar). See Kristabda.
INACCESSIBLE. [S]. Value = 9. See Durgd
and Nine.
INCARNATION. [SJ. Value = 10. See Avatdra
and Ten.
INDESTRUCTIBLE. [SI. Value = 1. See
Akshara and One.
INDETERMINATE EQUATION (Analysis of
an). See Kuttaganita and Indian mathematics
(History of).
INDETERMINATE. See Infinity.
INDIA (States of the present-day Indian
union). See Fig. 24. 27.
INDIAN ARITHMETIC. Alphabetical list
of words related to this discipline which
appear as entries in this dictionary:* Arithmetic.
*Bhdgahara, *Bhajya, *Bhinna, *Chhedana,
*Dashaguna, * Dashagunasamjna, *Dhulikarma,
*Ekadashardshika, *Ganana, *Ganita, *Ghana,
*Ghanamula, *Gunana, *High numbers, *Indian
mathematics (history of), ‘Indian methods of cal-
culation, ‘Infinity, *Kalasavarnana, *Labhda,
*Mudra, *Names of numbers, *Navarashika,
*Pahcharashika, *Pancha jati, *Parikarma,
*Pati, *Patiganita, *Rashi, *Rashividyd,
*Samkalita, *Samkhydna, *Samkhyeya,
*Saptarashika, *Sarvadhana, *Shatottaraganana,
*Shatottaraguna, *Shatottarasamjhd, *Shesha,
‘Square roots (How Aryabhata calculated
his), *Trairashika, * Varga, *Vargamula,
*Vyastrairdshika, * Vyavakalita.
INDIAN ASTROLOGY. This discipline used
to go by the name of *}yotisha, which literally
means the “science of the stars”. But today
this term is more commonly used to describe
astronomy. This naming dates back to the
time when astronomy was not yet considered
to be a separate discipline from arithmetic
and calculation.
Until early in the Common Era, astrology
was often confused with astronomy, the latter
at that time having no other objective than to
serve the former. Knowledge of the stars and
their movements was a method of predicting
the future and determining the favourable
dates and times of any given human action:
consecration of a ritual sacrifice, commercial
transactions, setting out on a voyage, etc. [see
Frederic (1994)]. Thus, when an individual
was born, the astrologers, having determined
the exact time and the position of the planets
and the sun, established the horoscope of the
newly-born infant, which was considered
indispensable in ascertaining the child’s birth
chart.
‘Varahamihira stands out as one of the
most famous astrologers of Indian history. He
lived in the sixth century CE, and is known
principally for his work, Panchasiddhdntika
(the “Five * Astronomical Canons), which is
dated c.575 CE. But he also wrote many works
on astrology, divination and practical knowl-
edge. The most important of these is
Brihatsamhitd (the “great compilation”) which
covers many subjects: descriptions of heavenly
bodies, their movements and conjunctions,
meteorological phenomena, indications of the
omens these movements, conjunctions and
phenomena represent, what action to take and
operations to accomplish, signs to look for in
humans, animals, precious stones, etc. [see
Filliozat, in: HGS, I, pp. 167-8]. See Indian
astronomy (The history of), Ganita, Rashi,
Tanu and Yuga.
INDIAN ASTRONOMY (The history of).
Here is an alphabetical list of terms related
to this discipline which appear in this
dictionary: ‘Aryabhata, ‘Astronomical
canon, ‘Bhaskara, *Bhdskardcharya, ‘Bhoja,
*Bija, ‘Brahmagupta, *Chaturyuga, * Ganita,
‘Govindasvamin, ‘Great year of Berossus,
‘Great year of Heraclitus, ‘Haridatta, ‘Indian
astrology, ‘Indianity (fundamental mecha-
nisms of), ‘Indian mathematics (The history
of), *)yotisha, *Kaliyuga, *Kalpa, ‘Kamalakara,
* Karan a, * La 11a, *Mahayuga, *Nakshatra,
*Nakshatravidya, ‘Nilakanthasomayajin,
‘Numerical symbols, ‘Parameshvara, *Put-
umanasomayajin, *Samkhyana, ‘Shank-
aranarayana, ‘Shripati, *Siddhanta, *Tithi,
*Varamihira, *Yuga (definition), *Yuga (systems
of calculating), *Yuga, (astronomical specula-
tion on), * Yugapada.
INDIAN ASTRONOMY (The history of). If we
take the word “astronomy” in its wider sense, we
can traditionally distinguish three principal
periods. The first corresponds to the astronomy
of era and ritual: a lunar-solar era devoid of any
time-scale or era. The corresponding “material”
is characterised by the *nakshatra, division of
the sidereal sphere in twenty-seven or twenty-
eight constellations or asterisms according to
the twenty-seven or twenty-eight days of the
sidereal revolution of the Moon. The planets (it
is unlikely that they had all been discovered at
this time) only played a very small part in div-
ination. Between the third century BCE and the
first century CE, elements and procedures of
Babylonian astronomy appeared in Indian
astronomy. A unity of time appeared called the
*tithi, which is approximately the length of a day
or nychthemer, and corresponds to one thirtieth
of the synodic revolution of the Moon. It was at
this time that the planets came to the fore in div-
ination and were subjected to arithmetical
calculations, based on their synodic revolutions.
However, it was at the beginning of the sixth
century CE that Indian astronomy underwent its
most spectacular developments: scientific astron-
omy was inaugurated by the work of
‘Aryabhata, which dates back to c. 510 CE.
From the outset, this astronomy was based on
an astonishing speculation about the cosmic
cycles called *yugas, of a very different nature
from arithmetical cosmogonic speculations.
This speculation involves astronomical ele-
ments, where the *mahdyuga (or *chaturyuga), a
cycle of 4,320,000 years is presented as the
period at the beginning and the end of which the
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
464
nine elements (the sun, the moon, their apsis
and node, and the planets) should be found in
average perfect conjunction with the starting
point of the longitudes. Thus the length of the
revolutions, which had hitherto been considered
commensurable, were subjected to common
multiplication and general conjunctions.
It is precisely this which makes the specula-
tion so surprising and audacious, because this
fact is totally devoid of any physical value.
These supposed general conjunctions confer
absurd values upon average elements even by
the approximative standards of ancient astron-
omy. However, thanks to a veritable paradox, it
is this strange coupling of speculation and real-
ity that enabled Billard to develop a powerful
method of determining (with precision) hith-
erto unknown facts and a chronology which
had been despaired of due to the unique condi-
tions of the Indian astronomical text. It is
interesting to note that the speculative ele-
ments of this astronomy have been as useful to
contemporary historical science as they were
once harmful. For more details, see: Yuga (cos-
mogonical speculation on), Indian astrology,
Yuga (astronomical speculation about),
Indian mathematics (The history of),
Indianity (Fundamental mechanisms of).
[See Billard (1971)].
INDIAN ATOMISM. See Paramanu,
Paramanu Raja and Paramabindu.
INDIAN CALENDARS. India (which only
adopted the universal calendar in 1947) has
known a great many different eras during the
course of its history. Certain eras, mythical or
local, have existed on a very limited scale.
Others, however, have become so widely used
that they still exist today. The (non-exhaustive)
list in the following entry gives an idea of the vast
number of eras which have been used, and also
allow for a better understanding of the elements
of Chapter 24, where the documentation is often
dated in one of these eras.
INDIAN CALENDARS. An alphabetical list
of terms relating to these eras which can
be found in this dictionary: Bengali San,
*Buddhashakaraja, *Chalukya, *Chhedi,
*Gupta, *Harshakala, *Hijra, *Kaliyuga,
*Kollam, *Kristabda, *Lakshamana, *iaukik-
asamvat, *Maratha, * Nepali, * Parthian,
*Samvat, *Seleucid, *Shaka, * Simhasamvat ,
*Thdkuri, *Vikrama, *Vildyati, *Virasamvat.
[See Cunningham (1913); Frederic,
Dictionnaire, (1987); Renou and Filliozat
(1953)1.
INDIAN COSMOGONIES AND COSMOLO-
GIES. Here is an alphabetical list of terms
relating to these subjects which appear
as entries in this dictionary: Aditya,
*Adri, * Anuyogadvara Sutra, ‘Arithmetical-
cosmogonic speculations, *Ashtadiggaja,
*Avatara, *Bhuvana, *Chaturananavadana,
*Chaturdvipa, *Chaturmahdraja, *Chaturyuga,
*Cosmognic speculations, *Dantin, *Day of
Brahma, *Diggaja, *Dikpala, *Dvaparayuga,
*Dvipa, *Gaja, *Go, *Graha, *Hastin, *High
numbers (the symbolic meaning of). *Indra,
*Jaina, *Kala, *Kaliyuga, *Kalpa, *Krita,
*Kritayuga, *Kunjara, *Lokapala, *Mahakalpa,
*Mahayuga, *Manu, ‘Mount Meru, ‘Ocean,
*Paksha, *Rahu, *Sagara, ‘Serpent (Symbolism
of the), *ShirshapraheIika, *Takshan, *Tretdyuga,
*Tribhuvana, *Triloka, *Vaikuntha, * Vasu,
*Vishnupada, *Vishva, *Vishvadeva, *Yuga.
INDIAN DIVINITIES. Here is an alphabetical
list of terms relating to this theme, wiiich can
be found as entries in this dictionary:Mrf/fytf,
*Agni, *Amara, *Aptya, *Arka, *Ashtamangala,
*Ashtamurti, *Ashva, *Ashvin, * Atman,
*Avatara, *Bhdnu, *Bharga, *Bhava, *Bhuvana,
*Bija, *Brahma, *Brahmasya, ‘Buddha
(Legend of), *Chandra, *Chaturmahdrdja,
*Chaturmukha, *Dashabala, *Dashabhumi,
*Deva, *Dhruva, *Dikpala, *Divdkara, *Durga,
* Dvatrimshadvaralakshana, *Dyumani, *Ekac-
hakra, *Ekadanta, *Ganesha, *Go, ‘High
numbers (Symbolic meaning of), *Hara,
*Haranayana, *Haranetra, *Haribdhu, *Indra,
‘Infinity (Indian mythological representation
of), *Isha, *Ishadrish, *Ishvara, *Jaina, *Kama,
*Karttikeya, *Kdrttikeydsya, *Kaya, *Keshava,
*Krishna, *Kumdrasya, *Kumaravadana,
*LokapaIa, ‘Lotus, *Mahadeva, *Martanda,
*Netra, *Pahchabana, *Pahchanana, *Para-
brahman, *Pavana, *Pindkanayana, *Pitamaha,
*Ravi, *Ravibdna, *Rudra, * Rudra-Shiva,
*Rudrasya, *Sagara, *Sahasramshu, *Sahasra-
kirana, *Sahasraksha, *Sahasranama, *Sbakra,
*Shakti, *Shatarupa, *Shikhin, * Shiva,
*Shukranetra, *Shula, *Shulin, *Sura, *Surya,
*Tapana, *Triambaka, *Tribhuvaneshvara,
*Tripurasundari, *Trishuld, *Trivarna, * Tryak-
shamukha, *Tryambaka, *Turiya, *Vaikuntha,
*Vaishvanara, *Varuna, * Vasu, * Vdyu, *Vrindd,
*Yama.
INDIAN DOCUMENTATION (Pitfalls of).
The purpose of this entry is to warn readers
about texts which have absolutely no historical
worth whatsoever, which contemporary
Indianists - doubtless through bias towards
material or excessive admiration of Indian cul-
ture - have put forward, due to shocking
carelessness and lack of objectivity, in order to
claim that the invention of zero and Indian posi-
tional numeration date back to the most ancient
times. These documents are either fakes, or
works resulting from recent compilations, or
even ancient texts w r hich successive generations
reproduced whilst constantly correcting and
revising them over the course of time.
Amongst these documents figures the man-
uscript of Bakshali, discovered in 1881 in the
village of Bakshali in Gandhara, near Peshawar,
in present-day Pakistan. The author of this
mathematical text is unknown. It is written in
Sanskrit (in verse and prose) and is mainly con-
cerned with algebraic problems. This
document is interesting from the point of view
of the history of Indian numeration because it
contains many examples of numbers written
using the sign zero and the ‘place-value
system, as well as several numerical entries
expressed in ‘numerical symbols. According to
certain historians of mathematics, this manu-
script dates back to “the fourth, or possibly
even the second century CE”. This document
undeniably constitutes an invaluable source of
information about ‘Indian mathematics, but
the manuscript itself, in its present form, could
not possibly be as old as is claimed. The reason
for this is that the numerals, like the characters
used for writing, are written in the Sharada
style, of which we know both the origin and the
date of its first developments. See Indian styles
of writing and Sharada numerals. See also
Fig. 24. 38 and 40A.
To give the second or fourth century CE as
the date of this document would be an evident
contradiction: it would mean that a northern
derivative of Gupta writing had been devel-
oped two or three centuries before Gupta
writing itself appeared. Gupta only began to
evolve into the Sharada style around the ninth
century CE. In other words, the Bakshali manu-
script cannot have been written earlier than the
ninth century CE. However, in the light of cer-
tain characteristic indications, it could not
have been written any later than the twelfth
century CE. Nevertheless, when certain details
are considered, which probably reveal
archaisms of styles, terminologies and mathe-
matical formulations, it seems likely that the
manuscript in its present-day form constitutes
the commentary or the copy of an anterior
mathematical work. See Sharada numerals,
Indian styles of writing and Indian mathe-
matics (The history of).
Other so-called “proofs” put forward to
demonstrate that zero and the place-value
system were discovered well before the
Common Era include the texts of the * Puranas
(particularly Agnipurana, Shivapurana and
Bhavishyapurdna).
In the Puranas, great importance is placed
in decimal numeration. Thus, in Agnipurana,
the eighth text, during an explanation of the
place-value system, it is written that “after the
place of the units, the value of each place
( *sthana ) is ten times that of the preceding
place". Similarly, in the Shivapurana, it is
explained that usually “there are eighteen posi-
tions ( *sthana ) for calculation", the text also
pointing out that “the Sages say that in this
way, the number of places can also be equal to
hundreds". These cosmological-legendary texts
have often been dated from the fourth century
BCE, and some have even been dated as far
back as 2000 years BCE. These dates, however,
are totally unrealistic, because these texts are
from diverse sources and they are the fruit of
constant reworkings carried out within an
interval of time oscillating between the sixth
and the twelfth centuries CE.
In fact, the Puranas only seem to have
become part of traditional Indian writings
after the twelfth century CE. This is a charac-
teristic trait of the Indian mentality which, in
order to give more weight to explanations of
mythology and legends and to support its ten-
dency to sanctify, immortalise and distort
certain elements of knowledge, often invokes
an authority from scripture which assumes
antiquity. See Indianity. Of course these texts
do stem from a relatively ancient source; but
this source, which has accrued diverse rubrics
in quite recent times, has been steadily and
constantly reworked.
Here is a typical passage: ravivare cha sande
cha phalgune chaiva pharvari shashtish cha sik-
sati jneya tad udaharam idrisham
Translation: Namely, for example, that
ravivare (Sunday) means sande, [the month of]
phalguna (February) pharvari, and sixty siksati.
Ref: Bhavishyapurdna (III, Pratisargaparvan, I,
line 37); ed. Shrivenkateshvar, Bombay, 1959,
p. 423) (Personal communication of Billard).
The text refers to a “barbaric language” which is
none other than Old English! This would sug-
gest that the English race already existed four
thousand years ago, and were contemporaries of
the Sumerians. This demonstrates just how far
biased authors can go with their unreliable
dating of documents. The above line (which was
doubtless added at the time of the British domi-
nation of India) is clearly out of context. Thus
we can see the difficulties we are faced with
when dealing with Indian documentation.
465
INDIAN MANUSCRIPTS
[See Datta, in BCMS, XXI, pp. 21ff.; IA,
XVII, pp. 33-48; Datta and Singh (1938); Kaye,
Bakhshali (1924); Smith and Karpinski (1911)].
INDIAN MANUSCRIPTS (First material
of)- See Potiganita and Indian styles of writ-
ing (Material of).
INDIAN MATHEMATICS (The history of).
What we know of this discipline only really
dates back to the beginning of the Common
Era, as no documents written in Vedic times
survive. Of course this does not mean that
Indian mathematical activities only com-
menced at this time. However, vital information
about geometry can be found from this time in
the kalpasutra, a collection of Brahmanic rites
including the Shulvasutra (or "Aphorisms on
lines”), dedicated to the description of the rules
of construction for altars and the measure-
ments of sacrificial altars.
Three versions of these texts exist; these are
called Baudhdyana, Apastamba and Katydyana.
The best known is the Apastamba version, in
which a similar statement to Pythagoras’s theo-
rem can be found as well as some problems
similar to those in Elements by Euclid. Thus, to
build altars of a predetermined size, a square
equal to the sum of the difference of two others
had to be built. The altars, which were con-
structed out of bricks, had to be constructed in
certain dimensions and with a determined
number of bricks, and in certain cases had to
undergo transformations to increase their sur-
face area by a quantity specified in advance.
Some historians think that Indian mathe-
matics of this era only constituted a utilitarian
science. However, there is no evidence to prove
this theory. As documentation currently
stands, only the obtained mathematical results
are known. The concise and essentially techni-
cal style of the texts in question did not leave
room for even a summary description of the
corresponding reasoning and methodology.
During the “classic" era (third to sixth
century CE), there was a veritable renaissance
in all fields of learning, especially in arith-
metic and calculation, which underwent rapid
expansion at this time. Moreover, it is proba-
bly at the beginning of this period that the
impressive Indian speculations on high num-
bers were developed.
In Vedic literature, there is already evi-
dence of skilful handling of quantities as large
as ten to the power seven or ten to the power
ten, the * Vedas mentioning names of numbers
from *eka (= 1) to *arbuda (= ten million). In
the texts *Vdjasaneyi Samhita, *Taittiriya
Samhita and *Kathaka Samhita (written at the
start of the Common Era), there are numbers
as high as *parardha, which, according to con-
temporary values, represents a billion.
Before the third century CE, however, there
was no known text as long as the * Lalitavistara
Sutra. It is a text about the life of the prince
Gautama Siddhartha (founder of Buddhist doc-
trine and thereafter named Buddha), which
tells of how Buddha, whilst still a boy, becomes
master of all sciences. It tells of the evaluation of
the number of grains of sand in a mountain,
which evokes the famous problem of
Archimedes’s Sand-Reckoner. What is significant
is that the speculation goes way beyond the
limits of numbers considered by Greek mathe-
maticians: in one passage, when Buddha arrives
at the number which today we write as "1" fol-
lowed by fifty-three zeros, he adds that the scale
in question is only one counting system, and
that beyond it there are many other counting
systems, and cites all their names without
exception. See Buddha (Legend of), Names of
numbers and High numbers.
When the * Lalitavistara Sutra was written
(before 308 CE), Indian arithmetical specula-
tion had reached and surpassed the number
ten to the power 421! After this time, equally
vast quantities are found in *Jaina cosmologi-
cal texts, which, speculating on the dimensions
of a universe believed to be infinite in terms of
both time and space, easily reach and surpass
numbers equivalent to ten to the power 200.
See Jaina, Shirshaprahelikd, Anuyogadvara
Sutra. This means that, since the third century
CE, the Indian mind had an extraordinary pen-
chant for calculation and handling numbers
which no other civilisation possessed to the
same degree. See Arithmetic, Calculation,
Arithmetical speculations and Arithmetical-
cosmogonical speculations.
In fact, long before the Lalitavistara Sutra
and Jaina speculations, astrological and astro-
nomical considerations had led the Indians to
be deeply interested in mathematics. This took
place between the third century BCE and the
first century CE, under the influence of Greek
astronomers and after the deployment of
India’s cultural, maritime and commercial
activities with the West during this period. This
was the time when astronomical procedures of
Babylonian origin were introduced to Indian
astronomy: a period characterised notably by
the appearance of the unit of time called *tithi,
of similar length to the nychthemer (the “day”)
and consequently corresponding to 1/30 of the
synodic revolution of the moon. It was also at
this time that planets came into divination and
became subjects for arithmetical calculation,
based on their apparent synodic revolutions.
This is how mathematics, which was essentially
applied to religion, came to be used in astron-
omy at the time of the Gupta dynasty.
The beginning of the third period of this
history roughly corresponds to the end of the
“classic” era, around the end of the fifth cen-
tury CE and the beginning of the sixth century,
thus coinciding with the epoch of the *Aryab-
hatiya. The Aryabhatiya is the name of the work
by the mathematician and astronomer *Aryab-
hata, one of the most original, productive and
significant scholars in the history of Indian sci-
ence. This work (written c. 510 CE) is the first
Indian text to display deep knowledge of
astronomy, and is arguably the most advanced
in the history of ancient Indian astronomy,
which at this time developed amazing specula-
tions about cosmic cycles called *yugas. See
Yuga (Astronomical speculations about) and
Indian astronomy (The history of).
This work also deals with trigonometry and
gives a summary of the principal Indian mathe-
matical knowledge at the beginning of the sixth
century: rules for working out square and cubic
roots; rules of measurement; elements and for-
mulas of geometry (triangle, circle, etc.); rules of
arithmetical progressions; etc. Here is another
important particularity of the Aryabhatiya:
whilst Ptolemy’s trigonometry was based princi-
pally on a relationship between the chords of a
circle and the angle at the centre which subtends
each one of them, Aryabhata’s trigonometry
established a relationship of a different nature
between the chord and the arc of the centre,
which is the sine Junction as a trigonomic ratio.
That is one of the fundamental contributions of
Aryabhata. The work also gives a sine table with
the "approximate value” (asanna) of the number
7t (pi): K - 62,832/20,000 - 3. 1416 [see Shukia
and Sarma, (1976) and Sarma (1976)]. See
Aryabhata.
Aryabhata invented a unique numerical
notation, the conception of which required a
perfect knowledge of zero and the place-value
system. He also used a remarkable procedure
for calculating square and cube roots, which
would be impossible to carry out if the envis-
aged numbers w'ere not expressed in written
form, according to the place-value system,
using nine distinct numerals and a unique
tenth sign performing the function of zero. See
Aryabhata (Numerical notations of). Aryab-
hata’s numeration, Patiganita, Indian
methods of calculation and Square roots
(How Aryabhata calculated).
Of course, Aryabhata was not the first to
use zero and the place-value system: the
*Lokavibhaga, or “Parts of the universe”, con-
tains numerous examples of them more than
fifty years before the Aryabhatiya was written: it
is a *Jaina cosmological work, which is very pre-
cisely dated Monday 25 August 458 CE in the
Julian calendar. Moreover, this is the oldest
known Indian text to use zero and the place-
value system, except that the text only uses the
system of *Sanskrit numerical symbols. See
Numerical symbols (Principle of the numera-
tion of). However, bearing in mind the perfect
conception of the examples taken from the
*Lokavibhaga and the concern about vulgarisa-
tion which is clearly expressed, and moreover
taking into consideration the fact that this text
was probably the Sanskrit translation of an ear-
lier document (most likely written in a Jaina
dialect), it seems very likely that these major
discoveries were made in the fourth century CE.
This system had become widespread amongst
the learned in India by the end of the sixth cen-
tury. After the beginning of the seventh
century, it had gone beyond the frontiers of
India into the Indian civilisations of Asia.
As a consequence, calculation and the
science of mathematics made substantial
progress, as did astronomy, the most spec-
tacular developments of which took place
after the start of the sixth century CE. See
Indian astronomy (The history of).
Amongst the many successors of Aryabhata
was Bhaskara I, his faithful disciple and fer-
vent admirer, who wrote a Commentary on
the Aryabhatiya in 629 CE. This gives invalu-
able information about the events which
took place during the century which sepa-
rated him from his preceptor. Moreover, this
work reveals that Bhaskara had fully mas-
tered mathematical operations which
employed the nine numerals and zero using,
for example, the Rule of Three. He dealt with
arithmetical fractions with ease, expressing
them in a very similar manner to our own,
although he did not use the horizontal line
(which was introduced by the Arabic-Muslim
mathematicians). Bhaskhara’s work also
gives many clues about the development of
algebra during that time.
*Brahmagupta was a contemporary of
Bhaskhara. In 628 CE he wrote an astronomical
text called Brahmasphutasiddhanta (“Revised
system of Brahma”). In 664 CE he wrote a
text on astronomical calculation called
Khandakhadyaka. He contradicted some of
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
466
Aryabhata’s accurate ideas about the rotation
of the earth, thus delaying the progress of cer-
tain aspects of astronomy. However, his work
marked progress in fields such as algebra. He is
without a doubt the greatest astronomer and
mathematician of the seventh century CE. He
made frequent use of the place-value system,
and described methods of calculation which
are very similar to our own using nine numer-
als and zero. Amongst his most important
contributions are his systematisation of the sci-
ence of negative numbers, his generalisation of
Hero of Alexandria’s formula for calculating
the area of a quadrilateral, as well as his expla-
nations of general solutions of quadratic
equations. The progress of Indian mathematics
was stimulated by the development of astron-
omy initiated by Aryabhata. Indian
astronomers used all sorts of mathematical
techniques and theories in this discipline. See
Indian astronomy (The history of) and Yuga
(Astronomical speculations on).
Through resolving indeterminate equa-
tions, which depend entirely upon knowing the
properties of whole numbers, the Indians
arrived at discoveries which went far beyond
those of other races of Antiquity or the Middle
Ages, and which modern science only arrived at
through the efforts of Euler [Woepcke (1863)].
Indian algebra never took the decisive step
which would have elevated it to the same level
as modern algebra. Instead of using symbols
such as a, b, x,y, etc., which are independent of
the real quantities that they represent, it never
occurred to the Indian mathematicians to use
symbols other than the first syllables of words
which denoted the operations concerned.
Moreover, the presentation of and solutions to
their various mathematical problems were usu-
ally written in verse and consequently
subjected to the rules of Sanskrit metric. This
explains why their algebraic symbols remained
for so long wrapped up in verbiage which was
subject to diverse interpretations. See Sanskrit
and Poetry and the writing of numbers.
Brahmagupta’s successors included the
‘Jaina mathematician ‘Mahaviracharya c. 850
CE, in Kannara in southern India. He wrote a
work entitled Ganitasarasamgraha: “This work
deals with the teachings of Brahmagupta but
contains both simplifications and additional
information. First he explains the mathemati-
cal terminology that he uses, then he deals with
arithmetical operations, fractions, the Rule of
Three, areas, volumes and in particular calcula-
tions with practical applications. He gives
examples of solutions to problems. Although
like all Indian versified texts it is extremely
condensed, this work, from a pedagogical
point of view, has a significant advantage over
earlier texts” [Filliozat (1957-64)].
Other astronomers or mathematicians who
corrected or significantly improved the work of
their predecessors include ‘Govindasvamin (c.
830), ‘Shankaranarayana (c. 869), *Lalla
(ninth century CE), *Shripati (c. 1039), *Bhoja
(c. 1042), ‘Narayana (c. 1356), *Parameshvara
(c. 1431), ‘Nilakanthasomayajin (c. 1500) and
‘Shridharacharya (date uncertain). After
Aryabhata and Brahmagupta, however, one
of the greatest Indian mathematicians of
the Middle Ages was without a doubt
‘Bhaskaracharya, who is usually referred to as
Bhaskara II, to avoid confusion with Bhaskara
I. Born in 1114, the son of Chudamani
Maheshvar, the astronomer in charge of the
observatory of ‘Ujjain, he finished writing his
Siddhdntashiromani in 1150. This work is
divided into four parts, the first two being
devoted to mathematics and the second two to
astronomy. These are respectively: the *Lildvati
(named after his daughter), in which he
explains the principle rules of arithmetic; the
*Bijaganita ( *bija means “letter” or “symbol”,
and *ganita means “calculation”), which is
about algebra; Grahganita, or “Calculation of
the Planets”; and finally the Goladhyaya, or
“Book of the Spheres”. In the field of astron-
omy, Bhaskaracharya “repeats his predecessors
but criticises them, even Brahmagupta, who he
agrees with most often ... he compares the
gravitational forces of the stars to the winds, in
distinguishing these winds from the atmos-
phere and its deplacements. Mathematically,
he accounts for the movements by a developed
theory of epicycles and eccentrics. One of the
most interesting aspects of his work is that he
analyses the movement of the sun, not only in
considering the difference of the longitudes
from one day to another, but even dividing the
day into several intervals, and considers the
movement in each one of them to be uniform”
(Filliozat in: HGS, I]. The mathematical sec-
tions are mainly the study of linear or
quadratic equations, indeterminate or other-
wise; measurements, arithmetical and
geometrical progressions, irrational numbers,
and many other arithmetical questions of an
algebraic, trigonomic or geometric nature.
Thus we have the names and the principle
contributions of some of the most renowned fig-
ures in the history of Indian science. See Infinity,
Infinity (Indian concepts of) and Infinity
(Indian mythological representation of).
INDIAN METHODS OF CALCULATION.
When they first started out, Indian mathe-
maticians carried out their arithmetical
operations by drawing the nine numerals of
their old *Brahmi numeration on the soft soil
inside a series of columns of an abacus drawn
in advance with a pointed stick. If a certain
order lacked units the corresponding column
was simply left empty. See Dhulikarma. This
archaic method was used later by the Arabic
arithmeticians, particularly those of the
Maghreb and Andalusia, who had adopted the
nine Indian numerals but who did not tend to
use zero or carry out their calculations without
the aid of columns. However, these mathe-
maticians did not only write out their
calculations on the ground: they normally
used a wooden board covered in dust, fine
sand, flour or any kind of powder, and wrote
with the point of a stylet, the flat end of which
was used to erase mistakes. This board might
be placed on the ground, on a stool or a table,
or sometimes the board was equipped with its
own legs, like the “counting tables” which
were later used in Arabic, Turkish and Persian
administrations. Sometimes this board consti-
tuted part of a type of kit, and was thus
smaller and could be carried in a case. See Pali
and Patiganita.
In the fifth century CE, the first nine
Indian numerals taken from the Brahmi nota-
tion began to be used with the place-value
system and were completed by a sign in the
form of a little circle or dot which constituted
zero: this system was to be the ancestor of our
modern written numeration. See The place-
value system, Position, Zero (Indian
concepts of).
Thus the Indian mathematicians radically
transformed their traditional methods of cal-
culation by suppressing the columns of their
old abacus and using the place-value system
with their first nine numerals whilst continu-
ing to use the board covered in dust. This step
thus marked the birth of our modern written
calculation.
To start with, although the corresponding
techniques had been liberated from the abacus,
they were still a faithful reproduction of the old
methods of calculation: they were still carried
out, as always, by successive corrections, con-
tinually erasing the results at each stage of the
calculation, and this limited human memory
whilst also preventing them from finding out
the errors they had committed on the way to
arriving at the final result. This method was
used with various variations by
‘Mahaviracharya (850 CE), ‘Shripati (1039
CE), ‘Bhaskaracharya (1150 CE) and even by
‘Narayana (1356 CE). Alongside this technique,
the Indian mathematicians (and the Arabic
arithmeticians after them) developed a way of
carrying out operations without any erasing, by
writing the intermediary results above the final
result. This, of course, was a great advantage
because they could see if they had made a mis-
take in their calculations if the final result
turned out to be wrong, yet brought with it the
inconvenience of a lot more writing and more
difficulty in deciphering the results, and this is
why this method of calculation using nine
numerals and zero remained beyond the under-
standing of the layman for so long.
Moreover, it was impossible for these
methods to progress further without a radical
transformation of the writing materials which
were being used. The use of chalk and black-
board, long before the use of pen and paper
became widespread, made the task much less
onerous because the intermediary results
could either be preserved or rubbed out with a
cloth. ‘Bhaskaracharya (1150 CE) used his
*pati to work out highly advanced methods of
calculation (notably multiplication which he
referred to as sthanakkhanda, which literally
means: the procedure of “separating the posi-
tions”). Even before him, the mathematician
*Brahmagupta, in his Brahmasphutasiddhanta
(628 CE), had described four methods of mul-
tiplication which were even more advanced
and are more or less identical to those we use
today. See also Square roots (How Aryabhata
calculated his).
INDIAN METRIC. Here is an alphabetical list
of terms related to this discipline, which
appear as entries in this dictionary :*Akriti,
*Anushtubh, *Ashti, *Atidhriti, *Atyashti,
*Dhriti, *Gayatri, *Jagati, *Kriti, * Poetry and
writing of numbers. *Prakriti, ‘Sanskrit,
*Serpent (Symbolism of the), ‘Numerical sym-
bols, * Vikriti.
INDIAN MYTHOLOGIES. Here is an alphabet-
ical list of words relating to this theme, which
appear as article headings in this dictionary:
*Agni, *Ahi, *Ananta, *Aptya, *Arjunakara,
*Ashva, *Ashtadiggaja, *Ashvin, *Atri, *Avatara,
‘Buddha (Legend of), *Brahmdsya,
*Chaturdvipa, *Chaturmukha, *Dantin, *Dasra,
* Dhartarashtra, *Dhruva, *Diggaja, *Dvipa,
*Gaja, ‘High numbers (The symbolic meaning
of), *Haribahu, *Hastin, *Indrarishti, *Indu,
‘Infinity (Indian mythological representation
467
INDIAN NUMERALS
of), *Jagat, *Jaladharapatha, *Kala,
*Karttikeya, * Karttikeyasya, *Kumarasya,
*Kumdravadana, *Kumud, *Kumuda, *Kuhjara,
*Lokapala, *Manu, *Mount Meru, *Mukha,
*Muni, *Murti, *Naga, *Naga, *Nasatya,
*Nrtpa, *Paksha, *Pandava, *Patala, *Pavana,
*Pundarika, *Purna, *Pushkara, *Putra, *Rahu,
*Ravana, *Rdvanabhuja, *Ravanashiras, *Rishi,
*Sahasrarjuna, *Saptarishi, *Senaninetra,
*Shanmukhabahu, * Shatarupd , *Sheshashirsha,
*Shukranetra, *Soma, *Trimurti, *Tripura,
*Trishiras, *Uchchaishravas, *Vaikuntha, *Vasu,
*Vayu, *Vishnupada.
INDIAN NUMERALS (or numerals of Indian
origin). List of the principal series of numerals
that originated in India (graphical signs which
derived from Brahmi numerals): *Agni,
‘Andhra, Balbodh, ‘Balinese, Batak, ‘Bengali,
‘Bhattiprolu, Bisaya, ‘Brahmi, Bugi, ‘Burmese,
‘Chalukya, ‘Cham, Chameali, ‘Dogri,
‘Dravidian, ‘Eastern Arabic, ‘European
(apices), ‘European ( algorisms ), ‘Ganga,
‘Ghubar, ‘Grantha, ‘Gujarati, ‘Gupta,
‘Gurumukhi, Jaunsari, ‘(Ancient) Javanese,
‘Kaithi, ‘Kannara, *Kawi, ‘Khudawadi,
Khutanese, Kochi, ‘Kshatrapa, Kului,
‘Kushana, Kutchean, ‘Kutila, Landa,
(Ancient) Laotian, Mahajani, ‘Maharashtri,
‘Maharashtri-Jaina, ‘Maithili, ‘Malayalam,
Mandeali, ‘Manipuri, ‘Marathi, ‘Marwari,
‘Mathura, Modi, ‘Mon, ‘Mongol, Multani,
‘Nagari, ‘Nepali, ‘Old Khmer, Old Malay,
‘Oriya, ‘“Pali", ‘PalJava, ‘Punjabi, ‘Rajasthani,
‘Shaka, Shan, ‘Sharada, ‘Shunga, Siamese,
‘Siddham, ‘Sindhi, ‘Sinhala, Sirmauri, Tagala,
‘Takari, ‘Tamil, ‘Telugu, ‘Thai-Khmer,
‘Tibetan, Tulu, ‘Valabhi, and ‘Vatteluttu
numerals.
For origins and graphical evolution, see Fig.
24.61 to 69. For genealogy, classification and
geographical distribution, see Fig. 24.52 and
53. For all numerical notations of both Ancient
and Modern India, see Numerical notation.
See also Indian styles of writing and Indian
written numeral systems (Classification of).
INDIAN RELIGIONS AND PHILOSOPHIES.
Here is an alphabetical list of terms related to
this theme which appear as entries in this
dictionary: *Abhra, *Abja, *Adi, *Aditya,
*Agni, *Akasha, *Amara, *Anala, *Aptya,
*Arka, *Ashtadiggaja, *Ashtamangala,
*Ashtamurti, *Ashtavimoksha, *Ashva, *Ashvin,
*Atman, *Avatara, *Bhanu, *Bharga,
*Bhava, *Bhuta, *Bhuvana, *Bija, *Bindu,
*Brahma, *Brahmasya, *Chandra, *Chatura-
shrama, *Chaturmahardja, *Chaturmukha,
*Chaturvarga, *Chaturyoni, *Dahana, *Dantin,
*Darshana, *Dashabala, *Dashabhumi,
*Dashahard, *Dashdhavatdra, *Dasra, *Deva,
*Dharma, *Dhruva, *Diggaja, * Dikpala , *Dish,
*Divakara, *Divyavarsha, *Dravya, *Drishti,
*Durga, *Dvaita, *Dvandvamoha, *Dvatrim-
shadvaralakshana, *Dvija, *Dvipa, *Dyumani,
*Ekachakra, *Ekadanta, *Ekddashi, *Ekdgrata,
*Ekakshara, *Ekdntika, *Ekatattvabhydsa,
*Ekatva, ‘Eleven, *Gagana, *Gaja, *Ganesha,
*Gati, *Go, *Guna, *Hara, *Haranayana,
*Haranetra, *Haribahu, *Hastin, ‘High num-
bers (The symbolic meaning of), *Hotri,
*Hutashana, ‘Indian atomism, *Indra,
*Indradrishti, *Indriya, ‘Infinity, ‘Infinity
(Indian concepts of), ‘Infinity (Indian mytho-
logical representations of), *Isha, *Ishadrish,
*Ishvara, *}agat, *Jaina, *Jala, *Jvalana, *Kala,
*Kama, *Karaniya, Karttikeya, Karttikeyasya,
*Kdrtlikeya, * Karttikeyasya, *Kaya, *Keshava,
*Kha, *Krishanu, *Kumdrasya, *Kumara-
vadana, *Kumud, *Kumuda, *Kuhjara, *Loka,
*Lokapala, ‘Lotus, *Mahdbhuta, *Mahadeva
*Mahapapa, *Mahayajha, ‘Mantra, *Manu,
*Martanda, *Matrikd, *Nasatya, *Netra,
*Panchabana, *Pahchdbhijha, *Pahchabhuta
*Pahchachakshus, *Pancha Indriyani, *Pahch-
aklesha, Pahchalakshana, *Panchanana,
*Parabrahman, *Paramdnu, *Pdtdla, *Pavaka,
*Pavana, *Pinakanayana, *Pitamaha, *Prana,
*Prithivi, *Pundarika, *Pura, *Purna, *Puran-
alakshana, *Pushkara, *Raga, *Rdma, *Rasa,
*Ratna, *Ravi, *Ravibdna, *Ravichandra,
*Rudra, *Rudra-Shiva, *Rudrasya, *Sagara y
*Sahasrakirana, *Sahasraksha, *Sahasrdmshu,
*Sahasrandma, *Samkhya, *Samkh yd,
*Sdmkhya, *Sdmkhyd, *Samkhycya, *Samsara,
*Saptamdtrika, *Shadanga, *Shadayatana,
*Shaddarshana, *Shakra, *Shakti, * Shatarupd ,
*Shatapathabrdhmana, *Shatkasampatti,
*Shikhin, * Shiva, *Shruti, *Shukranetra, *Shula,
*Shulin, *Shunya, *Shunyata, *Siddha, *Siddhi,
*Soma, *Sura, *Surya, *Ta!lakshana, *Tapana,
*Tattva, *Triambaka, *Tribhuvaneshvara,
*Trichivara, *Triguna, * Tripura, *Tripurasun-
dari, *Triratna, *Trishula, *Tri varga, *Trivarna,
*Trividya, *Tryakshamukha , * Tuny a, *Udarchis,
*Vahni, *Vaikuntha, *Vaishvanara, *Vajra,
*Varuna, *Vasu, *Vishaya, *Vrindd, *Yama,
* Yoni.
INDIAN STYLES OF WRITING (The materi-
als of). The Indians have used various
materials in the history of their writing, start-
ing with stone, which, like nearly all other
civilisations, has served for the writing of
official inscriptions and important commemo-
rative texts. Stone has often been replaced, at
later times, with copper and other metals.
Parchment has also been used, but only really
in central Asia, religious reasons probably pre-
venting its use in India. Tree-bark was used,
mainly in Assam and southern regions, upon
which scribes wrote in ink.
In Kashmir and the whole northwest of
India, as well as in the regions of the
Himalayas, ink and brush were (and still are)
used on birch-bark. This manuscript writing
was called bhurjapattra, and its use was men-
tioned by Quintus Curtius in this region at the
time of Alexander the Great: Libri arborum
teneri, baud secus quam chartae, litterarum notas
capiunt (“The tender part of the bark of trees
can be written on, like papyrus”) [quoted in
Fevrier (1959)1. Wooden boards were also
used, upon which characters were not carved,
but written in ink. Cotton was another writing
support, as reported in the same region by
Nearchus, Alexander’s admiral.
As for manuscripts (the oldest known
examples dating back to the first century CE),
palm-leaves were the most popular supports in
India and Southeast Asia. These were used
since ancient times, in regions of Nepal,
Burma, Bengal, as well as in southern India,
Ceylon, Siam, Cambodia and Java. Its popular-
ity was due to its availability and the ease with
which it could be used: “The leaves chosen for
writing were picked young, when they had not
yet unfurled. The middle vein was removed
and they were left to dry out under pressure.
To join them together, they were placed
between two boards or between two big dried
nervures. Then a thread passed through all the
leaves to join them together. Only one instru-
ment was needed to pierce, slice, prepare, join
and write a book. The extraordinary simplicity
of such material certainly played an important
role in the diffusion of Indian culture” [Fevrier,
(1959)]. One of the ways of writing on a palm
leaf was to engrave the characters using a
pointed instrument: "It is undeniable that the
characters traced with a point appear pale and
unclear, but when sprinkled with dust, they
become black, and the dust does not stick to
the rest of the leaf because its surface is natu-
rally smooth.” Another writing instrument is
the calamus, a type of reed whose blunted end
is dipped in a type of dye; this has been used
since time immemorial in Bengal, Nepal and
all southern regions oflndia. Thus the writing
materials used in India and Southeast and
Central Asia are as varied as the styles of writ-
ing themselves. These conditions account for
the great diversity oflndia writing styles: this
diversity has not come about by chance, as the
nature of the writing materials has had a pro-
found influence over the appearance of the
corresponding styles of writing. See Indian
styles of writing.
INDIAN STYLES OF WRITING. The various
styles of writing which are currently in use in
India, Central and Southeast Asia all derive
more or less directly from the ancient Brahmi
writing, as it is found in the edicts of Emperor
Asoka and in a whole series of inscriptions
which are contemporaries of the Shunga,
Kushana, Andhra, Kshatrapa, Gupta, Pallava,
etc., dynasties. This writing underwent many
successive and relatively subtle modifications
over the course of the centuries, which led to the
development of various completely individual
styles of writing. The apparently considerable
differences are due to either the specific charac-
ter of the languages and traditions to which they
have been adapted, or the regional customs and
the writing materials used. See Indian styles of
writing (The materials of the).
These styles of writing can be put into
three groups (Fig. 24. 28): the group of styles of
writing of northern and central India and of
Central Asia (Tibet and Chinese Turkestan):
the group of styles of writing of southern India;
and finally the group of styles of writing known
as “oriental”. Naturally, the writing of the first
nine numbers has undergone a similar evolu-
tion over the centuries: all the series of
numerals from 1 to 9 currently in use in India
and Central and Southeast Asia derive from the
ancient Brahmi notation for the corresponding
numbers and can be placed in the same groups
as those for the styles of writing (Fig. 24. 52).
For all the corresponding varieties, see Indian
numerals.
INDIAN THOUGHT. Here is an alphabetical
list of words related to this theme, which
appear as entries in this dictionary: *Abhra,
*Abja, *Adi, * Adilya, *Adri, *Agni, *Ahi,
*Akshara, *Arnara, *Anala, * Atlanta, *Anga,
* Anuyogadvara Sutra, *Aptya, ‘Arithmetical-
cosmogonical speculations, *Arjundkara ,
*Arka, *Asamkhyeya, *Asha, *Ashtadanda,
* Ashtadiggaja . * Ashtamangala, *Ashtamurti,
* Ashtanga, *Ashtavimoksha, *Ashva, *Ashvin,
* Atman, * Atrinayanaja, *AUM, *Avatdra,
*Bana, *Bhanu, *Bharga, *Bhava, *Bhuta,
*Bhuvana, *Bija, *Bindu, *Brahma, *Brah~
mdsya, ‘Calculation, *Chakskhus, *Chandra,
*Chaturananavddana, *Chaturdvipa, *Chatur-
maharaja, *Chaturmukha, *Chaturyoni,
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
468
*Chaturyuga, *Cosmogonic speculations,
*Dahana, *Dantin, *Darshana, *Dashabala,
*Dashabhumi, *Day of Brahma, *Dcva,
*Dharani, *Dharma, *Dhruva, *Diggaja,
*Dikpala, *Dish, *Divakara, *Dravya,
*Drishti, *Durga, * Dvadashadvdrashastra ,
*Dvaparayuga, * Dvatrimshadvaralakshana,
*Dvija, *Dvipa, *Dvipa, *Dyumani, ‘Eight,
*Eka, *Ekachakra, *Ekadanta, *Ekddashi,
*Ekagratd, *Ekakshara, *Ekatva, Ekavachana,
‘Eleven, ‘Fifteen, ‘Five, ‘Four, ‘Fourteen,
*Gagana, *Gaja, *Ganesha, *Gati, *Gavya,
*Go, *Graha, *Hara, *Haranayana,
*Haranetra, *Haribahu, *Hastin, ‘High num-
bers (The symbolic meaning of), *Hotri,
*Hutdshana, ‘Indian astrology, ‘Indian atom-
ism, ‘Indian documentation (Pitfalls of),
‘Indianity (Fundamental mechanisms of),
*Indra, *Indradrishti, *Indriya, *Indu, ‘Infinity,
‘Infinity (Indian concepts of), ‘Infinity
(Indian mythological representation of), *Isha,
*Ishadrish, *Ishu, *Ishvara, *Jagat, *Jaladhara-
patha, *Jvalana, *Jyotisha, Kabubh, *Kala,
*Kalamba, *Kaliyuga, *Kalpa, *Kama, * Karan -
iya, *Karttikeya, *Karttikeyasya, *Kavacha,
*Kaya, *Keshava, *Kha, *Krishanu, *Krita,
*Kritayuga, *Kshapeshvara, *Kumarasya, *Kum-
aravadana, *Kumud, *Kumuda, *Kuhujara,
*Loka, *Lokapala, ‘Lotus, *Mahadeva,
*Mahakalpa, *Mahayuga, *Mangala, ‘Mantra,
*Manu, *Margana, *Mdrtanda, *Mriganka,
*Mukha, * Marti, ‘Mysticism of infinity,
‘Mysticism of zero, *Ndga, *Netra, ‘Nine,
‘Numeral alphabet, magic, mysticism and div-
ination, ‘Numerical symbols, ‘Ocean, ‘One,
*Paksha, *Pahchabdna, *Pahchabhijha, *Pahc-
hachakshus, *Pahcha Indriyani, *Pahchaklesha,
*Pahchalakshana, *Pahchanana, *Parabrah-
man, *Patdla, *Pavaka, *Pavana, *Pina-
kanayana, *Pitdmaha, *Pundarika, *Pura,
*Purna, *Putra, *Raga, *Rahu, *Rasa, *Ratna,
*Ravanabhuja, *Ravi, *Ravibana, *Rudra,
*Rudra-Shiva, *Rudrasya, *Sdgara, *Saha-
srakirana , *Sahasraksha, *Sahasramshu,
*Sahasranama, *Samkhya, *Samkhya, *Sam-
khya, *Sdmkhyd, *Samkbydna, *Samkhyeya,
‘Sanctification of a concept, *Sanskrit,
*Saptabuddha, *Sarpa, *Sayaka, ‘Seven,
*Shakra, *Shakti, *Shankha, *Shanku,
*Shanmukhabdhu, *Shara, *Shashadhara,
*Shashanka, *Shashin, *Shatarupa, *Shikhin,
*Shirshaprahelikd, *Shitamshu, *Shitarashmi,
* Shiva, *Shukranetra, *Shula, *Shulin, *Shunya,
*Shunyatd, ‘Six, ‘Sixteen, *Soma, *Sud -
hamshu, *Sura, *Surya, ‘Symbolism of
numbers, ‘Symbols, *Takshan, *Tallakshana,
*Tapana, ‘Ten, ‘Thirteen, ‘Thirty-three,
‘Thousand, ‘Three, *Tretayuga, *Triambaka,
*Tribhuvana, *Tribhuvaneshvara, *Trikaya,
*Triloka, *Trimurti, * Tripit aka, * Tripura,
*Tripurasundari, *Trishula, *Trivarna,
*Tryakshamukha, *Tryambaka, * Tuny a,
‘Twelve, ‘Twenty, ‘Twenty-five, ‘Two,
*Udarchis, *Uppala, *Utpala, *Vachana,
*Vahni, *Vaikuntha, * Vaishvanara, *Vajra,
*Varuna, *Vasu, *Vidhu, *Vishika,
*Vishnupada, *Vishva, *Vishvadeva, *Vrinda,
*Yama, *Yuga, *Yuga (Astronomical specula-
tion on), *Yuga (Cosmogonical speculations
on), ‘Zero.
INDIAN WRITTEN NUMERAL SYSTEMS
(Graphical classification of). The aim of this
article is to give a quick recapitulation of the
various numerical notations formerly or cur-
rently used in the Indian sub-continent, in
order to identify the palaeographic type of
each one. The following references to figures
are mainly the ones which can be found in
Chapter 24. More or less all of the numerical
notations which are currently in use in India,
Central Asia and Southeast Asia (see Fig. 24.
61 to 69) derive from the ancient Brdhmi nota-
tion (Fig. 24. 29 to 31 and 70), which is found
in the edicts of Emperor Asoka and a whole
series of contemporary inscriptions of the
Shunga, Kushana, Andhra, Kshatrapa, Gupta,
Pallava, etc. dynasties (Fig. 24. 29 t 38 and
70). This original notation (which surely
derives from an earlier ideographical nota-
tion) has undergone several subtle graphical
modifications over the course of the centuries
(Fig. 24. 70), which led to the development of
various types of notations which are all highly
individual like Gupta (Fig. 24. 38),
Bhattiprolu and “Pali”. See also Andhra
numerals, Bhattiprolu numerals, Brahmi
numerals, Chalukya numerals, Ganga
numerals, Gupta numerals, Kshatrapa
numerals, Kushana numerals, Mathura
numerals, Pali numerals, Pallava numerals,
Shaka numerals, Shunga numerals and
Valabhi numerals.
For the graphical origin of Brdhmi numer-
als, see Fig. 24. 57 to 24. 59. For notations
derived from Brdhmi, see Fig. 24. 52. For their
graphical evolution, see Fig. 24. 61 to 24. 69.
The apparently considerable differences
between these notations are due to either the
specific character of the languages and tradi-
tions to which they belong to which the
corresponding writing would have been
adapted, or to the regional habits of the scribes
and the nature of the writing material used.
See Indian styles of writing.
The notations can be divided into three
broad groups (see Fig. 24. 52):
1. - The group of notations from Central
India, from Northern India, from Tibet and
Chinese Turkestan. These notations are the
ones which come from Gupta writing. These
can be divided in turn into five sub-groups:
1. 1. - The sub-group of notations derived
from Nagari. This group is made up of nota-
tions issued from Nagari numerals (Fig. 24.3,
39 and 72 to 74), amongst which are:
Maharashtra; Marathi (Fig. 24.4); Balbodh;
Modi; Rajasthani; Marwari; Mahajani; Kutila;
Bengali (Fig. 24.10); Oriya (Fig. 24.12);
Gujarati (Fig. 24.8); Maithili (Fig. 24.11);
Manipuri; Kaithi (Fig. 24. 9); etc.
The Arabic notations “Hindi” and Ghubar
also belong to this sub-group, as well as the
European Apices and Algorisms: Arabic numerals
both from the East and the Maghreb (Fig.
25.3 and 25.5), derive more or less directly from
Nagari numerals. The numerals that we use
today, and the European numerals of the Middle
Ages (Fig. 26.3 and 10), derive from the Ghubar
numerals of the Maghreb (Fig. 25.5). See
Eastern Arabic numerals, Bengali numerals,
European numerals (Apices), European
numerals (Algorisms), Ghubar numerals,
Gujarati numerals, Kaithi numerals, Kutila
numerals, Maharashtri numerals, Maharas-
htrijaina numerals, Maithili numerals,
Manipuri numerals, Marathi numerals,
Marwari numerals, Nagari numerals, Oriya
numerals and Rajasthani numerals.
1.2. - The sub-group of notations derived from
Sharada writing. This is composed of notations
derived from the numerals of the same name
(Fig. 24. 14 and 40), including: Takari (Fig. 24.
13); Dogri (Fig. 24. 13); Chameali ; Mandeali;
Kului; Sirmauri; Jaunsari; Sindhi (Fig. 24. 6);
Khudawadi (Fig. 24. 6); Gurumukhi (Fig. 24.
7); Punjabi (Fig. 24. 5); Kochi; Landa; Multani;
etc. See Dogri numerals, Gurumukhi numer-
als, Khudawadi numerals, Punjabi numerals,
Sharada numerals, Sindhi numerals,
Sirmauri numerals and Takari numerals.
1.3. - The sub-group of notations from Nepal.
This includes modern Nepali (Fig. 24. 15),
which derives from the ancient Siddham nota-
tion (Fig. 24. 42) which itself comes from
Gupta but under the influnce of Nagari. See
Nepali numerals and Siddham numerals.
1.4. - The sub-group of Tibetan notations.
This contains Tibetan notations (Fig. 24. 16),
which all derive from Siddham, and which are
notably related to ancient Mongol writing
(Fig. 24.42). See Tibetan numerals and
Mongol numerals.
1.5. - The sub-group of notations from
Central Asia. This contains notations of
Chinese Turkestan, which also all derive
from Siddham.
2. - The group of notations from Southern
India. These are notations which come from
Bhattiprolu (Fig. 24. 43 to 24. 46), distant
cousin of Gupta. They can be subdivided into
four groups:
2. 1. - The sub-group of Telugu notations.
This is made up of Telugu and Kannara nota-
tions (Fig. 24. 20, 21, 47 and 48).
2.2. - The sub-group of Grantha notations.
This contains Grantha, Tamil and Vatteluttu
notations (Fig. 24. 17 and 24. 49).
2.3. - The sub-group of Tulu notations. This
contains Tulu and Malayalam notations
(Fig. 224. 19).
2.4. - The sub-group of Sinhalese notations .
In this group Sinhala notation can be found
(Fig. 24. 22).
See Dravidian numerals, Grantha numer-
als, Kannara numerals, Malayalam numerals,
Sinhala numerals, Tamil numerals, Telugu
numerals and Vatteluttu numerals.
3. - The group of eastern notations. These
are the notations of Southeast Asia, which are
all derived from “Pali” writing, which itself
comes from the same source as Gupta and
Bhattiprolu (Fig. 24. 43 to 46). These in turn
can be subdivided into seven groups:
3.1. - The sub-group of Burmese notations.
This contains ancient and modern Burmese
notations (Fig. 24. 23).
3.2. - The sub-group of Old Khmer notations.
In this group there is the ancient notation of
Cambodia (Fig. 24. 77, 78 and 80).
3.3. - The sub-group of Cham notations. This
contains the notation of Champa (Fig. 24. 79
and 80).
3.4. - The sub-group of Old Malay notations.
This group contains the writing style once used
in Malaysia (Fig. 24. 80).
3.5. - The sub-group of Old Javanese
notations. This group contains “Kawi” writing
which was once used in Java and Bali (Fig. 24.
50 and 24. 80).
3.6. - The sub-group of present-day Thai-
Khmer writing. This includes Shan, Laotian and
Siamese, as well as the notation which is cur-
rently used in Cambodia, Laos and Thailand
(Fig. 24. 24).
3. 7. - The sub-group of current Balinese nota-
tions. This sub-group is made up of the current
469
INDIAN WRITTEN NUMERAL SYSTEMS
Balinese, Buginese, Tagala, Bisaya and Batak
notations (Fig. 24.25). See Balinese numerals,
Burmese numerals, Cham numerals, Ancient
Javanese numerals, Kawi numerals, Thai-
Khmer numerals and Old Khmer numerals.
For an overview of all these notations, see Fig.
24.52. For their geography, see Fig. 24.27 and
53. For their mathematical classification, see
Indian written numeral systems (The mathe-
matical classification of).
INDIAN WRITTEN NUMERAL SYSTEMS
(The mathematical classification of). Here is a
quick summary of the mathematical structure of
the various notations which were once used, or
are still in use, in the Indian sub-continent. The
numerical notations which derive from Brahmi
(see Indian written numeral systems
(Graphical classification of)) are not the only
ones to be used in the Indian sub-continent. In
northwest India, after Asoka’s time until the sixth
or seventh century CE, a style of writing was used
which was imported by Aramaean traders. This
was known as Karoshthi (Fig. 24. 54). See
Karoshthi numerals. There is also the system
which was found in Mohenjo-daro and Harappa
(in present-day Pakistan), which was used from c.
2500 to 1500 BCE by the ancient Indus civilisa-
tion, long before the Aryans arrived on Indian
soil. See Proto-Indian numerals.
From a mathematical point of view, accord-
ing to the classification of numerations in
Chapter 23, these different systems (which gen-
erally have a decimal base) can be divided into
three broad categories:
A. - The category of additional numera-
tions. These are systems which are based upon
the additional principle, each numeral possess-
ing its own value, independent of its position
in numerical representations. This category
can be subdivided into three types:
A.l. - The first type of additional numera-
tions. These are numerations which (like the
Egyptian hieroglyphic system for example)
attribute a particular numeral to each of the
numbers 1, 10, 100, 1,000, 10,000, etc., and
which repeat these signs as many times as nec-
essary to record other numbers (Fig. 23.30).
The ancient *Indusian numeration no doubt
belonged to this type.
A. 2. - The second type of additional numer-
ation. These are numerations which (like the
Roman system for example) attribute a spe-
cific numeral to each of the numbers 1, 10,
100, 1,000, etc., as well as to 5, 50, 500, etc.,
and which repeat these signs as many times
as necessary to record other numbers (Fig.
23.31). There is no known example of this
type in India.
A. 3. - The third type of additional numera-
tion. These are numerations which (like the
Egyptian hieratic system for example)
attribute a particular sign to each unit of each
decimal order (1, 2,3,... 10, 20, 30, . . . 100,
200, 300, . . ., etc.) and which use combina-
tions of these different signs to write other
numbers (Fig. 23.32). This is the type that all
notations derived from Brahmi belong to, at
least initially (Fig. 24.70). Thus the following
notations belong to this sub-category: Andhra
notation (Fig. 24.34 and 36); Bhattiprolu nota-
tion; Chalukya notation (Fig. 24.45); Ganga
notation (Fig. 24.46); Gupta notation (Fig.
24.38); Kshatrapa notation (Fig. 24.35);
Kushana notation (Fig. 24.33); Mathura nota-
tion (Fig. 24.32); Ancient Nagari notation
(Fig. 24.39B); Ancient Nepali notation (Fig.
24.41); Pallava notation (Fig. 24.34 and
24.36); Valabhi notation (Fig. 24.44); etc.
Alphabetical notations also fall into this cate-
gory (which use vocalised consonants of the
Indian alphabet, to which a numerical value is
assigned in a regular order, and which are still
used today in various regions of India, from
Tibet, Nepal, Bengal or Orissa to
Maharashtra, Tamil Nadu, Kerala, Karnataka
and Sri Lanka, and from Burma to Cambodia,
in Thailand and in Java); notably that of
Aryabhata (the difference being that the latter
had a centesimal base, not a decimal one).
One exception is Katapayadi numeration
(which seems to have been invented by
Haridatta), which was alphabetical but based
on the place-value system. See Numeral
alphabet, Aryabhata’s numeration and
Katapayadi numeration.
B. - The category of hybrid numerations.
These are numerations which use both multi-
plication and addition in their representations
of numbers. This category can be divided into
five types:
B.l. - The first type of hybrid numeration.
These are numerations which (like the
Babylonian system) attribute a particular
numeral to each of the numbers 1, 10, 100,
1,000, etc., using an additive notation for num-
bers inferior to one hundred and a multiplicative
notation for the hundreds, the thousands, etc.,
and which represents other numbers through
combinations which use both the additive and
multiplicative principles (Fig. 23.33). Aramaean
numeration belongs to this group (Fig. 23.17) as
well as *Kharoshthi numeration which is
derived from the former (Fig. 24.54).
B.2. - The second type of hybrid numeration.
These are numerations which function exactly
like the Sinhalese system (Fig. 23.18 and
24.22): a particular numeral is given to each
simple unit, as well as to each power of ten (10,
100, 1,000, etc.), and the notation of hundreds,
thousands, etc., follows the multiplicative rule
(Fig.23.34).
B.3. - The third type of hybrid numeration.
These are Mari numerations (Fig. 23.22),
which do not seem to exist in India.
B.4. - The fourth type of hybrid numeration.
These are Ethiopian numerations (Fig. 23.36),
which do not seem to exist in India.
B. 5. - The fifth type of hybrid numeration.
These are the numerations for which Tamil and
Malayalam numerations provide the models
(Fig. 23.20 and 21); these give a specific
numeral to each simple unit (1, 2, 3 . . .), as
well as to diverse multiples of each power of
ten (10, 20, 30, . . . 100, 200, 300 etc.), and
where the notation of tens, hundreds, thou-
sands, etc., is carried out using the
multiplicative principle (Fig. 23.37).
C. - The category of positional numera-
tions. These are numerations founded on the
principle according to which the basic value of
numerals is determined by their position in the
writing of the numbers, and which thus
requires the use of a zero (Fig. 23.27). This cat-
egory can be subdivided into two types:
C.l. - The first type of positional
numerations. These are Babylonian, Chinese or
Maya (Fig. 23. 23, 24, 25 and 38), which are
not found in India.
C.2. - The second type of positional numera-
tions. These are the numerations (Fig. 23.28),
which belong to the one which was invented in
India over fifteen centuries ago and which is
the origin of all decimal positional notations
which are currently in use (Fig. 24.3 to 16 and
20 to 26), including our own (Fig. 23.26) and
the one which is still used in Arabic countries
(Fig. 24.3). This system has a decimal base,
and nine distinct numerals which give no
visual indication as to their value, which repre-
sent the nine significant units (from w'hich our
signs 1, 2, 3, 4, 5, 6, 7, 8, 9 are) derived; it also
possesses a tenth sign, called *shunya (zero),
w-hich is written as a little dot or circle (Fig.
24.82 and *Zero, Fig. D. 11), and is the ancestor
of our zero, w'hose function is to mark the
absence of units in any given order, and w r hich
possesses a veritable numerical value: that of
“nil” (Fig. 23.27). The fundamental character-
istic of this system is that it can express all
numbers in a simple and coherent way,
whether they are whole, fractional, irrational
or transcendental (Fig. 23.28). Thus the Indian
place-value system (for that is what it is) is the
first of the category of the most evolved writ-
ten numerations in history (Fig. 28.29). The
following are the notations w'hich belong to
this category:
Modern Nagari (Fig. 24.3, 39 A and 39 C);
Marathi (Fig. 24.4); Punjabi (Fig. 24.5); Sindhi
(Fig. 24.6); Gurumukhi (Fig. 24.7); Gujarati
(Fig. 24.8); Kaithi (Fig. 24.9); Bengali (Fig.
24.10) ; Maithili (Fig. 24.11); Oriya (Fig.
24.12); Takari (Fig. 24.13); Sharada (Fig. 24.14
and 40); modern Nepali (Fig. 24.15); Tibetan
(Fig. 24.16); Telugu (Fig. 24.20 and 47);
Kannara (Fig. 24.21 and 48); Burmese (Fig.
24.23 and 51); Thai-Khmer (Fig. 24.24);
Balinese (Fig. 24.25); modern Javanese (Fig.
24.26); ancient Javanese (Fig. 24. 50); Mongol
(Fig. 24.42); the “Hindi" form of Arabic writ-
ing (Fig. 24.3 and 25.5); the “Ghubar” form of
Arabic writing, whilst it was used to represent
numbers with zero, without the columns of
the abacus drawn in the dust (Fig. 25.3); the
“Algorism" form of European writing (Fig.
26.10) ; etc.
Thus the discovery of Indian positional
numeration not only allowed the simple
and perfectly rational representation of
absolutely any number (however large or
complex), but also and above all a very easy
way of carrying out mathematical operation;
this discovery made it possible for anyone to
do sums. The Indian contribution to the his-
tory of mathematics was essential, because it
united calculation with numerical notation,
thus enabling the democratisation of the art
of calculation.
For the graphical classification of the
various numerical notations, see *Indian
written numerations (The graphical classifi-
cation of). For the Sanskrit names, usage,
conditions and discovery of positional
numeration, see: Anka, Sthana, Ankak-
ramena, Ankasthana, Sthdnakramad, Names
of numbers, High numbers, Sanskrit,
Numerical symbols, Numeration of numer-
ical symbols, Katapayadi numeration,
Aryabhata’s numeration.
For zero and its graphic or symbolic repre-
sentations, see Zero, Shunya, Numeral 0.
For corresponding methods of calcula-
tion, see Patiganita, Indian methods of
calculation.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
470
For the subtleties relating to zero and the
place-value system in Sanskrit poetry, see
Poetry, zero and positional numeration.
INDIVIDUAL SOUL. [$]. Value = 1. See
Atman. One.
INDO-EUROPEAN NAMES OF NUMBERS.
See Chapter 2, especially Fig. 2. 4 A to 4 J and
2. 5, where the Sanskrit names of numbers are
compared to those of other languages of Indo-
European origin. See Fig. D. 2 of the entry
entitled Aryabhata’s numeration.
INDRA. IS]. Value = 14. “Strength”, "Courage”,
“Power". The name of one of the principal gods
of Vedic times and of the Brahm anic pan-
theon. He represents the source of cosmic life
that he transmits to the earth through the
intermediary of rain. His strength lies in the
seminal fluid of all beings, this god being said
to be “made of all the gods put together". He is
eternally young, because he rejuvenates himself
at the start of each manvantara, which means
each of the fourteen “ages” of our world which
make up a *kalpa. Thus Indra = 14. See Yuga,
Manu and Fourteen.
INDRADRISHTI. [SJ. Value = 1,000. “Eyes
of Indra”. One of this god’s attributes is
*Sahastaksha, “of the thousand eyes”. See
Indra, Thousand.
INDRIYA [S]. Value = 5. “Power”. This is due to
the Buddhist physical and mental powers, which
are divided into five groups: the foundations
(i dyatana ); the natures {bhava)\ the senses
( vedana ); the spiritual powers (bald)] and the
supramundane powers. The same word also
means the “five roots” ( *paiicha indriya), which,
as positive agents, enable a person to lead a
moral life: faith ( shraddendriya ), energy (viyen-
driya ), memory ( smritindriya ), meditation
(samddhindriya), and wisdom (prajnendriya) [see
Frederic, Dictionnaire (1987)]. See Five.
INDU. [S]. Value = 1. “Drop". This represents
the moon, and alludes to the “dew" ( chan -
drakanta), the mythical pearl said to have been
made from concentrated moonbeams. The
moon being worth 1, this symbolism is self-
explanatory. This word should not be confused
with *bindu (“point”) which is a synonym for
zero. See One.
INDUSIAN NUMERALS. See Proto-Indian
numerals.
INDUSIAN NUMERATION. See Proto-
Indian numerals.
INFERIOR WORLD. [S]. Value = 7. See Pdldla
and Seven.
INFINITELY BIG. See High numbers.
Asamkhyeya , High numbers (Symbolic mean-
ing of).
INFINITELY SMALL. See Low numbers,
Para manu, Shunya, Shunyatd, Zero and
Infinity.
INFINITY (Indian concepts of). Amongst the
Sanskrit words for zero is *ananta, which liter-
ally means “infinity”: Ananta is an immense
serpent, who, in Indian cosmology and
mythology, represents the serpent of infinity,
eternity and the immensity of space. *Vishnu
is said to rest on the serpent in between cre-
ations. See Serpent (Symbolism of the), High
numbers and Infinity (Indian mythological
representation of).
In Indian mysticism, the concept of zero
and that of infinity are very closely linked.
Thus words such as *ambara, *kha, *gagana,
etc., meaning “space”, “sky” or the “canopy of
heaven” came to represent zero. See Zero,
Shunya, Akdsha, Vishnupada and Puma.
Of course, Indian mathematicians knew
perfectly well how to distinguish between
these two notions, which are the inverse of
one another, for to their mind, division by
zero was equal to infinity. This was the case at
least since the time of *Brahmagupta (628
CE), who defined infinity with the term
*khachheda, literally “the quantity whose
denominator is zero” [see Datta and Singh
(1938), pp. 238-44]. In *Lildvati,
*Bhaskaracharya wrote the following about
the same concept, which he refers to as *kha-
hara, which literally means “division by zero”
[see Datta and Singh (1938), pp. 238-44]: “In
this quantity which has zero as divisor, there
is no [possible] modification, even though
several [quantities] can be extracted or intro-
duced; in the same way, no changes can be
carried out on the constant and infinite God
[*Brahma] during the period of the destruc-
tion or creation of worlds, however many
living species are projected forward or are
absorbed.” This is what Ganesha wrote on the
subject in Grahaldghava (c. 1558 CE): “The
Khachheda is an indefinite quantity, unlimited
and infinite; it is impossible to know how
high this quantity is. It can be modified by
neither the addition nor the subtraction of
limited [= finite] quantities, for in the prelimi-
nary operation of reducing all the fractional
expressions to the same denominator, which
it is necessary to do beforehand in order to be
able to calculate their sum or their difference,
the numerator and the denominator of the
finite quantity both disappear." So Indian
scholars, at least since Brahmagupta’s time,
knew that division by zero equalled infinity:
a/0 = ».
To their mind, this “quantity” remained
unchanged if a finite number was added to it or
subtracted from it; thus:
a/0 + k = k + a/0 = a/0
and
a/0 - k = k - a/0 = a/0.
This means that the Indians, at least as early as
the beginning of the seventh century CE, knew
these mathematical formulas that we use
today:
®° + k = k + «> = °°
and
°° — k — k — °° — °°.
Brahmagupta, however, (and several of his suc-
cessors) committed the error of thinking that
w'hen zero was divided by itself the result was
zero, when in reality the result is an “indefinite
quantity”. Bhaskaracharya, who made the nec-
essary corrections to the erroneous assertions
of his predecessors, and who quite rightly
affirmed that a number other than zero
divided by zero gives an infinite quotient, him-
self committed an error by saying that the
product of infinity multiplied by zero is a finite
number. However, this in no way detracts from
the merits of Indian civilisation which was so
advanced in comparison with all the other
civilisations of Antiquity and the Middle Ages.
See Infinity, Infinity (Indian mythological
representation of) and Indian mathematics
(The history of).
INFINITY (Indian mythological representa-
tion of). It seems that the lemniscate which
today represents the concept of infinity, was
introduced for the first time in 1655 by the
English mathematician John Wallis. Hindu
mythological iconography contains a very simi-
lar symbol representing the same idea,
although it seems that it was never used in the
domain of mathematics. This symbol is that of
Ananta, the famous serpent of infinity and
eternity, which is always represented coiled up
in a sort of figure of eight on its side like the
symbol <». See Ananta (in particular Fig. D. 1),
Puma and Vishnupada.
This begs the following question: Did
Wallis know of the Indian mythological
symbol when he introduced this sign into the
list of mathematical conventions? The
answer is no; this graphism and its many
variants («\ 8, S, etc.) can be found in diverse
civilisations and many different epochs and
parts of the world, and the symbolism is sim-
ilar, if not identical, to that of the Indian
mythological representations. This symbol-
ism can be found in many ancient
astrological, magical, mystical and divinatory
representations, for example in ancient and
mediaeval talismans, both Eastern and
Western, where the S is very common and is
meant to express, for the wearer of the
amulet, a sign favourable to eternal union and
infinite happiness. The sign which looks like a
figure of eight lying on its side can be found
painted on the walls of masonic lodges or
embroidered upon clothing. It is not there for
merely decorative purposes; it symbolises the
bonds which unite the members of a social
body: the interlacing expresses the sentiment
united until death [see Chevalier and
Gheerbrant (1982)]. This symbol can also be
found in the manuscripts of mediaeval
alchemists, where three Ss signify the abun-
dance of rain water, as well as its Constance.
The S can also be connected to the celestial
wheel of the Romanised god of Ancient Gaul,
and to talismans which have celestial meaning
in Greek-Roman magical traditions [see
Marques-Riviere (1972)]. For the Assyrians,
hawu was also in the form of an S, like the ser-
pent of eternal life. This symbol was later used
by the Hebrews to represent the “bronze ser-
pent” before it was destroyed by Hezekiah
(2 Kings XVIII, 4). This is the serpent made
by Moses to save the Israelites who had
spoken against the Lord, and who had been
bitten by the fiery serpents sent by Yahweh
(Numbers, XXI, 6)[see GLE, IX, p. 770].
The interlace is often a symbol for water or
for the vibration of the air. In many cosmogo-
nies, the interlace symbolises the very nature of
creation, energy and all existence. In Celtic art,
it symbolises the notion of ourobouros : the end-
less movement of evolution and involution
through the muddle of cosmic and human
facts. The ourobouros is the serpent which bites
its own tail ; this symbolises self-evolution, or
self-fertilisation and, consequentially, eternal
return. This evokes the *samsara (or the Indian
471
INFINITY AND MYSTICS M
cycle of “rebirth”), which is an indefinite circle
of rebirths, of continual repetition. Thus the
serpent gradually came to be represented by a
circular graphism. Sometimes this circle has
been dissected by two perpendicular diameters
in order to show the inter-relationship between
the sky and the earth. The sign which looks like
an X or a cross symbolises the earth with its
four horizons. Thus the circle dissected by the
cross is none other than the celestial-terrestrial
opposition of the mysticism of the serpent. See
Serpent (Symbolism of the).
Palaeography proves that this dissected
circle is, cursively speaking, the S or 8 sign
denoting a vast quantity or eternity. This is
very significant when we look at the shapes of
Roman numerals. The Roman numerals that
we know today look like Roman letters: 1 (1), V
(5), X (10), L (50), C (100), D (500), M (1,000).
In reality, however, these symbols are not
the original ones used to write the numbers. In
fact, Roman numerals derive from the ancient
practice of counting using a “tally” system
which led to numbers being represented by the
following symbols:
i v x y %
1 5 10 50 100
Originally, the unit was represented by a verti-
cal line, the number 5 by an acute angle, the
number 10 by a cross, 50 by an acute angle dis-
sected by a vertical line and 100 was a cross
dissected by a vertical line. We can easily see
how the primitive signs for 1, 5 and 10 became
the letters I, V and X. The sign for 50 originally
looked like an arrow pointing downwards. This
evolved into what looked like a T on its head
before finally being mistaken for the letter L.
As for the representation of 100, this initially
evolved into a sign which looked like this:
Cjb Then, in order to save time, this symbol
was cut in half so that it looked like the letter C,
or its mirror image. This is also the initial of
the Latin word for hundred, Centum. To create
a sign for 1,000, the Romans decided to use the
symbol for 10 (the cross) and draw a circle
around it. Then, for 500, they cut the sign for a
thousand in half: ^ . This sign would later be
mistaken for the letter D. The circle dissected
by a cross (1,000) evolved into various shapes,
which were replaced by the M due to the Latin
word Mile (see Fig. 16. 26 to 34): Thus we can
see how, graphically, the circle dissected by a
cross became a sign which was shaped like an S
or an 8 on its side. In Latin, the term Mile
„CD — GD
CD<— CD — co
, / \>-cl3< C,D
/ CIO
CO -
OO-XJ
i,ooo \ ^9 — Y
\ rh — eh —
corresponded to the highest number in spoken
numeration and, by extension, in everyday lan-
guage, it meant “vast number” and “the
incalculable”. In his Natural History (XXXIII,
133), Pliny wrote that the Romans had no
names for powers of ten superior to a hundred
thousand, and so referred to a million as decies
centena milia (“ten hundred thousand”). The
snake as the sign for infinity, in its various
forms, has been connected to ideas such as the
sky, the universe, the axis of the world, the night
of beginnings, the primordial substance, the
vital principle, life, eternal life, sexual energy,
spiritual energy, vestiges of the past, the seed of
times to come, cyclical development and resorp-
tion, longevity, extreme fertility, the incalculable
quantity, abundance, immensity, totality,
absolute stability, endless movement, etc. See
High numbers (The symbolic meaning of).
INFINITY AND MYSTICISM. See Infinity
(Indian mythological representation of) and
Serpent (Symbolism of the).
INFINITY. All confusion must be avoided
between infinity and indefinite. Indefinite comes
from the Latin indefinitus, signifying “vague”.
This word also has other possible meanings.
The first is the opposite of “defined”, that
which is “unspecified", which remains “unde-
termined”, like death for example: “That which
is certain in death is softened a little by that
which is uncertain: it is an indeterminate
length of time which has something of infinity
and of what we call eternity.” The second
meaning expresses the opposite of that which
is “finite”; it is a quantity which, whilst remain-
ing finite, is susceptible to unlimited expansion
or growth. This is the meaning of indefinite
progress. The third sense can be found in this
extract from Descartes: “Each body can be
divided into infinitely small parts. I do not
know if their number is infinite or not, but cer-
tainly, to the best of our knowledge, it is indefi-
nite” I Oeuvres , XI. 12]. This is the opposite of
that which is infinite: here the indefinite is
“that which is only infinite from a certain point
of view-, because w'e cannot see its end”
IFoulquie (1982)).
On the other hand, a fourth meaning
makes this word a synonym of infinity. This is
potential infinity, illustrated by this quote from
Pascal: “The eternal silence of these infinite
spaces fill me with fear” [Pen sees, 428]. This
extract inspired the following commentary
from Paul Valery: “This phrase, which is so
powerful and magnificent that it is one of the
most famous ever to have been uttered, is a
Poem, and by no means a thought. Eternal and
Infinite are symbols of non-thought. Their
value is entirely emotional. They act on our
sensitivity. They provoke the peculiar sensation
of the inability to imagine" [ Variete , La Pleiade,
I, 458]. Thus potential infinity is “that which,
being effectively finite, has the potential for
limitless growth” [Foulquie]. In terms of either
potential or reality, infinity has posed one of
the most serious problems to the human mind
in all the history of civilisation. Confronting
infinity' has been a little like meeting Cerberus
at the entrance to the Underworld. There is one,
final number, but it is beyond the power of mortals
to reach it; this power belongs only to the gods who
are the only ones who may count the stars and the
firmament. Such is the leitmotiv of both ancient
and modern religions. It bears witness to
humanity’s constant obsession with this con-
cept. It demonstrates not only our ability to
count numbers “to the end", but also to learn
the true meaning of that which conceals the
rather vague notion of the unlimited: “We imag-
ine some kind of finite range, then w'e disregard
the limits of the range, and we have the idea of
an infinite range. In this way, and perhaps in
this way alone, we are able to conceive of an infi-
nite number, an infinite duration, etc. Through
this definition, or rather this analysis, we can see
to what extent our notion of infinity is vague and
imperfect; it is only really the notion of the
indefinite, if we understand by this word a vague
quantity to which no limits have been assigned,
and not. as one could understand by another
meaning of this word, a quantity for which there
are limits, yet these limits have not been speci-
fied" [D’Alembert, Essai sur les elements de
philosophic, Eclairc., XIV].
This explains why comparisons are some-
times made w'hich are reminiscent of religious
metaphors and parables. The grains of sand of
the desert, the drops of water of the ocean or
the stars in the sky are evoked, without the
realisation that such comparisons are puerile,
as they only involve the domain of the finite.
In everyday use, infinity is only understood by
its negation. In fact, the word “infinity”
derives from the Latin infinitus, “that which
has no end”, “that which never ends". It is the
negation of the finite, in the sense that infinity
is “that which can never be reached”.
[See Blaise (1954); Bloch and von Wartburg (1968);
Chantraine (1970); Du Cange (1678); Ernout and
Meillet (1959); F.stienne (1573); Gaffiot; GLF
(1971); Littre (1971); Robert (1985)).
It is precisely this limited conception which
prevented the Greek mathematicians from
making progress in this domain. Historically, it
was in Greece, after Pythagoras’s discoveries,
that the evolution of this concept began with
the undisputed statement that “infinity is
something which cannot be measured”. The
problem, according to Bertrand Russell, repre-
sented “in one form or another, the basis of
nearly all the theories of space, time and of
infinity which persisted from that time up until
the present day”.
Descartes was one of the first European
scholars to establish infinity as a fundamen-
tal reality. This notion later became a
perfectly precise, objective concept, present-
ing no basic problems such as those often
conferred upon it by the profane. The symbol
for infinity («>) seems to have appeared for
the first time in 1655 in a list of mathematical
signs compiled by the English mathematician
John Wallis.
Mathematically, infinity is that which is
bigger than any other quantity, and no finite
number can be added to it. Flechier com-
pared infinity to God, as God is “infinitely
powerful and thus infinitely free”. Zero is the
opposite of infinity: it is infinitely small, the
variable quantity which is inferior to all posi-
tive numbers, however small they might be.
Infinity, or the impossibility of counting all
the numbers, remains a mathematical hypoth-
esis', it is one of the fundamental axioms upon
which contemporary mathematics is based.
See Infinity (Indian concepts of) and
Infinity (Indian mythological representa-
tions of).
INFINITY. I erm used as a synonym for
“potential infinity”. See Infinity. See also
Indian mathematics (The history of).
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
472
INFINITY. Term used as a synonym for the
‘‘indeterminable”. See Infinity and High num-
bers. See also Serpent (Symbolism of the).
INFINITY. Term used as a synonym of the
"unlimited”. See Infinity, High numbers and
High numbers (The symbolic meaning of).
See also Serpent (Symbolism of the).
INFINITY. Term used as a synonym of the
eternity and immensity of space. See Atlanta,
Infinity (The Indian mythological represen-
tation of), High numbers (The symbolic
meaning of) and Infinity.
INFINITY. Term used to represent the number
ten to the power fourteen. See Ananta and
High numbers (The symbolic meaning of).
INFINITY. [S]. Value = 0. See Infinity, Akasha,
Ananta, Vishnupada, Shunya and Zero.
INNATE REASON. As a symbol for a large
quantity. See High numbers (Symbolic mean-
ing of).
INNUMERABLE. See Abhabagamana,
Asamkhyeya, High numbers and Infinity.
INNUMERABLE. Term used as the name for
the number ten to the power 140. See
Asankhyeya.
INSIGNIFICANCE. See Low numbers,
Shunyatd and Zero.
INTERLACING. See Infinity (Indian mytho-
logical representation of) and Serpent
(Symbolism of the).
IRYA. IS]. Value = 4. “Position”. Allusion to the
four principal positions of the human body
(positions: lying flat on one’s stomach, lying
flat on one’s back, standing up or sitting
dotvn). See Four.
ISHA. [SJ. Value = 11. This is the shortened
form of Ishana, one of the names of *Rudra.
the symbolic value of which is eleven. See
Rudra-Shiva and Eleven.
iSHADRISH. [S], Value = 3. The "eyes of
Hara”. See Isha, Haranetra and Three.
ISHU. [S]. Value = 5. “Arrow”. See Shara
and Five.
ISHVARA. [S], Value = 11. “Lord of the uni-
verse”, “Supreme divinity”. One of the
attributes of *Shiva, who is an emanation of
*Rudra, whose name has the symbolic value of
eleven. See Rudra-Shiva and Eleven.
ISLAND-CONTINENT. IS]. Value = 7. See
Dvipa and Seven.
ISLAND-CONTINENTS (The four). See
Chaturdvipa.
ISLAND-CONTINENTS (The seven). See
Sapta and Dvipa.
iSVI (Calendar). See Kristabda.
J
JAGAT. [S]. Value = 3. “Universe”,
“Phenomenal world”. Here this word is taken
in the sense of three “worlds”. See Loka, Triloka
and Three.
JAGAT. [S]. Value = 14. “World”. Here the
word is taken in the sense of the fourteen
chosen lands of the Buddhism of the
Mahayana (including *Vaikuntha). See
Bhuvana and Fourteen.
JAGATf. IS]. Value = 1. “Earth”. See Prithivi
and One.
JAGATi. IS]. Value = 12. In Sanskrit poetry,
this is the metre which is made up of a verse of
four times twelve syllables. See Indian metric
and Twelve.
JAGATf. [S], Value = 48. In Sanskrit poetry,
this is the metre which is made up of a verse of
four times twelve syllables. See Indian metric.
JAHNAVIVAKTRA. [S]. Value 1,000. The
“mouths ofjahnavi”. The name Jahnavi denotes
the river Ganges (Ganga), considered to be the
daughter of Jahnu. According to legend, Jahnu
drank the river because it disturbed his prayer,
but the water came out of his ears. The Sanskrit
name for “thousand” ( *sahasra ) often means
“multiplicity” and “multitude”. The swampy
delta of this river is divided into many hun-
dreds of branches, and so these “mouths” came
to represent the quantity thousand because they
are so numerous. See Thousand and High
numbers (Symbolic meaning of).
JAINA RELIGION. See Indian religions and
philosophies.
JAINA. This is the name of an Indian religious
sect. This religion seems to have been founded
around the sixth century BCE by a “sage”
(muni) named Vardhamana, better known as
Jina. Jaina philosophy and logic is accompa-
nied by a very strict moral doctrine, born out
of several concepts including nayavada (a
highly developed science of the knowledge of
the real from its most diverse aspects) and
syadvada (which consists of a relativist vision
which is meant to adjust the affirmation and
negation of things to their moving reality).
Nature is divided into "categories”, which are
classed in different orders depending on the
point of view from which they are considered.
In one of these “categories”, there are “prin-
ciples” and “masses of beings", the most
important of which are the soul, matter, the
cause of movement, the cause of the halting of
movement and space (*dkasha). Matter is of
atomic structure. Each “atom” of corpororal
nature is uncreated, indivisible and indestruc-
tible, whilst possessing particular tastes, smells
and colours. As for time, it is considered a sub-
stance without space, yet according to Jaina
philosophy, it is made up of an infinite number
of "temporal atoms” ( *kaldnu ). These diverse
theories are accompanied by a highly devel-
oped cosmological vision of the universe
(*loka) in which the universe is represented as
a man made up of three worlds, his head form-
ing the superior world, his body the middle
world and his legs the inferior world. These
three worlds are surrounded by a triple atmos-
pheric cover, made up of air, vapour and ether
(* akasha), beyond which is nothing but empty
space (* shunyatd). This universe is organised
around a hollow vertical axis, inside which live
all "mobile” living beings.
Each world is divided into several stages:
the inferior world; the middle world, which
includes our world and the island-continents
(* dvipa, * chaturdvipa): and finally the supe-
rior world, situated above *Mount Meru, the
mythical mountain of Hindu and Brahmanic
cosmology, which is said to be the centre of
the universe where the gods live. The summit,
which constitutes the “chignon” of the cosmic
man, is said to be occupied by liberated souls.
As for the ages of the world, the Jainas accept
Brahmanic classification. Thus the fifth age
(the age which we are living in) would have
begun in 523 BCE and be characterised by
pain. This would be followed by a sixth and
last “age”, 21,000 years long, at the end of
which the human race would undergo terrible
mutations.
However, the world would not disappear,
for, according to Jaina doctrine, the universe is
indestructible. This is because it is infinite, in
terms of both time and space. It was in order to
define their vision of this impalpable universe,
situated in the unlimited and the eternal, that
the Jainas began their impressive numerical
speculations and thus created a science which
was characteristic of their way of thinking: a
“science” which, by using incredibly high num-
bers and constantly expanding the limits of
* asamkhyeya (the "incalculable”, the “impossi-
ble to count") finally allowed them to get
within reach of the world of infinite numbers.
[See Frederic, Dictionnaire (1987)]. See
Anuyogadvara Sutra, Names of numbers,
High numbers and Infinity .
JALA. [S]. Value = 4. Synonymous with *apa,
“water". This symbolism is explained by the
Brahmanic doctrine of the “elements of the rev-
elation" ( *bhuta ). According to this
philosophy, the universe is the result of the
interaction of five “powers” (nritya) personified
respectively by *Brahma, *Vishnu, *Rudra,
Maheshvara and *Shiva. These powers are: cre-
ation (shrishti), conservation (stithi), creative
emotion (tirobhava), destruction ( shangara )
and rest (anugraha). On account of these five
“powers", the universe is thus the result of the
transformation and interaction of the “five ele-
ments" (*pahchabhuta). These elements are
respectively: ether (* akasha), water (*apa), air
(*vayu), fire (*agni) and water (* prithivi). Ether,
the most subtle of the five elements, is consid-
ered to be the condition of all corporal
extension and the receptacle of all matter
which manifests itself in the form of any one of
the other four elements. Ether is thus space,
the “element which penetrates everything”, the
*shunya, the “void”. In other words, according
to this philosophy, *dkdsha (ether) is the
immobile and eternal element which is the
essential condition for all manifestation, but
which, by its very essence, is indescribable, and
cannot be mixed with any material thing. Thus
this element is not meant to participate
directly with the “material order of nature”,
which comes from *prakriti (the supposed orig-
inal material substance of the universe).
Hence we are dealing with “natural order”
which is very similar to the doctrine of the
great philosophers of Ancient Greece
(Pythagoras, Plato, Aristotle, etc.). This doc-
trine states that: the various phenomena of life
can be attributed to the manifestations of the
elements which determine the essence of the
forces of Nature, who carries out her work of
generation and destruction using these ele-
ments: water, air, fire, and earth. Each one of
these elements is created by the combination
of two primordial constituents: water comes
from coldness and humidity, air comes from
humidity and heat, fire is made by heat and
dryness, and earth comes from dryness and
cold. Each one of these is representative of a
state, liquid, gas, igneous and solid. In each of
these groups is a collection of fixed conditions
of life, and the groups together form a cycle,
which begins with the first element (water) and
ends with the last (earth), after passing
through the intermediary stages (air and fire).
This gives a quaternary order of nature, which
473
JALA DH A RAPATHA
corresponds to both the human temperaments
and to the stages of human life: winter, spring,
summer, autumn; midnight to dawn, dawn to
midday, midday to dusk, dusk to midnight;
phlegmatic, sanguine, bilious and choleric;
childhood, youth, maturity and old age; learn-
ing, blossoming, culminating, declining; etc.
[Chevalier and Gheerbrant (1982)]. It is thus
on this basis that water (Jala) came to symbol-
ise the number four in Sanskrit. This
quaternary symbolism is also responsible for
the fact that the value of four has often been
attributed to the word for “ocean” (*sdgara).
See Four and Sahara.
JALADHARAPATHA. [S]. Value = 0. “Voyage
on the water”. Allusion to *Ananta, the serpent
with a thousand heads, who floats on the pri-
mordial waters, or the “ocean of
unconsciousness”, during the space of time
which separates two succesive creations of the
world. This symbolism thus corresponds to the
identification of infinity with zero, because
Atlanta is none other than the serpent of infin-
ity and eternity. See Zero.
JALADHI. [S]. Value = 4. “Sea". See Sagara,
Four, Ocean.
JALADHI Name given to the number ten to
the power fourteen (= a hundred billion). See
Names of numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Source: * Li I avail by Bhaskaracharya (1 ISO CE).
JALANIDHI. [S]_ Value = 4. “Sea”. See Sagara,
Four. See also Ocean.
JAMBUDVIPA. “Isle of the Jambu tree”. Name
in Hindu cosmology for the whole of the
Indian subcontinent, w'hich is situated (accord-
ing to a characterised representation of the
structure of the universe) to the south of
*Mount Meru.
JAVANESE NUMERALS (Ancient). See Kawi
numerals.
JAVANESE NUMERALS (Modern). Currently
in use in the island ofjava, in Bali, Madura and
Lombok, as well as in the Sounda islands. The
corresponding system functions according to
the place-value system and possesses zero (in
the form of a little circle). Apart from the
numerals 0 and 5 (whose graphical origin is
evident), this notation actually corresponds to
a relatively recent graphical creation, the shape
of the numerals having (curiously) become
similar in appearance to some of the letters of
the contemporary Javanese alphabet. The
Javanese people formerly used a notation
which derived from Brahmi numerals, which
belongs to the group of numerals know'n as
“Pali”. See Fig. 24.26 and 52. See also Indian
written numeral systems (Classification of)
and Kawi numerals.
JEWEL. [S]. Value = 3. See Ratna and Three.
JEWEL. IS]. Value = 5. See Ratna and Five.
JEWEL. IS]. Value = 8. See Mangala and Eight.
JEWEL. IS]. Value = 9. See Ratna and Nine.
JEWEL. [S]. Value = 14. See Ratna and
Fourteen.
JINA. Name of the founder of the religious sect
of the *Jainas.
JINABHADRA GANI. Indian arithmetician
who lived at the end of the sixth century. His
works notably include Brihatkshctrasamdsa,
where he gives an expression for the number
224,400,000,000 in the simplified Sanskrit
system using the place-value system (see Datta
and Singh (1938) p. 79]. See Indian mathe-
matics (The history of).
JVALANA. [S]. Value = 3. “Fire”. See Agni and
Three.
JYOTISHA. Sanskrit name attributed to
astronomy, once it was considered to be a sepa-
rate discipline from arithmetic and calculation.
This name, how'ever, (which literally means
“science of the stars”) w'as long attributed to
astrology. See Indian astrology, Ganita and
Indian astronomy (The history of).
JYOTISHAVEDANGA. “Astronomical Element
of Knowledge”. Name of an ancient text on
astrology, notably concerning the determina-
tion of the exact dates of the sacrifices of the
Brahman cult [see Billard (1971)]. See Jyotisha,
Indian astrology and Indian astronomy (The
history of).
K
KACHCHAYANA. Grammarian from Sri
Lanka who is believed to have written the
Vydkarana, a Pali grammar divided into eight
parts. He probably lived during the eleventh
century CE. Here is a list of the principal names
of numbers mentioned in this w'ork:
*Koti ( = 10 7 ), *Pakoti (= 10 H ), * Kotippakoti (=
10 21 )> *Nahuia (= 10 28 ), *Ninnahuta (= 10 35 ),
*Akkhobhini (= 10 42 ), *Bindu (= 10 49 ), *Abbuda
(= 10 56 ), *Nirabbuda (= 10 63 ), *Ahaha (= 10 7t> ),
* Ababa (=10 77 ), *Atata (= 10 84 ), * Sogandhika
(= 10 91 ), *Uppala (= 10 98 ), *Kumuda (= 10 105 ),
*Pundarika (= 10 112 ), *Paduma (= 10 119 ),
*Kathdna (= 10 126 ), *Mahdkathana (= 10 u3 ),
*Asankhyeya (= 10 140 ).
See Names of numbers and High numbers
Source: Vydkarana by Kachchayana [see JA, 6th
Series, XVII, 1871, p. 411. line 51-52).
KAITHi NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Gupta,
Nagari, Kutila and Bengali numerals. Currently
in use in Bihar state, in the east of India, and
sometimes used in Gujarat state. The corre-
sponding system functions according to the
place-value system and possesses zero (in the
form of a little circle). See also Fig. 24.9, 52
and 61 to 69. See Indian written numeral sys-
tems (Classification of).
KAKUBH. [S]. Value = 10. “Horizon". See Dish
and Ten.
KALA. [ S] . Value = 3. “Time”. In Brahman
mythology, time is personified by the terrible
Kala, Lord of Creation and Destruction. He is
often identified as Shiva holding his Trident
( *lrishuld ), w'hich symbolises the three aspects
of the revelation (creation, preservation,
destruction), as well as the three primordial
properties ( *guna ) and the three states of con-
sciousness. Here, the word is synonymous with
*trikala, “three times”. See Guna, Shula,
Triguna and Three.
KALACHURI (Calendar). See Chhedi.
KALAMBA. [S]. Value = 5. “Arrow”. See Shara
and Five.
KALANU. “Temporal atom”. In *Jaina philoso-
phy, time (*kala) is made up of an infinite
number of temporal atoms (atom = *anu).
KALASAVARNA. Word used in arithmetic to
denote “fundamental operations” carried out
on fractions. See Parikarma.
KALIYUGA (Calendar). Calendar of fictitious
times, which is sometimes referred to in Hindu
religious texts and Indian astronomical texts. It
begins on the 18 February of the year 3101
BCE. Characteristically, the beginning of this
calendar is traditionally related to a theoretical
starting point of celestial revolutions corre-
sponding to a supposed general conjunction in
average longitude w ith the starting point of the
sidereal longitudes of the sun, the moon and
the planets (the ascending apogees and node of
the moon being respectively at 9(T and 180° of
these longitudes). To find the corresponding
date in our calendar, simply subtract 3,101
from a date in the Kaliyuga calendar. See
Kaliyuga, Indian calendars and Yuga
(Astronomical speculation).
KALIYUGA. Name of the last of the four
cosmic calendars w'hich make up a * m ah ay uga.
This cycle, said to be 432,000 human years
long, is the “iron age”, during which living
things only live for a quarter of their existence
and the forces of evil triumph over good: we
are living in this age now', and it is meant to
end with a pralaya (destruction by fire and
water). See Yuga (Definition), Yuga (Systems
of calculating) and Yuga (Cosmogonical spec-
ulations about).
KALPA. Unit of cosmic time which, according
to Indian speculations, corresponds to the
length of 1,000 *mahayugas. Thus one Kalpa
corresponds to 4,320,000,000 human years.
See Yuga (Definition).
KALPA. [Astronomy]. According to
Brahmagupta (628 CE), the kalpa cycle, or
period of 4,320,000,000 years, is delimited by
two perfect conjunctions in real longitude of the
totality of elements, each one accompanied by a
total eclipse of the sun at exactly six o’clock in
*Ujjayini. See Kalpa (first entry above) and Yuga
(Astronomical speculations about).
KALPAS (Cosmogonical speculations about).
In Buddhist cosmogony, the term kalpa
denotes an infinite length of time. The kalpa is
made up of four periods: the creation of
worlds, the lifespan of existing worlds, the
destruction of worlds and the duration of the
existence of chaos. During the period of cre-
ation the different universes are formed with
their living beings. The second period sees the
appearance of the sun and the moon, the dif-
ferentiation between the sexes and the
development of social life. During the phase of
destruction, fire, water and w'ind destroy every-
thing apart from the fourth dhydna. Chaos
represents total annihilation. These four
phases make up one “big" kalpa (* mahakalpa)\
Each one of them is made up of twenty “little”
kalpas, which themselves are broken dow n into
fire, bronze, silver and golden ages. During the
entire creation phase of a “little” kalpa , the life
expectancy of humans increases by one year
per century until it reaches 84,000 years. In a
parallel fashion, the human body grows to a
height of 84,000 feet. During the “little” kalpa s
period of disappearance, which is made up of
successive phases of plague, war and famine,
human life is shortened to ten years and the
human body returns to the height of one foot.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
474
[This article is taken from the Dictionnaire
dc la sagesse orient ale, Friedrichs, Fischer-
Schreiber, Erhard and Diener (1989)]. See
Kalpa (First entry) and Day of Brahma.
KAMA. [S J. Value = 13. Name of the Hindu
divinity of Cosmic Desire and Carnal Love
whose action decides the laws of the reincarna-
tion of living beings {*samsara). Kama presides
over the thirteenth lunar day. See Thirteen and
Pahchabdna.
KAMALAKARA. Indian astronomer of the
seventeenth century CE. His works notably
include Siddhantatattvaviveka, in which the
place-value system with Sanskrit numerical
symbols is frequently used [see Dvivedi
(1935)]. See Numerical symbols, Numerical
symbols (Principle of the numeration of),
and Indian mathematics (The history of).
KANKARA. Name given to the number ten to
the power thirteen (= ten billion). See Names
of numbers and High numbers.
KANNARA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi and Bhattiprolu
numerals. Currently used by the Dravidians of
Karnataka state and part of Andhra Pradesh.
They are also called Kannada (or even Karnata)
numerals. The corresponding system today
uses the place-value system and zero (in the
from of a little circle). For ancient numerals,
see Fig. 24.48. For modern numerals, see
Fig. 24.21. See also Fig. 24.52 and 24.61 to 69.
See Indian written numeral systems
(Classification of).
KARA. [S], Value = 2. “Hand”. This is because
of the symmetry of the two hands. See Two.
KARAHU. Name given to the number ten to
the power thirty-three. See Names of numbers
and High numbers.
Source: *Lalitavistara Sutra (before 308 CE).
KARANA. Name of the astronomical formula
employing, for example, in the workings of real
longitudes, the interpolation - generally linear
- of tabulated values. See Indian astronomy
(The history of) and Yuga (Astronomical
speculation on).
KARANAPADDHATI. See Putumanasomayajin.
KARANIYA. [SJ. Value = 5. “That which must
be done". This refers to the five major obser-
vances of *Jaina religion, which constitute the
basic rules of their philosophy: not to harm
living beings {ahimsa); not to be false (sunrita);
not to steal (asteya); carnal discipline (j brah -
machdrya); and detachment from earthly
possessions (aparigraha). See Five.
KARNATA NUMERALS. See Kannara
numerals.
KARNIKACHALA. One of the names of
* Mount Meru. See Adri, Dvipa, Purna, Patala,
Sdgara, Pushkara , Pdvana and Vdyu.
KARTTIKEYA. Hindu divinity of war and the
planet Mars, son of Shiva, often likened
to *Kumara.
KARTTIKEYASYA. [S]. Value = 6. “Faces of
*Karttikeya”. Allusion to the six heads of this
divinity. See Six.
KATAPAYA (Spoken numeration). See
Katapayadi numeration.
KATAPAYADI NUMERATION. Method of
writing numbers using the letters of the
Indian alphabet. In this system, the numerical
attribution of of syllables corresponds to
the following rule, according to the regular
order of succession of the letters of the Indian
alphabet (see Fig. 24. 56): the first nine letters
( ka , kha, ga, gha, na, cha, chha, ja and jha )
represent the numbers 1 to 9, whilst the tenth
{na) corresponds to zero; the following nine
letters ( ta , tha, da, dha, na, ta, tha, da, dha )
also receive the values 1 to 9, whilst the
following letter {na) has the value of 0; the
next five {pa, pha, ba, bha, ma) represent
the first five units; and the last eight {ya, ra,
la, va, sha, sha, sa and ba) represent the
numbers 1 to 8.
Thus each simple unit is represented by
two, three or four different letters: numeral 1
by one of the letters ka, ta, pa or ya (hence kat-
apaya is the name of the system); 2 by kha, tha,
pha or ra\ 3 by ga, da, ba, la; 4 by gha, dha, bha
or va; 5 by na, na, ma or sha; 6 by cha, ta or sha;
7 by chha, tha or sa; 8 by ja, da or ha; 9 by jha
or dha; and 0 by ha or na. This system is infi-
nitely simpler than Aryabhata’s.
The complete key is given in the following
lines, which are an extract from Sadratnamald
by Shankaravarman: Nahdvachashacha shuydni
Samkhya katapayadayah
Mishre tupdnta hal samkhya Na cha chinty-
ohalasvarah
Translation: "[The letter] na and a, as well
as the vowels, are zero. [The letters] starting
with ka, ta,pa,ya, represent the numbers [from
1 to 9). When two consonants are joined, only
the last one corresponds to a number. And a
consonant which is not joined to a vowel is
insignificant.” [See El, VI, p. 121; Datta and
Singh, p. 70]. In other words, in this system,
the vowels and the consonants which are not
vocalised have no numerical value; and groups
such as ksha, tva, ktya, etc., often considered as
unitary in Indian alphabets, receive respec-
tively the same value as the letters sha, va,ya,
etc. The letters ha and na, represent zero. Thus
the vocalised consonants are the only “numer-
als” in the system, their numerical value being
entirely independent of the vocalisations in
question. This means that, unlike Aryabhata’s
system, there is no difference between syllables
such as sa, si, su, se, so, sai, etc. In fact, this
system constitutes a simplification of Aryab-
hata’s alphabetical numeration. See Aryabhata
and Aryabhata’s numeration.
Historically, the first author who is known
to have used this system employing the
name of katapayadi is the astronomer
Shankaranarayana, author of a work entitled
iaghubhahaskariyavivarana written in 869 CE.
This date is given by the author himself, and is
expressed as the *Shaka year 791, which is 791
+ 78 = 869 CE.
However, the latter did not invent kata-
payddi, because the system had already
appeared, under the name of varnasamjhd,
“from syllables”, in Grahacharanibandhana by
the astronomer Haridatta, for which there is
overwhelming evidence to suggest that he was
the inventor of this system. First, there is no
mention is made of the system by his predeces-
sors; secondly, in his work (where he makes
frequent reference to Aryabhata), he takes the
trouble to give all the details (like the inventor
of a new system who feels obliged to explain it
to readers who are used to using a different
method); finally, it is Haridatta who is the first
and last person to explain the system, which
suggests that afterwards it became common
knowledge. [Personal communication of
Billard]. According to a tradition in Kerala,
Haridatta wrote his text in 684 CE [see Sarma
(1954), p. v]. However, this date does not
seem to correspond to a significant piece of
evidence found in the work of astronomer
Shankaranarayana, where he is paying homage
LETTERS USED
for the numeral 1
SR
z
tr
Zf
ka
ta
pa
ya
for the numeral 2
t=T
z
■err
\
kha
tha
pha
ra
for the numeral 3
IT
1
K a
da
ba
la
for the numeral 4
'El
s
TT
=T
gha
dha
bha
va
for the numeral 5
Z
R
ST
ha
na
ma
sha
for the numeral 6
=ET
FT
tr
cha
ta
sha
for the numeral 7
$
ST
TT
chha
tha
sa
for the numeral 8
TT
5
*
M
da
ha
iTi
jha
dha
for the numeral 0
«T
ha
na
fig. 2 4 d . 7 Letter-numerals of the Katapayadi system ”. Ref. : Datta and Singh ( 1938); Jacquet
(1835); Pihan (1860); Renou and Filliozat (1953); Sarma (1954)
475
KATAPAYADI NUMERATION
to his illustrious predecessors, and quotes their
respective names, using the word *kramad,
which means “in the order":
1. Aryabhata [c. 510 CEJ
2. Varahamihira [c. 575 CE]
3. Bhaskara [c. 629 CE]
4. Govindasvamin [c. 830 CE]
5. Haridatta.
Thus Haridatta is placed after Govindasvamin, of
whom Shankaranarayana was a disciple. Such a
list is very rare for an Indian scholar; chronology
was not generally of much interest to them. It
seems even more remarkable in light of the fact
that Indian astronomical texts are usually very
sparing with historical facts, and it is very rare to
find a reference to another text. If mention is
made of earlier authors, the list is usually in
order, to aid the rhythm of the versification. This
is the only known example of such a list accompa-
nied by a chronological indication. In short, if
Haridatta ’s work was written before 869 (the date
of Shankaranarayana’s text), it probably dates
back to c. 850 CE. (Personal communication of
Billard.) This means that katapayadi numeration
was not created until the middle of the ninth cen-
tury, three centuries after Aryabhata. Through
abandoning the method of successively vocalis-
ing the consonants of the Indian alphabet, and
replacing each value which was equal to or higher
than ten with a zero or one of the nine numerals,
the inventor of the katapayadi system trans-
formed Aryabhata’s system into a place-value
system equipped with a zero.
The proof of this is in the following mention
in an anonymous text from the tenth century,
where there is frequent use of the katapayadi
notation: vibhavonashakdbdam . . .
"The Shaka date decreased by 444 . . Ref.:
Grahachdranibandhanasamgraha , line 17;
Billard (1971), p. 142.
This mention contains the expression
vibhavona (= vibhava + una ) which means “444
(= vibhava) decreased by". Bearing in mind the
principle of this notation, where the value of a
consonant is independent of its vocalisation,
and where the numbers are expressed in
ascending order starting with the smallest
u nits, the number 444 is written as follows:
vibhava (= va.bha.va).
According to the values of the numeral letters
ln the katapayadi system, this gives the follow-
ing (Fig. D. 7):
(va) (bhd) (va) = 4 + 4 x 10 + 4 X 10 2
4 4 4 =444.
Thus the numeral letters are combined and
are never modified by vowels; these can be
inserted wherever necessary, as they have no
numerical value. As for the principle of the
notation, which is the rule of position applied
to any of the nine letter-units and the two
letter-zeros, it follows the ascending powers of
ten starting with the smallest unit, as it does
with the numerical symbols. Here is another
example found in an astronomical table of
Haridatta's Grahachdranibandhana (II, 14),
giving a trigonometric function for Saturn Isee
Sarma(1954), p. 12]:
dhanadhya dha.nd.dhya
= dhiradhya = dhi.rd.dhya
= dha.na .ya = dha.ra.ya
9 0 1 9 2 1
= 109 = 129
This is more proof that Aryabhata was fully
aware of zero and the place-value system, but
by confining himself as he did to his system of
vocalisation, he made it impossible for his
numeration to be positional. (See Ankanam
vamato gatih.)
It is surprising to note the numbers of let-
ters that could be used to record the same
numeral. In fact, this system, like the notation
which inspired it, offered many possibilities to
the mnemonics of numbers. Moreover, it was
perfectly capable of meeting the needs of the
rules of versification or prosody, with the
advantage of being especially useful when
reproducing abundant tables of trigonometric
functions in a much shorter form than the
system of Numerical symbols. Added to the
possibility of expressing a given numeral with
many different letters was the ability to
vocalise these letters without changing the
values they expressed. Thus it was always pos-
sible to find several intelligible words to
express a number. This is doubtless the reason
why this system came to be used, in various
forms, in southern India (the notation in this
case being applied to letters of the Grantha,
Tulu, Telugu (etc.) alphabets).
KATHAKA SAMHITA. Text derived from the
Yajurveda “black". It figures amongst the texts
of Vedic literature. Passed down through oral
transmission since ancient times, it only found
its definitive form at the beginning of
Christianity. See Veda. Here is a list of the main
names of numbers mentioned in this text:
*Eka (= 1), * Dasha (= 10), *Sata (= 10 2 ),
*Sahasra (= 10 3 ), *Ayuta (= 10 4 ), *Prayuta (=
10 5 ), *Niyuta (= 10 6 ), *Arbuda (= IQ 7 ),
*Nyarbuda (= 10 8 ), *Samudra (= 10 9 ), *Vadava
(= 10 9 ), *Madhya (= 10 10 ), *Anta (= 10 11 ),
*Parardha (=10 12 ).
See Names of numbers and High numbers.
Ref.: Kathaka Samhita, XVII, 10 [see Datta
and Singh (1938), p. 10],
KATHANA. Name given to the number ten to
the power 119 See Names of numbers and
High numbers.
Ref.: *Vyakarana (Pali grammar) by
Kachchayana (c. eleventh century).
Gutturals
IT
3?
ka
kha
g°
gha
ha
Aryabhata's system 1
i
2
3
4
5
Katapayadi system 2
i
2
3
4
5
Palatals
=5r
$
TT
;n
cha
chha
) a
jha
ha
Aryabhata’s system
6
7
8
9
10
Katapayadi system
6
7
8
9
0
Cerebrals
7
7
T
5
TTT
ta
tha
da
dha
na
Aryabhata’s system
n
12
13
14
15
Katapayadi system
i
2
3
4
5
Dentals
n
ST
5
R
R
ta
tha
da
dha
na
Aryabhata’s system
16
17
18
19
20
Katapayadi system
6
7
8
9
0
Labials
tr
TK
R
R
R
pa
pha
ba
bha
ma
Aryabhata’s system
21
22
23
24
25
Katapayadi system
1
2
3
4
5
Semivowels
R
1
C5
ya
ra
la
va
Aryabhata’s system
30
40
50
60
Katapayadi system
1
2
3
4
Sibilants
ST
TT
R
sha
sha
sa
Aryabhata’s system
70
80
90
Katapayadi system
5
6
7
Aspirates
ha
Aryabhata’s system
100
Katapayadi system
8
1. See Fig. D.2, p. 448
2. See Fig. D.7, p. 474
FIG. 24D.8
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
476
KAVACHA. Literally “Charm, armour”. This is
the name for Tantric talismans and amulets.
See Numeral alphabet, magic, mysticism and
divination.
KAWI NUMERALS. Signs derived from
‘Brahmi numerals, through the intermediary
ofShunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali” and
Vatteluttu numerals. Formerly used (since
the seventh century CE) in Java and Borneo.
These are the numerals of Old Javanese
writing. The corresponding system uses
the place-value system and zero (in the form
of a little circle). See Fig. 24.50, 52, 61 to
69 and 80. See also Indian written numeral
systems (Classification of).
KAYA. IS]. Value = 6. "Body”. Allusion to the
*trikdya, the “three bodies” that a Buddha can
assume simultaneously, and which are often
associated with the “three spheres” of Buddhas’
existence. Thus the symbolic sum: 3 + 3 = 6.
See Six.
KESHAVA. [S]. Value = 9. This concerns one of
the epithets of *Vishnu (and ‘Krishna). The
symbolism is due to the fact that keshava is
another name for the month of margashirsha,
the ninth month of the *chaitradi year.
See Nine.
KHA. IS]. Value = 0. Word meaning “space”.
This symbolism is explained by the fact that
space is nothing but a “void". See Shunya
and Zero.
KHACHHEDA. Sanskrit term used to denote
infinity. Literally “divided by zero” (from *kha,
“space” as a symbol for zero, and chheda, “the
act of breaking into many parts”, “division”).
Thus it is the “quantity whose denominator is
zero”. The term is used notably in this sense by
‘Brahmagupta in his Brahmasphutasiddanta
(628 CE). See Chhedana, Infinity (entries
beginning with), Zero and Indian mathemat-
ics (The history of).
KHAHARA. Sanskrit word for infinity.
Literally "division by zero”. Notably used by
‘Bhaskaracharya. See Khachheda.
KHAMBA. Name given to the number ten to
the power thirteen (= ten billion). See Names
of numbers and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
KHAROSHTHI ALPHABET. See Fig. 24. 28.
KHAROSHTHI NUMERALS. Numerals
derived from the numerical notations of west-
ern Semitic civilisations. This is attested
notably in the edicts of Asoka written in
Aramaean Indian. The corresponding system
does not use the place-value system, nor does it
possess zero. See Indian written numeral sys-
tems (Classification of). See Fig. 24.54.
KHARVA. Name given to the number ten to
the power ten (ten thousand million). See
Names of numbers and High numbers.
Sources: *Kitab fi tahqiq i ma li'l hind by al-Biruni
(c. 1030 CE); *Lilavati by Bhaskaracharya (1150
CE); *Ganitakaumudi by Narayana (1350 CE);
'Trishatika by Shridharacharya (date unknown).
KHARVA. Name given to the number ten to
the power twelve (= one billion). See Names of
numbers and High numbers.
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
KHARVA. Name given to the number ten to
the power thirty-nine. See Names of numbers
and High numbers.
Source: *Rdmayana by Valmiki (in the first
centuries of the Common Era).
KHMER NUMERALS. For modern numerals,
see Thai-Khmer numerals. For ancient numer-
als, see Old Khmer numerals.
KHUDAWADI NUMERALS. Signs derived
from ‘Brahmi numerals, through the interme-
diary of Shunga, Shaka, Kushana, Andhra,
Gupta and Sharada numerals, and constituting
a slight variation of Sindhi numerals. Once
used by the merchants of Hyderabad (a town of
Sind, built on the delta of the Indus, to the east
of Karachi, not to be confused with the other
Hyderabad, capital of Andhra Pradesh). The
corresponding system functions according to
the place-value system and possesses zero (in
the form of a little circle). See Indian written
numeral systems (Classification of) and
Fig. 24.6, 52 and 61 to 69.
KING. [SI. Value = 16. See Bhupa, Nripa
and Sixteen.
KITAB FI TAHQIQ I MA LI’L HIND. Arabic
work by al-Biruni, which constitutes one of the
most important pieces of evidence about
Indian civilisation at the beginning of the
eleventh century. See al-Biruni.
KOLLAM (Calendar). Beginning in 825 CE,
created by the sovereign of the town of the
same name situated in Kerala near to
Travancore, on the Malabar coast. To find the
corresponding date in the Common Era, add
825 to a date expressed in Kollam years. This
calendar is also called Parashurama. It is rarely
used. See Indian calendars.
KOTI. Name given to the number ten to the
power seven (= ten million). See Names of
numbers and High numbers.
Sources: *Ramdyana by Valmiki (in the first cen-
turies CE); *Lalitavistara Sutra (before 308 CE);
* Aryabhatiya (510 CE); * Ganitasarasamgraha by
Mahaviracharya (850 CE); *Kitab fi tahqiq i ma li’l
hind by al-Biruni (c. 1030 CE); * Vyakarana (Pali
grammar) by Kachchayana (eleventh century CE);
*Lildvati by Bhaskaracharya (1150 CE);
*Ganitakaumudi by Narayana (1350 CE); * Trishtika
by Shridharacharya (date unknown).
KOTIPPAKOTI. Name given to the number
ten to the power twenty-one (= quintillion).
See Names of numbers and High numbers.
Source: * Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
KRAMAD (KRAMAT). Word meaning “in the
order”. See Sthdna, Sthanakramad and
Ankakramena.
KRISHANU. [S]. Value = 3. “Fire”. See Agni
and Three.
KRISHNA. S eeAvatdra.
KRISTABDA (Calendar). Name given to the
Christian calendar. It is also referred to as isvi
or imraji. See Indian calendars.
KRITA. IS]. Value = 4. The name of the first of
four cosmic cycles (*kritayuga) which make up
a *chaturyuga (or *mahayuga) in Brahman cos-
mogony. The symbolism is not due to the fact
that the kritayuga was the “first age" of the
world, but because it inaugurated a new chatu-
ryuga. Thus it began a new cosmic cycle
composed of four periods corresponding to the
life of a universe. See Yuga and Four.
KRITAYUGA. Name of the first of the four
cosmic eras which make up a *mahayuga (or
*chaturyuga). This cycle, said to last 1,728,000
human years, is regarded as the “golden age”
during which humans have an extremely long
life and everything is perfect. See Yuga.
KRITI. [S]. Value = 20. In Sanskrit poetry, this
is a metre of four times twenty syllables. See
Indian metric and Twenty.
KSHAPESHVARA. [S]. Value = 1. “Moon". See
Abja and One.
KSHATRAPA NUMERALS. Signs derived
from ‘Brahmi numerals, through the interme-
diary of Shunga, Shaka and Kushana numerals.
Contemporaries of the western Satraps (second
to fourth century CE). The corresponding
system did not function according to the place-
value system and moreover did not possess
zero. See Indian written numeral systems
(Classification of). See also Fig. 24.35, 52,
24.61 to 69 and 70.
KSHAUN1. IS]. Value = 1. “Earth”. See Prithivi
and One.
KSHEMA. [S]. Value = 1. “Earth”. See Prithivi
and One.
KSHETRAGANITA. Term used in early times
meaning geometry. See Ganita.
KSHITI. Literally “earth”. Name given to the
number ten to the power twenty (= a hundred
quadrillions). See Names of numbers. For an
explanation of this symbolism, see High num-
bers (Symbolic meaning of).
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
KSHITI. [SI. Value = 1. “Earth”. See Prithivi
and One.
KSHOBHA. Name given to the number ten to
the power twenty-two (= ten quintillions). See
Names of numbers and High numbers.
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
KSHOBHYA. Literally “Movement". Name given
to the number ten to the power seventeen. This
name might have been attributed to such a high
number because of the “endless movement” of
the waves, since "ocean” ( *samudra , *jaladhi ) was
also sometimes used to represent large quanti-
ties. See Names of numbers and High numbers.
Source: * Lalitavistara Sutra (before 308 CE).
KSHONI. Literally “earth". Name given to the
number ten to the power sixteen. See Names of
numbers. For an explanation of this symbolism,
see High numbers (Symbolic meaning of).
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
KSHONI. [S]. Value = 1. “Earth”. See Prithivi
and One.
KU. [S]. Value = 1. “Earth”. See Prithivi
and One.
KUMARA. See Kumarasya, Kumaravadana
and Karttikeya.
KUMARASYA. [S]. Value = 6. “Faces of
‘Kumara”. Allusion to the six heads of
‘Karttikeya. See Kumara and Six.
KUMARAVADANA. [S]. Value = 6. “Faces of
‘Kumara”. Allusion to the six heads of
‘Karttikeya. See Kumara and Six.
KUMUD. Literally “(pink-white) lotus”. Name
given to the number ten to the power twenty-
one (= quintillion). See Names of numbers.
For an explanation of this symbolism, see High
numbers (Symbolic meaning of).
Source: * Lalitavistara Sutra (before 308 CE).
477
KUMUDA
KUMUDA. Literally “pink-white lotus". Name
given to the number ten to the power 105. See
Names of numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Source: *Vydkarana (Pali grammar) by
Kachchayana (eleventh century CE).
KUNJARA. [S], Value = 8. “Elephant". See
Diggafe and Eight.
KUSHANA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga and Shaka numerals.
Contemporaries of the Kushana dynasty (first
to second century CE). The corresponding
system did not function according to the place-
value system and moreover did not possess
zero. See Indian written numeral systems
(Classification of). See also Fig. 24.33, 52,
24.61 to 69 and 70.
KUTILA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary of
Shunga, Shaka, Kushana, Andhra, Gupta and
Nagari numerals. Formerly used in Bengal and
Assam. The corresponding system was based on
the place-value system and possessed zero (in
the form of a little circle). These numerals were
the ancestors of Bengali, Oriya, Gujarati, Kaithi,
Maithili and Manipuri numerals. See Indian
written numeral systems (Classification of).
See Fig. 24.52 and 24.61 to 69.
KUTTAKAGANITA. In algebra, the name
given to the part related to the analysis of inde-
terminate equations of the first degree. See
Indian mathematics (The history of).
L
LABDHA. Term used in arithmetic to denote
the quotient of a division. Synonym: labdhi.
See Bhdgahdra, Chhedana and Shesha.
LAGHUBHASKARIYAVIVARANA. See
Shankaranarayana.
LAKH. Name given to the number ten to the
power five (= a hundred thousand). See Names
of numbers and High numbers.
Source: * ialitavistara Sutra (before 308 CE).
LAKKHA. Name given to the number ten to
the power five (= a hundred thousand). See
Names of numbers and High numbers.
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
LAKSHA. Name given to the number ten to
the power five (= a hundred thousand). See
Names of numbers and High numbers.
Source: *Ganitasarasamgraha by Mahaviracharya
(850 CE); *Kitab fi tahqiq i ma li'l hind by al-Biruni
(c. 1030 CE): *I.ilavati by Bhaskaracharya (1150
CE); *Ganitakaumudi by Narayana (1350 CE);
*Trishtikd by Shridharacharya (date unknown).
LAKSHAMANA (Calendar). This calendar
begins in the year 1118 CE. To find the corre-
sponding date in the Common Era, add 1118 to
the date expressed in the Laksharnana calendar.
Formerly used in the region of Mithila (north
of Bihar). See Indian Calendars.
LALITAVISTARA SUTRA. “Development of
games'. Sanskrit text on the Buddhism of the
Mahayana, written in verse and prose, about
the life of Buddha, as he is said to have
recounted it to his own disciples, where there is
constant reference to numbers of gigantic pro-
portions. This text is in fact a relatively recent
compilation of ancient stories and legends. It is
clearly later than the *Vdjasaneyi Samhita
(written at the start of the Common Era) but
not later than the beginning of the fourth cen-
tury, because the Lalitavistara Sutra was
translated into Chinese by Dharmaraksha in
the year 308 CE. Here is a list of some of the
names of high numbers mentioned in the text:
Lakh (= 10 s ), *Koti (=10 7 ), *Nahut (= 10 9 ),
*Ninnahul (= 10 11 ), *Khamba (= 10 13 ),
*Vi skhamba (= 10 15 ), *Abab (= 10 17 ), * Attala
(= 10 19 ), *Kumud (= 10 21 ), *Gundhika (= 10 23 ),
*Utpala (= 10 25 ), *Pundarika (= 10 2/ ), *Paduma
(= 10 29 ).
Here is another list of high numbers men-
tioned in the text (legend of Buddha):
*Ayuta 10 9 ), *Niyuta (= 10 11 ), *Kankara
(= 10 13 ), *Vivara (= 10 15 ), *Kshobhya (= 10 17 ),
*Vivaha (= 10 19 ), *Utsanga (= 10 21 ), *Bahula
(= 10 23 ), *Ndgabala (= 10 25 ), *Titilambha
(= 10 27 ), *Vyavasthanaprajnapati (= 10 29 ),
*Hetuhila (= 10 31 ), *Karahu (= 10 33 ),
*Hetvindriya (= 10 35 ), *Samaptalambha (= 10 37 ),
*Gananagati (= 10 39 ), *Niravadya (= 10 41 ),
*A iudrabala (= 10 43 ), *Sarvabala (= 10 45 ),
*Visamjnagati (= 10 47 ), *Sarvajna (= 10 49 ),
*Vibhutangama (= 10 51 ), *Tallakshana (= 10 53 ),
* Dhvajagravati (= 10 99 ), * Dh va jagranishdma n i
(= 10 145 ), etc.
See Names of numbers and High num-
bers. [See Lai Mitra (1877); Datta and Singh
(1938), pp. 10-11; Woepcke (1863)].
LALLA. Indian astronomer who lived in the
ninth century CE. His works notably include an
interesting astronomical text entitled
Shishyadhivriddhidatantra, in which there is
abundant usage of the place- value system
recorded by means of ‘Sanskrit numerical sym-
bols [see Billard (1971), p. 10]. See Numerical
symbols (Principle of the numeration of), and
Indian mathematics (The history of).
LAUKIKAKALA (Calendar). See
Laukikasamvat.
LAUKIKASAMVAT (Calendar). Beginning in
3076 BCE, this calendar was formerly used in
Punjab and Kashmir. To find the correspond-
ing date in the Common Era, take away 3076
from a date expressed in the Laukikasamvat cal-
endar. This calendar also goes by the names of
Laukikakala, Lokakala, Saptarishikdh, etc. See
Indian calendars.
LEGEND OF BUDDHA. See Buddha (Legend
of) and Lalitavistara Sutra.
LILAVATL “The (female) player”. Name of a
mathematical work from the twelfth century
CE written in a highly poetic style. See
Bhaskaracharya and Indian mathematics
(The history of).
LINGA (LINGAM). Literally “sign". Erected
stone, in the shape of a prism or cylinder, phal-
lic in appearance, which, in Hinduism,
represents the universe and fundamental
nature, complement of the *yoni, the “feminine
vulva”, which is symbolised by a stone lying on
its side and represents manifest energy [see
Frederic, Diction naire (1987)].
LOCHANA. [S3. Value = 2. The (two) “eyes”.
See Two.
LOKA. [S], Value = 3. “World”. Division of the
Hindu universe. There are three loka : the earth
( bhurloka ), the space between the earth and the
sun ( bhuvarloka ), and the space between the
sun and the pole star ( svarloka ). In Buddhism,
there are also three lokas and these represent
the “spheres" of existence which make up the
universe: kamaloka (the world of sensations).
rupaloka (world of shapes or forms), and aru-
paloka (the formless, immaterial world) [see
Frederic Dictionnaire (1987)]. See Triloka and
Three.
LOKA. IS]. Value = 7. “World”. Here the allu-
sion is to another classification which tells of
the existence of seven superior worlds: bhurloka
(the earth); bhuvarloka (the space between the
earth and the sun, supposedly the home of the
*muni, the *siddha, etc.); the svarloka (the sky
of *Indra); maharloka (where Bhrigu and many
other “saints" are said to reside); janaloka (the
land of the three sons of ‘Brahma); taparloka
(home of the vairaja)', and satyaloka or brah-
maloka (the domain of Brahma). These seven
superior worlds defend themselves against
seven *pdtdla (“inferior worlds”) [see Frederic
Dictionnaire, (1987)]. See Seven.
LOKA. [S]. Value = 14. “world”. See Bhuvana
and Fourteen.
LOKAKALA (Calendar). See Laukikasamvat.
LOKAPALA. IS]. Value = 8. “Guardian of the
horizons". In Hindu mythology, this is the
name of the eight divinities who are guardians
of the eight “horizons” and the eight points of
the compass, who are represented as warriors
in armour riding elephants. See Diggaja,
Dikpala, Dish and Eight.
LOKAVIBHAGA. “Parts of the Universe”. A
‘Jaina cosmological text which possesses the
very exact date of Monday, August 25th of the
year 458 CE in the Julian calendar. It is the
oldest Indian text known to be in existence
which contains zero and the place-value system
expressed in numerical symbols [see Anon.
(1962), chapter IV, line 56, p. 79]. See Anka,
Ankakramena, Sthdna, Sthanakramad and
Ankasthdna.
LORD OF THE UNIVERSE. [S]. Value = 11.
See Ishvara, Rudra-Shiva and Eleven.
LOTUS. This flower is the most famous symbol
in all of Asia. It symbolises the pure spirit leav-
ing the impure vessel of the body. It is the very
image of divinity, which remains intact and is
never soiled by the troubled waters of this
world. A whole symbolism has developed
around the lotus, according to its colour, the
number of petals, and whether it is open, fresh-
blown or in bud [see Frederic, Le Lotus (1987)].
Thus it is not surprising that Indian arithmetic
is full of related vocabulary and that such sym-
bolism was often used to express very high
numbers. In many texts, the words *padma,
*paduma, *utpala, *pundarika (also spelt *pun-
darika), *kumud and *kumuda (which all
literally mean “lotus”) express numbers such
as: ten to the power four, ten to the power nine,
ten to the power fourteen, ten to the power
twenty-one, ten to the power twenty-five, ten
to the power twenty-seven, ten to the power
twenty-nine, ten to the power ninety-eight, ten
to the power 105, ten to the power 112 or ten to
the power 119. See High numbers (The sym-
bolic meanings of).
LOW NUMBERS. See Paramdnu.
LUMINOUS. [S]. Value = 1. See Chandra
and One.
LUNAR MANSION. [S]. Value = 27. See
Nakshatra and Twenty-seven.
M
MADHYA. Literally “Milieu". Name given to
the number ten to the power ten (= ten
DICTIONARY OF INDIAN NUMERIC AL SYMBOLS
478
thousand million). See Names of numbers.
For an explanation of this symbolism, see High
numbers (The symbolic meaning of).
Sources: * Vdjasantyi So mbit a (beginning of the
Common lira); * Taitiriya Samhitd (beginning of
the Common Era); *Pahchavimsha Brahma rut
(date unknown).
MADHYA. Literally "Milieu”. Name given to
the number ten to the power eleven (= thou-
sand million). See Names of numbers. For an
explanation of this symbolism, see High num-
bers (The symbolic meaning of).
Sources: * Kdlhaka Samhitd (beginning of the
Common Era).
MADHYA. Literally "Milieu”. Name given to
the number ten to the power fifteen (= trillion).
See Names of numbers. For an explanation of
this symbolism, see High numbers (The sym-
bolic meaning of).
Source: *Kitab fi tahqiq i ma li'l hind by al-Biruni (c.
1030 CE).
MADHYA. Literally “Milieu". Name given to
the number ten to the power sixteen (= ten tril-
lion). See Names of numbers. For an
explanation of this symbolism, see High num-
bers (The symbolic meaning of).
Source: *Lilavati by Bhaskaracharya (1150 CE);
*Ganitakaumudi by Narayana (1350 CE);
* Trishatikd by Shridharacharya (date uncertain).
MADHYAMIKA. Name given to the adepts of the
Buddhist doctrine called the "Middle Path”. This
doctrine does not separate the reality and non-
reality of things, and even considers the latter as a
type of “vacuity” ( *shunyatd ). This is why its
adepts are sometimes called the *shunyavadin, the
“vacuists”. See Shunya and Zero.
MAGIC. See Numeral alphabet, magic, mysti-
cism and divination.
MAHABHARATA. Name of a great Indian
epic. See Arjunakara, Dhdrlardshlra, Nripa
and Vasu.
MAHABHUTA. [S]. Value = 5. "Great ele-
ment”. This term is used by Hindus to denote
collectively the five elements of the revelation.
It is thus synonymous with the word *bhuta,
which can denote any one of the elements.
Another generic term for the five elements is
Avarahakha, made up of the five letters w'hich
symbolise each one of them: A (earth); Va
(water); Ra (fire); Ha (wind); and Kha (ether).
See Pahchabhuta and Five.
MAHABJA. Literally "great moon”. Name
given to the number ten to the power twelve (=
billion). See Abja and Names of numbers. For
an explanation of this symbolism, see High
numbers (The symbolic meaning of).
Source: *Ganitakaumudi by Narayana (1350 CE).
MAHADEVA. [SJ. Value = 11. "Great god”.
One of the names for *Rudra, whose symbolic
value is eleven. See Rudra, Shiva and Eleven.
MAHAKALPA. "Great * kalpa”. According to
arithmetical-cosmogonical speculations, this
term denotes a unit of cosmic time which is even
bigger than the kalpa (= 4,320,000,000 human
years). It is the equivalent of twenty “little” kalpa
or ordinary kalpa. Thus one rnahakalpa =
86,400,000,000 human years. See Yuga.
MAHAKATHANA. Literally "great *kathdna".
Name given to the number ten to the power 126.
See Names of numbers and High numbers.
Source: *Vydkarana (Pali grammar) bv
Kachchayana (eleventh century CE).
MAHAKHARVA. Literally "great *kharva”.
Name given to the number ten to the power
thirteen (= ten billion). See Names of numbers
and High numbers.
Source: ‘Ganitasdrasamgraha by Mahaviracharya
(850 CE).
MAHAKSHITI. Literally: “great earth”. Name
given to the number ten to the power twenty-
one (= quintillion). See Names of numbers.
For an explanation of the symbolism, see High
numbers (Symbolic meaning of).
Source: ‘Ganitasdrasamgraha by Mahaviracharya
(850 CE).
MAHAKSHOBHA. Literally “great earth”.
Name given to the number ten to the pow r er
twenty-three (= a hundred quintillions). See
Names of numbers and High numbers.
Source: * Ganitasdrasamgraha by Mahaviracharya
(850 CE).
MAHAKSHONI. Literally "great earth.” Name
given to the number ten to the power seven-
teen ( = a hundred trillion). See Names of
numbers and High numbers.
Source: * Ganitasdrasamgraha by Mahaviracharya
(850 CE).
MAHAPADMA. Literally "great (pink) *!otus”.
Name given to the number ten to the power
twelve (= billion). See Padma , High numbers
and Names of numbers.
Sources : Kitab ft tahqiq i ma Hi hind by al-Biruni (c.
1030); ‘Uldvati by Bhaskaracharya (1150 CE).
MAHAPADMA. Literally “great (pink) *lotus”.
Name given to the number ten to the power
fifteen (= trillion). See Padma, High numbers
and Names of numbers.
Source : *Ganitasdrasamgraha by Mahaviracharya
(850 CF.).
MAHAPADMA. Literally “great (pink) *lotus”.
Name given to the number ten to the power
thirty-four. See Padma, High numbers and
Names of numbers.
Source : ‘Ramayana by Valmiki (in the early cen-
turies CE).
MAHAPAPA. [S]. Value = 5. “Great sin”.
Allusion to the *Pahchaklesha, the “five
impurities”, which, in Hindu and Buddhist
philosophies, constitute the five main obstacles
denying the faithful the Way of Realisation
(bodhi) : greed, anger, thoughtlessness,
insolence and doubt. See Five.
MAHARASHTRI NUMERALS. Signs derived
from *Brahmi numerals, through the
intermediary of Shunga, Shaka, Kushana,
Andhra, Gupta and Nagari numerals. Formerly
used in Maharashtra State. The corresponding
system was based on the place-value system
and possessed zero (in the form of a little
circle). These numerals were the ancestors of
Marathi, Modi, Marwari, Mahajani and
Rajasthani numerals. See Indian written
numeral systems (Classification of). See
Fig. 24.52 and 24.61 to 69.
MAHARASHTRI-JAINA NUMERALS. Signs
derived from *Brahmi numerals, through the
intermediary of Shunga, Shaka, Kushana,
Andhra, Gupta and Nagari numerals. Formerly
used by the *Jainas ( Shvetambara ). The
corresponding system was based on the place-
value system and possessed zero (in the form of
a little circle). See Indian written numeral
systems (Classification of). See Fig. 24.52 and
24.61 to 69.
MAHASAROJA. Literally “great *saroja\
Name given to the number ten to the power
twelve (= billion). See Names of numbers and
High numbers.
Source : ‘Trishatikd by Shridharacharya (date
uncertain).
MAHASHANKHA. Literally “great conch”.
Name given to the number ten to the power
nineteen (= ten quadrillions). See Shankha,
Names of numbers and High numbers.
Source : * Ganitasdrasamgraha by Mahaviracharya
(850 CF).
MAHAVIRACHARYA. *Jaina mathematician
who lived in the ninth century. His works
notably include Ganitasdrasamgraha, where
there is frequent use of the place-value system,
written not only in numerical symbols, but also
with nine numerals and the sign for zero. Here
is a list of the principle names for numbers
mentioned in Ganitasdrasamgraha :
*Eka (= 1), * Dasha (= 10), *Shata (= 10 2 ),
Sahasra (= 10 3 ), *Dashasahasra (= 10 4 ),
*Laksha (= 10 5 ), *Dashalaksha (= 10 6 ), *Koti (=
10 7 ), *Dashakoti (= 10 8 ), *Shatakoti (= 10 9 ),
‘Arbuda (= 10 10 ), *Nyarbuda (= 10 u ), *Kharva
(= I0 12 ), ‘Mahakharva (= 10 13 ), *Padma (=
10 14 ), *Mahapadma (= 10 15 ), *Kshoni (= 10 16 ),
‘Mahakshoni (= 10 17 ), *Shankha (= 10 J8 ),
‘Mahashankha (= 10 19 ) r *Kshiti (= 10 2 °),
‘Mahakshiti (= 10 21 ), ‘Kshobha (= 10 22 ),
‘Mahakshobha (= 10 23 ).
See Names of numbers, High numbers,
Numeration of numerical symbols, Zero and
Indian mathematics (The history of).
Source : GtsS, I, p. 63-68 [See Datta and Singh
(1938), p. 13; Rangacarya (1912)].
MAHAVRINDA. Literally “great *vrinda n .
Name given to the number ten to the power
twenty-two (= ten quintillions). See Names of
numbers, Vrinda and High numbers (The
symbolic meaning of).
Source : ‘Ramayana by Valmiki (first centuries CE).
MAHAYAJNA. [S). Value = 5. “Great
sacrifice". This is the name for the five daily
sacrifices that all orthodox Hindus must make
: prayer and devotion ( hapujd ), the placing of
offerings in various places {baliharana),
offerings to the shades of ancestors
( pitriyajha ), the offering of a ritual meal
( manushyayajha ) and a sacrifice in honour of
the fire which cooks the food [see Frederic
Diction na ire (1987)]. See Five.
MAHAYUGA. “Great period”. This is the
largest cosmic cycle of Indian speculations.
Considered as the “Great age", this cycle is
made up of four successive periods ( *kritayuga ,
*tretayuga, * dvdparayuga, *kaliyuga)\ this is
why it is also called the *chaturyuga (“four
ages”). It is said to be made up of 4,320,000
human years. See Yuga.
MAHI. [SJ. Value = 1. This term (which is
also the name of a river in Rajasthan) means
“curds”, the first product derived from
milk. Milk itself is the first and most
important of the “gifts of the Cow” ( *gavya ),
which is the first nourishment by which
all others potentially exist. Thus this
symbolism embraces the idea of the cow as a
whole and even, in an esoterical sense, the
sacred Cow of the Hindus, which is identified
with the whole world, because the Cow
dispenses life. As "Earth” is a numerical
symbol for the value 1, Mahi = “curds” = X.
See Prithivi, Go and One.
MAHIDHARA. IS]. Value = 7. “Mountain”. See
Mount Meru, Adri and Seven.
MAHORAGA. “Great serpents”. Category of
demons in the form of cobras. See Serpent
(Symbolism of the).
MAIN OBSERVANCE. [SJ. Value = 5. See
Karaniya and Five.
MAJTHILI NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
479
MAI.AYALAM NUMERALS
of Shunga, Shaka, Kushana, Andhra, Gupta,
Nagari and Bengali numerals. Currently found
mainly in the north of Bihar State. The
corresponding system is based on the place-
value system and possesses zero (in the form of
a little circle). See Indian written numeral
systems (Classification of). See Fig. 24.11, 52
and 24.61 to 69.
MALAYALAM NUMERALS. Signs derived
from *Brahmi numerals, through the
intermediary of Shunga, Shaka, Kushana,
Andhra, Pallava, Chalukya, Ganga, Valabhi,
Bhattiprolu and Grantha numerals. Currently
used by the Dravidians of Kerala State on the
ancient coast of Malabar, to the southwest of
India. The corresponding system is not based
on the place-value system and has only
possessed zero since a relatively recent date.
See Indian written numeral systems
(Classification of). See also Fig. 24.19, 52 and
24.61 to 69.
MANDARA. One of the names for Mount
Meru. See Mount Meru, Adri, Dvipa, Puma,
Pdlala, Sagara, Pushkara, Pavana and Vayu.
MANGALA. [SJ. Value = 8. “Jewel”, “thing
which augurs well". Here the allusion is to
*ashtamangala, the eight “things which augur
well”, Buddhist symbols which represent the
veneration of the “Master of the world” (and,
by extension, Buddha). These are: the parasol
(symbol of royal dignity meant to protect
against misfortune); the two fish (signs of the
Indian master of the universe); the conch
(symbol of victory in combat); the lotus flower
(symbol of purity); the container of lustral
water (filled with Amrita, the nectar of
immortality); the rolled flag (sign of victorious
faith); the knots of eternal life; and the wheel of
the Doctrine ( Dharmachakra ). See Eight.
MANIPURI NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Gupta,
Nagari, Kutila and Bengali numerals. Currently
in use in Manipur State, to the east of Assam
and next to the border of Burma. The
corresponding system functions according to
the place-value system and possesses zero (in
the form of a little circle). See Indian written
numeral systems (Classification of). See also
Rg. 24.52 and 24.61 to 69.
MANTRA. Sacred formula which constitutes
the digest, in a material from, of the divinity
which it is meant to invoke. See Numeral
alphabet, magic, mysticism and divination
and Mysticism of letters.
MANU. [Sj. Value = 14. Literally “human”. This
is the name given in traditional legends to
the Progenitor of the human race as a symbol
of the thinking being and considered as the
intermediary between the Creator and
the human race. According to the * Vedas, the
manus constituted the first divine legislators who
fixed the rules of religious ceremonies and ritual
sacrifices. According to the *purdnas, there were
fourteen successive manus, sovereigns living in
ethereal worlds where they are meant to direct
the conscious life of humankind and its ability to
think. Thus: manu - 14. (The mam of the present
era is the seventh: named Vaivashvata, “Born of
the Sun”) See Fourteen.
MANUAL ARITHMETIC. See Mudrd.
MANUSMRITI. Important religious work
considered to be the foundation of Hindu
society.
MARATHA (Calendar). This calendar begins
in the year 1673 CE, and was founded by
Shivaji. To find the corresponding date in the
Common Era, add 1673 to a date expressed in
the Maratha calendar. Formerly used in
Maharashtra. See Indian calendars.
MARATHI NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Gupta,
Nagari and Maharashtri numerals. Currently
used in the west of India, in the state-province
of Maharashtra. The corresponding system is
based on the place-value system and possesses
zero (in the form of a little circle). See Indian
written numeral systems (Classification of).
See Fig. 24.4, 52 and 24.61 to 69.
MARGANA [S]. Value = 5. “Arrow”. See Shara
and Five.
MARTANDA. [S]. Value = 12. One of the
names of *Surya. See Twelve.
MARWARI NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary of
Shunga, Shaka, Kushana, Andhra, Gupta,
Nagari and Maharashtri numerals. Currently
used in the northwest of India (Rajasthan) and
in the Aravalli mountains, and between the
afore-mentioned mountains and the Thar desert
(Marusthali). The corresponding system is
based on the place-value system and possesses
zero (in the form of a little circle). See Indian
written numeral systems (Classification of).
See Fig. 24.52 and 24.61 to 69.
MASA. [S]. Value = 12. “Month”. Allusion to
the twelve months of the year. See Twelve.
MASARDHA. [SJ. Value = 5. “Season”. See
Ritu and Five.
MATANGA. [S]. Value = 8. “Elephant”. See
Diggaja and Eight.
MATHEMATICIAN. See Samkhyd.
MATHEMATICS. See Ganita, Ganitdnuyoga,
Arithmetic, Calculation and Indian
mathematics (The history of).
MATHURA NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana and Andhra
numerals. Contemporaries of a Shaka dynasty
(first to third century CE). These are attested
mainly in the inscriptions of Mathura (in Uttar
Pradesh). The corresponding system did
not use the place-value system or zero.
See Fig. 24.32, 52 and 24.61 to 69, and 70.
See also Indian written numeral systems
(Classification of).
MATRIKA. IS]. Value = 7. "Divine Mother".
Name given in Hinduism to the *saptamalrikd,
the seven aspects of shakti, “feminine energy”
of the divinities : aspects which are considered
to be the “mothers of the world”. Thus: mdtrika
= 7. See Seven.
MATTER (Indian concept of). See Indian
atomism, Jaina and Jala.
MEMBER. [SJ. Value = 6. See Anga and Six.
MENTAL ARITHMETIC. See Ganand.
MERIT. [S] . Value = 3. See Guna, Triguna and
Three.
MERIT. [SI. Value = 6. See Guna, Shadayatana
and Six.
MILIEU. As a name of a high number. See
Madhya.
MILLION (= ten to the power six). See
Dashalaksha, Niyuta, Prayuta and Names of
numbers.
MON NUMERALS. Signs derived from
*Brahmi numerals, through the intermediary
of Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali" and Vateluttu
numerals. Formerly used by the people of
Pegu before the Burmese invasion. The
corresponding system was not based upon the
place-value system and did not possess zero.
See Fig. 24.52 and 24.61 to 69. See also Indian
written numeral systems (Classification of).
MONGOL NUMERALS. Signs derived from
*Brahmi numerals through the intermediary
notations of Shunga, Shaka, Kushana, Adhra,
Gupta, Siddham and Tibetan numerals. Used
by the Mongols during the thirteenth and
fourteenth centuries. The corresponding
system functioned according to the place-value
system and possessed zero (in the form of a
little circle). See Fig. 24. 42, 52 and 24.61 to 69.
See Indian written numeral systems
(Classification of).
MONTH (Rite of the four). See Chaturmasya.
MONTH. [SJ. Value = 12. See Mdsa and
Tw'elve.
MOON. Used as a name for ten to the power
nine or ten to the power twelve. See Abja and
Mahabja. See also High numbers (The
symbolic meaning of).
MOON. IS]. Value = 1. See Abja, Atrinayanaja,
Chandra, Indu, Kshapeshvara, Mriganka,
Shashadhara, Shashanka, Shashin, Shitdmshu,
Shitarashmi, Soma, Sudhamshu, Vidhu and
One.
MORTAL SINS (The Five). See
Pahchanantarya .
MOUNT MERU. Mythical mountain in Hindu
cosmology and Brahman mythology. It has
many Sanskrit names : Ratnasanu, Sumeru,
Hcmadri, Mandara, Karnikachala, Devapdrvata,
etc. Mount Meru was meant to be the place
where the gods lived and met. It was said to be
situated at the centre of the universe, under the
Pole star, and also constituted the “axis of the
world”. *Indra lived at the summit, the head of
the *deva, whilst the slopes were peopled with
the *Trdyastrimsha, the thirty-three deva (gods).
Mount Meru plays an important role in
mythology and Brahman and Hindu
cosmological texts. Thus this mountain was
said to act as a pivot between the *deva and the
*asura (“anti-gods”) during the churning of the
sea of milk.
In corresponding representations, Mount
Meru, and all that is connected to it, is always
associated with the number seven. First there
is the concept of the “mountain" and, by
extension, that of “hill”, which is generally
symbolically connected with this number.
There are also the “seven oceans” ( *sapta
sdgara ). Then there are the “island-
continents” {* dvipa), each one flooded by one
of the seven oceans, which surround Mount
Meru. As for Mount Meru itself, it has seven
faces, each facing one of the seven seas and
one of the seven continents. It is above the
*patala, the seven underworlds or “inferior
worlds”, where the *ndga live, the master of
whom is the king Muchalinda, the chthonian
genie in the form of a cobra, depicted with
seven heads.
DICTIONARY O I INDIAN NUMERICAL SYMBOLS
480
Fig. 24D.9. Mount Meru, centre of the universe in Hindu and Brahmanic cosmology. Ref: Dubois de
Jancigny, L’Univers pittoresque, Hachette, Paris, 1846
Thus Mount Meru represents total stability
and the absolute centre of the universe, around
which the universe and the firmament revolve.
This image of Mount Meru connects it to one
of the universal images of the Pole star.
According to the legend, Mount Meru is
directly underneath the Pole star and is “on the
same axis". The symbolic correspondence
between Mount Meru and the number seven
comes from the fact that the Pole star, the
*Sudrishti, “That which never moves", is the
last of the seven stars of the constellation: the
Bear. According to Indian tradition, this
constellation is the personification of the seven
“great Sages” of Vedic times, the *Saptarishi,
who are thought to be the authors of both the
hymns and invocations of the Rigveda, and of
the most important Vedic texts.
In Sanskrit, the word for “mountain” is
*parvata, which appears in one of the names of
Mount Meru : Devapdrvata, “mountain of the
gods”. Because this sacred mountain was
associated with the number seven, the
mountain, daughter of Himalaya, sister of
Vishnu and wife of ‘Shiva also came to be
synonymous with this number: she was Kali,
the “Black", who represented the destructive
power of time ( *Kala ) and was considered in
the *Veda to be the seventh tongue of Agni,
“Fire”. It is perhaps not by chance that Manasa,
the Hindu tantric divinity, who symbolises the
destructive and regenerative aspects of Parvati,
has been considered as one of the sisters of
Muchalinda (the king of the naga with seven
cobra heads) and as the *Patdla Kumara,
divinity of the serpents and “princess of the
(seven) Underworlds".
Even ‘Surya, the Sun-god, traditionally
associated with the number twelve in the
system of ‘numerical symbols, has represented
the number seven : he is often represented as a
warrior flying through the sky on a chariot
pulled either by seven horses (*ashva), or by
Aruna, the horse with seven heads. See Adri,
Dvipa, Sagara, Patala, Pushkara, Pdvana,
Vdyu, Loka (= 7) and Seven.
MOUNTAIN. IS]. Value = 7. See Adri, Parvata ,
Seven and Mount Meru.
MOUTHS OF JAHNAVL [SI. Value = 1,000.
“Mouths of the Ganges". See Jdhnavivaktra
and Thousand.
MRIGANKA. [SJ. Value = 1. “Moon”. See Abja
and One.
MUCHALINDA (MUCHILINDA). Name of
the king of the *ndga. See Serpent (Symbolism
of the).
MUDRA. “Mark, sign”. In Indian mysticism,
mainly in esoteric Buddhism, this word
denotes the gestures made by the hands and is
meant to symbolise a mental attitude of the
divinities. They are mainly used during
ceremonies and prayers to invoke Buddha and
the power of his divinites.
MUDRA. Term denoting manual arithmetic
and digital calcultaion in Ancient Sanskrit
literature. See Chapter 3.
MUDRABALA. Literally : “Power of the
* mudra". Name given to the number ten to the
power forty-three. To those using it, such a high
number must have symbolically represented a
quantity which was as incalculable as the
powers concealed within the mystical gestures
called mudrd. See Mudra (first article), Names
of numbers and High numbers.
Source : * ialiiavistara Sutra (before 308 CE).
MUKHA. [SI. Value = 4. “Face”. Allusion to the
*chaturmukha (“Four Faces”), which refers to all
the of the Brahmanic (or Buddhist) divinities
who are represented as having four faces
(‘Brahma, ‘Shiva, etc.). See Four.
MULTIPLICATION. (Arithmetic]. See
Gunana, Patiganita and Indian methods
of calculation.
MUNI. [S]. Value = 7. “Sage”. This is an allusion
to the seven mythical sages of Vedic times.
Strictly speaking, the word muni, “sage”, is
much less strong than *Rishi, which denotes the
seven “Sages”. But the name began to be used as
a symbol for the number seven because of the
desired effect in the versification of expressions
using numerical symbols. See Seven, Sanskrit
and Poetry and the writing of numbers.
MURTI. [S3. Value = 3. "Form”. Allusion to the
“three forms" of Hindu triads (ftrimurti),
constituted by either three different divinities
(usually ‘Brahma, ‘Shiva and ‘Vishnu), or
three aspects of one single divinity. See Three.
MURTI. [SI. Value = 8. “Form”. Allusion to the
*ashtamurti, the “eight” most important
“forms” of ‘Shiva : *Rudra, who represents the
power of fire; *Bhava, water; Sharva, the earth;
ishana, the sun; Pashupati, sacrifice; Bhima, the
terrible; and Ugra and Mahddeva. See Rudra-
Shiva and Eight.
MUSICAL MODE. [S]. Value = 6. See Rdga
and Six.
MUSICAL NOTE. [S]. Value = 7. See Svara and
Seven.
MUSLIM INDIA. See Numeral alphabet and
composition of chronograms and Eastern
Arabic numerals.
MYSTICISM AND POSITIONAL NUME-
RATION. See Durga.
MYSTICISM OF HIGH NUMBERS. See High
numbers (The symbolic meaning of).
MYSTICISM OF INFINITY. See Infinity
(Mythological representation of) and Serpent
(Symbolism of the).
MYSTICISM OF LETTERS. See Akshara,
Numeral alphabet, magic, mysticism and
divination, Bija, Mantra, Trivarna, Vachana.
MYSTICISM OF NUMBERS. See Numerical
symbols, Symbolism of words with a
numerical value, Symbolism of numbers
(Concept of large quantity), Symbolism of zero
and High numbers (The symbolic meaning of).
MYSTICISM OF THE NUMBER FOUR. See
Naga, Jala, Ocean and Serpent (Symbolism
of the).
MYSTICISM OF THE NUMBER SEVEN. See
Mount Meru and Ocean.
MYSTICISM OF ZERO. See Shunya, Shunyatd ,
Zero, Zero (Indian concepts of), Zero and
Sanskrit poetry and Symbolism of Zero.
MYTHICAL PEARL. [S]. Value = 1. See Indu.
N
NABHA. [S]. Value = 0. “Sky, atmosphere”.
This symbolism is due to the fact that the sky
is considered to be the “void”. See Zero
and Shunya.
NABHAS. [SJ. Value = 0. “Sky, atmosphere".
See Nabha, Zero and Shunya.
NADI. Hindu word denoting the arteries of the
human body. See Numeral alphabet, magic,
mysticism and divination.
NAGA. [S]. Value = 7. “Mountain”. Literally,
“That which does not move”. This is an allusion
to ‘Mount Meru, the mythical mountain of
Hindu cosmology and Brahman mythology,
the dwelling and meeting place of the gods,
which is said to be situated at the centre of the
universe and thus constitute the axis of the
world. This symbolism comes from the fact
that the number seven plays an important role
in mythological representations related to
481
NAG A
Mount Meru, and because the Pole star,
situated directly above this mountain, is the
*Sudrishti, the divinity “who never moves". See
Adri, Mount Meru. Seven and Dhruva.
NAGA. IS], Value = 8. “Serpent". This
symbolism is due to the fact that the serpent
(especially the naga) is considered to be not
only a sun genie who owns the earth and its
treasures, but also an aquatic symbol. It is a
“spirit of the waters” that lives in the *pdtala or
“underworlds”. In Sanskrit, water is *jala, and
this word is used as a numerical symbol for the
number four. In their subterranean kingdom,
the naga reproduce in couples and evolve in the
company of the ndgini (the females), so
“water”, in this case, has been symbolically
multiplied by two, to give their generic name
the symbolic value of : 4 X 2 = 8. In traditional
Indian thought, the earth (to which the serpent
is also associated) corresponds symbolically to
the number four, being associated with the
square and its four horizons (or cardinal
points). As the naga is also aquatic, water (= 4)
has been symbolically added to give the
serpent its generic designation as a numerical
symbol with a value equal to eight ( naga =
earth + water = 4 + 4 = 8). See Eight, Serpent
(Symbolism of the) and Infinity.
For a documented example of this : see El,
XXXV, p. 140.
NAGABALA. Literally, “Power of the *ndga".
Name given to the number ten to the power
twenty-five. See Names of numbers. High
numbers and Serpent (Symbolism of the).
Source : * I.alitavistara Sutra (before 308 CE).
NAGARI ALPHABET. See Fig. 24.56. See also
Aryabhata’s numeration.
NAGARI NUMERALS. Signs derived from
*Brahmi numerals through the intermediary
notations of Shunga, Shaka, Kushana, Adhra
and Gupta. Today these are the most widely used
numerals in India, from Madhya Pradesh
(central province) to Uttar Pradesh (northern
province), Rajasthan, Haryana, Himachal
Pradesh (the Himalayas) and Delhi. These
numerals are also called Devanagari because they
are the most regular numerals of India. The
corresponding system is based on the place-
value system and possesses zero (in the form of a
little circle). However, this was not always the
case, as a considerable number of documents
written before the eighth century CE prove.
These signs were the ancestors of Siddham,
Nepali, Tibetan, Mongol, Kutila, Bengali, Oriya,
Kaithi, Maithili, Manipuri, Gujarati,
Maharashtri, Marathi, Modi, Marwari,
Mahajani, Rajasthani, etc. numerals, as well as
the “Hindi” numerals of the eastern Arabs, the
Ghubar numerals of North Africa, the apices and
algorisms of mediaeval Europe, not to mention
our own modern numerals. For ancient Nagari
numerals recorded on copper charters, see
Fig. 24.39 A and 75; for those recorded on
manuscripts, see Fig. 24.39 B; for inscriptions of
Gwalior, see Fig. 24.39 C and 24. 72 to 74. For
modem Nagari numerals, see Fig. 24.3. For
notations which derived from Nagari, see
Fig. 24.52. For the corresponding graphical
evolution, see Fig. 24.61 to 69. See also Indian
written numeral systems (Classification of).
NAGINI. Female of the *naga. See Naga and
Serpent (Symbolism of the).
NAHUT. Name given to the number ten to
the power nine. See Names of numbers and
High numbers.
Source : *I.a!itavistarn Sutra (before 308 CE).
NAHUTA. Name given to the number ten to
the power twenty-eight. See Names of
numbers and High numbers.
Source : *Vydkarana (Pali grammar) by Kachayana
(eleventh century CE).
NAIL. [S]. Value = 20. See Nakha and Twenty.
NAKHA. [S]. Value = 20. “Nail”. This is
because of the nails of the ten fingers and ten
toes. See Twenty.
NAKSHATRA. [S]. Value = 27. “Lunar
Mansion". This refers to the houses occupied
successively by the moon in its monthly cycle,
which in solar days lasts between twenty-seven
and twenty-eight days. For the representation
of the sidereal movements of the moon,
however, Indian astronomers usually used the
system of twenty-seven nakshatra marking
twenty-seven ideal equal divisions of the
ecliptic zone (each one equal to 13° 20'). This is
why the word came to symbolically signify the
number twenty-seven. See Twenty-seven.
NAKSHATRAVIDYA. Literally : “Knowledge of
the * nakshatra". Name given to “astronomy" in
the Chandogya Upanishad.
NAMES OF NUMBERS (up to thousand).
Here is a list of ordinary Sanskrit names
of numbers :
'Eka (= 1); *Dva (= 2); *Dve (= 2); *Dvi (=
2); *Trai (= 3); 'Traya (= 3); *7W (= 3); * Chatur
(= 4); *Pahcha (= 5); *Shad (= 6); *Shash (= 6);
*Shat (= 6); *Sapta (= 7); * Saptan (= 7); *Ashta
(= 8); *Ashtan (= 8); *Nava (= 9); *Navan (= 9).
* Dasha (= 10); * Dashan (= 10); *Ekadasha
(= 11); *Dvddasha (= 12); *Trayodasha (= 13);
*Chaturdasha (= 14); * Pahchadasha (= 15);
*Shaddasha (= 16); *Saptadasha (= 17);
*Ashtadasha (= 18); *Navadasha (= 19).
*Vimshati (= 20); *Ekavimshati (= 21);
*Dvavimshati (= 22); Trayavimshati (= 23);
* Chaturvimshati (= 24); * Pahchavimshati (= 25);
*Shadvimshati (= 26); * Saptavimshati (= 27);
Ashtavimshati (= 28); Navavimshati (= 29).
*Trimshat (= 30); *Chatvarimshat (= 40);
*Pahchashat (= 50); *Shashti (= 60); * Saptati
(= 70); * Ashiti (= 80); •Navati (= 90).
At the start of the Common Era, the
subtractive forms were also used for the
numbers 19, 29, 39, 49, etc. : * ekannavimshati
(= 20 - 1 = 19); * ekannatrimshati (= 30 - 1 =
29); etc.
[See *Taittiriya Samhita, VII, 2, 11; Datta
and Singh (1938), pp. 14-15].
*Shata (= 100). This is the classical Sanksrit
form of this number. However, at the
beginning of the Common Era, the Indo-
European form Sata was still used.
Ref.: There is evidence of the use of this
form in *Vajasaneyi Samhita, *Tailtiriya
Samhita, *Kathaka Samhita, * Panchavimsha
Brahmana and *Sankhyayana Shrauta Sutra.
Dvashata (= 200); Trishata (= 300);
Chatuhshata (= 400); etc. *Sahasra ( = 1,000);
Dvasahasra (= 2,000); Trisahasra (= 3,000);
Chatursahasra (= 4,000); etc.
See Sanskrit.
NAMES OF NUMBERS (Powers of ten above
thousand). After ten thousand, Sanskrit spoken
numeration assigns names to the various
powers of ten which differ considerably from
one author to another and from one era to
another; thus the same word can have several
numerical values depending on the source in
question. The use of these names was not
commonplace in India. However, they were very
familiar to scholars, since the following terms
are found in astronomical, mathematical,
cosmological, grammatical and religious texts,
as well as in legend and mythology.
In the following list, the letters in brackets
indicate the source of each word in question;
here are the letters, the sources they represent,
and the era in which they were written :
(a) Vajasaneyi Samhita. (b) Taittiriya
Samhita. (c) Kathaka Samhita. (d) Ramayana by
Valmiki. (e) Lalitavistara Sutra. (f)
Panchavimsha Brahmana. (g) Sankhyayana
Shrauta Sutra, (h) Aryabhatiya by Aryabhata, (i)
Ganitasdrasamgraha by Mahaviracharya. (j)
Kitab fi tahqiq i ma li'l hind by al-Biruni. (k)
Vyakarana, Pali grammar, by Kachchayana. (1)
Lilavati by Bhaskaracharya. (m) Ganitakaumudi
by Narayana. (n) Trishatika by Shridhara-
charya.
(a, b, c : beginning of the Common Era; d :
early centuries of the Common Era; e : before
308 CE; f, g : date uncertain; h : c. 510 CE; i ;
850 CE; j : c. 1030 CE; k : eleventh century CE; 1
: 1150 CE; m : 1356 CE; n : date uncertain).
Here is a (non-exhaustive) arithmetical list
of the Sanskrit names of high numbers :
TEN TO THE POWER 4: *Ayuta (a, b, c, f,
g, h, j, 1, m, n): * Dashasahasra (i).
TEN TO THE POWER 5; * Lakh (e);
*Lakkha (k); *!.aksha (i, j, 1, m, n); *Niyuta (a, b,
f, h); *Prayuta (c).
TEN TO THE POWER 6: * Dashalaksha (i);
*Niyula (c); * Prayula (a, b, f, g, h, j, 1, m, n).
TEN TO THE POWER 7: * Arbuda (a, b, c. f,
g, h); *Koti (d, e, h, i, j, k, 1, m, n).
TEN TO THE POWER 8: * Arbuda (1, m, n);
*Dashakoli (i); *Nyarbuda (a, b, c, f, g);
* Vyarbuda (j).
TEN TO THE POWER 9: *Abja (1, n);
*Ayuta (e); *Nabut (e); *Nikharva (g); *Padma
(j); *Samudra (a, b, c, f); *Saroja (m);
*Shatakoti (i); *Vddava (c); *Vrindd (h).
TEN TO THE POWER 10: * Arbuda (i);
*Kharva (j, 1, m, n); *Madhya (a, b, f);
*Samudra (g).
TEN TO THE POWER 11: *Anta (a, b, c, f);
*Madhya (c); *Nikharva (j, 1, m, n); *Ninnahut
(e); *Niyuta (e); * Nyarbuda (i); "Salila (g).
TEN TO THE POWER 12: *Antya (g);
*Kharva (i); *Mahabja (m); *Mahapadma
(j, 1); Mahasaroja (n); *Parardha (a, b, c, f);
*Shankha (d).
TEN TO THE POWER 13: *Ananta (g);
*Kankara (e); *Khamba (e); * Mahakharva (i);
*Nikharva (f); *Shankha (j); *Shanku (1, m, n).
TEN TO THE POWER 14: * Jaladhi (1);
* Padma (i); *Pakoti (k); *Pdrdvdra (m);
*Samudra (j); * Saritdpati (n); *Vddava (f).
TEN TO THE POWER 15: "Akshiti (f);
*Antya (1, m, n); * Madhya (j); *Mahdpadma (i);
*Viskhamba (e); *Vivara (e).
TEN TO THE POWER 16: *Antya (j);
* Madhya (1, m, n); * Kshoni (i).
TEN TO THE POWER 17: *Abab (e);
* Kshobhya (e); *Mahakshoni (i); *Parardha (j, 1,
m, n); * Vririda (d).
TEN TO THE POWER 18: *Shankha (i).
TEN TO THE POWER 19: *Attata (e);
*Mahdshankha (i); * Vivaha (e).
TEN TO THE POWER 20: *Kshili (i).
TEN TO THE POWER 21: *Kotippakoti (k);
*Kumud (e); *Mahakshiti (i); *Utsanga (e).
TEN TO THE POWER 22: *Kshobha (i);
*Mahavrinda (d).
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
48 2
TEN TO THE POWER 23: *Bahula (e);
*Gundhika (e); *Mahdkshobha (i).
TEN TO THE POWER 25: *Ndgabala (e);
* Utpala (e).
TEN TO THE POWER 27: *Pundar!ka (e);
*Titilambha (e).
TEN TO THE POWER 28: *Nahula (k).
TEN TO THE POWER 29 : *Padma (d);
*Paduma (e); *Vyavasthanaprajfiapati (e).
TEN TO THE POWER 31: * Hetuhila (e).
TEN TO THE POWER 33: *Karahu (e).
TEN TO THE POWER 34: *Mahdpadma (d).
TEN TO THE POWER 35: * Hetvindriya (e);
*Ninnahuta (k).
TEN TO THE POWER 37:
*Samaptalambha (e).
TEN TO THE POWER 39: * Gananagati (e);
*Kharm (d).
TEN TO THE POWER 41: * Niravadya (e).
TEN TO THE POWER 42: *Akkhobhim (k).
TEN TO THE POWER 43: *Mudrdbala (e).
TEN TO THE POWER 45: * Sarvabala (e).
TEN TO THE POWER 47: * Visamjnagati (e).
TEN TO THE POWER 49: *Bindu (k);
*Sarvajna (e).
TEN TO THE POWER 51: *Vi bhutangamd
(e).
TEN TO THE POWER 53: * Tallakshana (e).
TEN TO THE POWER 56: *Abbuda (k).
TEN TO THE POWER 63: *Nirabbuda (k).
TEN TO THE POWER 70: *Ahaha (k).
TEN TO THE POWER 77: *Ababa (k).
TEN TO THE POWER 84: *Atata (k).
TEN TO THE POWER 91: *Sogandhika (k).
TEN TO THE POWER 98: *Uppala (k).
TEN TO THE POWER 99: *Dhvajdgravati (e).
TEN TO THE POWER 105: *Kumuda (k).
TEN TO THE POWER 112: * Pundarika (k).
TEN TO THE POWER 119: *Kathdna (k);
*Paduma (k).
TEN TO THE POWER 126: *Mahakathdna
(k).
TEN TO THE POWER 140: *Asankhyeya (k).
TEN TO THE POWER 145:
* Dhvajagranishdmani (e). And so on until ten to
the power 421 (e).
See Sanskrit and Poetry and writing
of numbers.
Indian scholars did not specialise in just
one field of study; they embraced diverse
disciplines all at once, such as mathematics,
astronomy, literature, poetry, phonetics or
philosophy, and even mysticism, divination
and astrology. Thus it is not surprising that in
arithmetic, their fertile imaginations led them
to use subtle symbolism to name high
numbers. They gave a unique name to each
power of ten up to at least as high as ten to the
power 421. This is why their spoken numeration
had a mathematical structure with the potential
to lead them to the discovery of the place-value
system and consequently the “invention" of
zero. See High numbers. For an explanation of
the symbolism of these diverse words, see: High
numbers (The symbolic meaning of), Zero,
Numeration of numerical symbols and
Numerical symbols (Principle of the
numeration of).
NARAYANA. Indian mathematician c. 1356.
His works notably include Ganitakaumudi.
Here is a list of the principal names of
numbers mentioned in that work: *Eka (=
l),* Dasha (= 10), *Shata (= 10 2 ), *Sahasra (=
10 3 ), *Ayuta (= 10* 1 ), *laksha (= 10 s ), *Prayuta
(= 10 6 ), *Koti (= 10 7 ), *Arbuda (= 10 8 ), *Saroja
(= 10 9 ), *Kharva (= 10 l0 ), *Nikharva (= 10 u ).
*Mahapadma (= 10 12 ), *Shanku (= 10 13 ).
*Pdrdvdra (= 10 14 ), * Madhya (= 10 15 ), *Antya
(_ iqi 6 ), *Parardha (= 10 17 ).
See Names of numbers and High
numbers. [See Datta and Singh (1938), p. 13]
NASATYA. [S]. Value = 2. Name of one of the
two twin gods Saranyu and Vivashvant of the
Hindu pantheon (also called *Dasra and
Nasatya). The symbolism is through an
association of ideas with the “Horsemen”. See
Ashvin and Two.
NAVA (NAVAN). Ordinary Sanskrit names for
the number nine, which appear in the
composition of many words which have a
direct relationship with the concept of this
number. Examples: *Navagraha, *Navaratna,
*Navardshika and *Navardtri. For words which
have a more symbolic relationship with this
number, see: Nine and Symbolism of numbers.
NAVACH ATVARIMSH ATI . Ordinary Sanskrit
name for the number forty-nine. For words
having a symbolic link to this number, see
Forty-nine and Symbolism of numbers.
NAVADASHA. Ordinary Sanskrit name for the
number nineteen. For words which have a
symbolic link to this number, see : Nineteen
and Symbolism of numbers.
NAVAGRAHA. Literally : “nine planets”. This
relates to the nine planets of the Hindu
cosmological system: the seven planets
(* saptagraha) plus the demons of the eclipses
*Rahu and Ketu. See Graha and Paksha.
NAVAN. Ordinary Sanskrit name for the
number nine. See Nava.
NAVARASHIKA. [Arithmetic]. Sanskrit name
for the Rule of Nine. See Nava.
NAVARATNA. “Nine jewels”, “Nine precious
stones”. Collective name given to the nine
famous poets of Sanskrit expression who are
said to have lived in the court of King
Vikramaditya (namely : Dhavantari, the pearl;
Kshapanaka, the ruby; Amarasimah, the topaz;
Shanku, the diamond; Vetdlabhatta, the
emerald; Ghatakarpara, the lapis-lazuli;
Kalidasa, the coral; Vardhamihira, the sapphire;
and Vararuchi, not identified to any specific
stone). See Nava and Ratna (= 9).
NAVARATRI. Name of the nine-day Feast.
See Durga.
NAVATI. Ordinary Sanskrit name for the
number ninety.
NAYANA. [SI. Value = 2. “Eye”. See Netra (= 2)
and Two.
NEPALI (Calendar). Beginning in 879. To find
the corresponding date in the Common Era,
simply add 879 to a date expressed in this
calendar. Still used occassionally in Nepal. Also
called Newari. See Indian calendars.
NEPALI NUMERALS. Signs derived from
*Brahmi numerals through the intermediary
notations of Shunga, Shaka, Kushana, Adhra,
Gupta, Nagari and Siddham numerals.
Currently used mainly in the independent state
of Nepal. They are also called Gurkhali
numerals. The corresponding system is based
on the place-value system and has a zero (in the
form of a little circle). For ancient numerals, see
Fig. 24.41. For modern numerals, see Fig. 24.15.
See Fig. 24.52 and 24.61 to 69. See also Indian
written numeral systems (Classification of)-
NETHER WORLD. [S]. Value = 7. See Pdtdla.
Seven.
NETRA. [S]. Value = 2. “Eye”. See Two.
NETRA. [S]. Value 3. “Eye”. Symbol used only
in regions of Bengal, where this word is
generally used to denote the three eyes of
*Shiva. See Three.
NEWARi (Calendar). See Nepali.
NIHILISM. See Shunyata and Zero.
NIKHARVA. Name given to the number ten to
the power nine. See Names of numbers and
High numbers.
Source: *Sankhyayana Shraula Sutra (date uncertain).
NIKHARVA. Name given to the number ten to
the power eleven. See Names of numbers and
High numbers.
Sources : *Kitab fi tahqiq i ma li’l hind by al-Biruni
(c. 1030 CE); *Lilavati by Bhaskaracharya (1150
CE); * Ganitakaumudi by Narayana (1350 CE);
*Trishatika by Shridharacharya (date unknown).
NIKHARVA. Name given to the number ten to
the power thirteen. See Names of numbers
and High numbers.
Source: * Pahchavimsha Brahmana (date uncertain).
NIL, NULLITY. See Shunyata and Zero.
NILAKANTHASOMAYAJIN. Indian astronomer
c. 1500 CE. His works notably include
Siddhdntadarpana, in which the place-value
system with Sanskrit numerical symbols is used
frequently [see Sarma, Siddhdntadarpana].
See Numerical symbols, Numeration of
numerical symbols and Indian mathematics
(The history of).
NINE. Ordinary Sanskrit names: *nava,
Here is a list of the corresponding numerical
symbols: Abjagarbha, Aja, *Anka, Brihati,
*Chhidra, * Durga, Dvara, *Go, *Graha, *Keshava,
Khanda, Laddha, Labdhi, Nan da, Nidhdna, Nidhi,
Padartha, *Randhra, * Ratna, Tdrkshyadhvaja,
Upendra, Varsha. These words have the following
literal or symbolic meaning: 1. The Brahman
(Abjagarbha, Aja). 2. The name of the ninth
month of the chaitradi year (Keshava). 3. The
numerals of the place-value system (Anka). 4. The
“Inaccessible”, the “Divine Mother", in allusion
to a divinity of the same name ( Durgd ). 5. The
Jewels (Ratna). 6. The holes, the orifices ( Chhidra ,
Randhra). 7. The planets (Graha). 8. The radiance
(Go). 9. The “Cow” to denote the earth (Go). See
Numerical symbols.
NINETEEN. Ordinary Sanskrit name :
*navadasha. The corresponding numerical
symbol is *Atidhriti. Note that at the beginning
of the Common Era, and probably since
Vedic times, this number was also called
ekannavimshati, which literally means “one
away from twenty” [see Taittiriya Samhitd, VII,
2. 11]; but it is also used in its normal form
from this time [See Taittiriya Samhitd, XIV, 23;
Datta and Singh (1938), pp. 14-15].
NINETY. See Navati.
NINNAHUT. Name given to the number ten to
the power eleven. See Names of numbers and
High numbers.
Source : * Lalitavistara Sutra (before 308 CE).
NINNAHUTA. Name given to the number ten
to the power thirty-five. See Names of
numbers and High numbers.
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
NIRABBUDA. Name given to the number ten
to the power sixty-three. See Names of
numbers and High numbers.
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
483
NIRAVADYA
NIRAVADYA. Name given to the number ten
to the power forty-one. See Names of numbers
and High numbers.
Source : *ialitavistara Sutra (before 308 CE).
NIRVANA. According to Indian philosophers,
this is the supreme state of non-existence,
reincarnation and absorption of the being in
the Brahman. See Shunyata and Zero.
NIYUTA. Name giver, to the number ten to the
power five. See Names of numbers and High
numbers.
Sources : * Vajasaneyi Samhitd, * Taittiriya Samhitd
and * Kathaka Samhitd (from the star: of the first
millennium CE); * Pahchavimsha Brdhmana (date
uncertain); *Aryabhaliya (510 CE).
NIYUTA. Name given to the number ten to the
power six. See Names of numbers and High
numbers.
Source : * Kathaka Samhitd (start of the Common
Era).
NIYUTA. Name given to the number ten to the
power eleven. See Names of numbers and
High numbers.
Source : * Lalitavistara Sutra (before 308 CE).
NON-BEING. See Shunyata and Zero.
NON-EXISTENCE. See Shunyata and Zero.
NON-PRESENT. See Shunyata and Zero.
NON-PRODUCT. See Shunyata and Zero.
NON-SUBSTANTIALITY. See Shunyata and
Zero.
NON-VALUE. See Shunyata and Zero.
NOTHING. See Shunya and Zero.
NOTHINGNESS. See Shunyata and Zero.
NRIPA. [S]. Value = 16. “King”. This is an
allusion to the sixteen kings of the epic poems of
the *Mahabharata (Brihadbala, king of
Koshala; Chitrasena, king of the Gandharva;
Dhritarashtra, the blind king of Indraprastha;
Drupada, king of the Panchala; Jayadratha, king
of the Sindhu; Kartavirya, king of the Haihaya;
Kashipati, king of the Kashi; Madreshvara, king
of the Madra; king Pradtpa; Shatayupa, ascetic
king; Shishupala, king of the Chedi; Subala,
king of Gandhara; Vajra, king of Indraprastha;
Virata, king of the Matsya; Yavanadhipa, king of
the Yavana; and Yudhisthira, king of
Indraprastha). See Sixteen.
NUMBERS (Philosophy and science of). See
Samkhya, Samkhya, Samkhya, Samkhya,
Numerical symbols, Symbolism of words with
a numerical value, Symbolism of numbers,
Shunya, Shunyata, Zero, Infinity and Mysticism
of infinity.
NUMBERS (The science of). See Samkhydna. See
also Numbers (The philosophy and science of).
NUMERAL "0” (in the form of a little circle).
Currently the symbol used in nearly all the
numerical notations of India (the following
types of modern numerals : Nagari, Gujarati,
Marathi, Bengali, Oriya, Punjabi, Sindhi,
Gurumukhi, Kaithi, Maithili, Takari, Telugu,
Kannara, etc.), of central Asia (Nepali and
Tibetan numerals) and of Southeast Asia
(Thai-Khmer, Balinese, Burmese, Javanese, etc.
numerals). There is evidence of the use of this
sign since the seventh century CE in the
Indianised civilisations of Southeast Asia
(Champa, Cambodia, Sumatra, Bali, etc.). See
Fig. 24.3 to 13, 24.15, 16, 21, 24, 25, 26, 28, 39,
41, 42, 50, 51. 52, 78, 79 and 24.80. See Indian
written numeral systems (Classification of).
See also Circle and Zero.
NUMERAL "0" (in the shape of a point or dot).
This was formerly in use in the regions of
Kashmir and Punjab (Sharada numerals).
There is evidence of the use of this sign since
the seventh century CE in the Khmer
inscriptions of ancient Cambodia. Today, this
sign is still used in Muslim India in eastern
Arabic numeration (“Hindi" numerals). See
Fig. 24.2, 14, 40, 78 and 80. See Indian written
numeral systems (Classification of), Eastern
Arabic numerals, Dot and Zero.
NUMERAL “1”. (The origin and evolution of
the). See Fig. 24. 61.
NUMERAL “ 2 (The origin and evolution of
the). See Fig. 24. 62.
NUMERAL “3". (The origin and evolution of
the). See Fig. 24. 63.
NUMERAL “4”. (The origin and evolution of
the). See Fig. 24. 64.
NUMERAL “5”. (The origin and evolution of
the). See Fig. 24. 65.
NUMERAL “6”. (The origin and evolution of
the). See Fig. 24. 66.
NUMERAL “7”. (The origin and evolution of
the). See Fig. 24. 67.
NUMERAL “8”. (The origin and evolution of
the). See Fig. 24. 68.
NUMERAL “9”. (The origin and evolution of
the). See Fig. 24. 69.
NUMERAL (as a sign of written numeration).
See Anka and Signs of numeration.
NUMERAL ALPHABET AND
COMPOSITION OF CHRONOGRAMS.
Chronograms can be found on certain
monuments. These are short phrases written in
Sanskrit (or Prakrit), the words of which, when
evaluated then totalled according to the
numerical value of their letters, give the date of
an event which has already taken place or will
take place in the future. In Muslim India, the
same procedure was used frequently, this time
using the numeral letters of the Arabic-Persian
alphabet. They are commonly found on
epitaphs to express the date of death of the
person buried in the tomb. See Numeral
alphabet, Chronogram. Chronograms
(System of letter numerals).
NUMERAL ALPHABET AND SECRET
WRITING. Like all those who have used an
alphabetical numeration, the Indians,
Sinhalese, Burmese, Khmers, Thais, Javanese
and Tibetans alike have used it to write in a
secret code. We still use such systems today to
write information or incantatory or magic
formulas. In this way numerical series are
written to hide their meanings should they fall
into the hands of the profane or uninitiated.
Likewise, if the order of pages are numbered in
this way, it prohibits the profane from reading
the texts, thus keeping them secret in a
coherent manner; the initiated only has to put
the pages in the correct order before he reads
the text. See Numeral alphabet.
NUMERAL ALPHABET, MAGIC, MYSTICISM
AND DIVINATION. As with the Greeks, the
Jews, the Syrians, the Arabs and the Persians,
the Indian mystics, Magi and soothsayers used
their numeral alphabets as the basic
instruments of their magical, divinatory or
numerological interpretations or practices. A
whole mystical-religious practice, just like
gnosis, Judaeo-Christian Cabbala or Muslim
Sufism was created in this manner. This led
to all kinds of homilectic and symbolic
interpretations, to various predictive
calculations and to the creation of certain
*kavachas, talismans curiously resembling
Hebrew Cabbalistic pentacles and Muslim hen
from North Africa. The practice was based on a
doctrine of sound and the Sanskrit alphabet:
*bijas or “letter-seeds", where each syllable of
the alphabet characterised a divinity of the
Brahmanic pantheon (or of the pantheon of
tantric Buddhism in the schools in the North),
whom it was believed that one could evoke just
by pronouncing the letter. The sound, by
definition, was considered to be the creative
and evocative element par excellence. Hence the
mystical value attached to each letter in
association with the esoterical meaning of its
numerical equivalent. The external sound of
the voice is born in the secret centre of the
person in the form of the essence of the sound,
and passes through three vibratory processes
before becoming audible: para, pashyanti and
madhyama. Beginning subtly, the sound turns
into one of the forty-six letters of the Sanskrit
alphabet. As the sound is transmitted by the
*nddi, it becomes one or another of the
Sanskrit alphabetical letters. The matter, in
Hindu cosmology, is divided into five states of
manifestation : air, fire, earth, water, ether.
Each state corresponds to a Sanskrit letter as is
shown in the following table:
Air (* Vayu): ka, kha, ga, gha, na, a, a, ri,
ha, sha.ya.
Fire (*Agni): cha, chha,ja,jha, ha, /, /, ri,
ksa, ra.
Earth Prithivi): ta, tha, da, dha, na, u, u,
li, sha, va, la.
Water ( *Apa ): ta, tha, da, dha, na, e, ai, li,
sa.
Ether (*Akasha): pa, pha, ba, bha , ma, o,
au, am, ah.
We can now understand the principle of the
creation of a *mantra, which is a combination
of sounds which have been carefully studied in
terms of their secret values. It is not worth
trying to make intelligible sense of a mantra
because this is not its aim; just as certain
numerical combinations enabled Cabbalists to
invent ingenious secret names, names which
are impossible to translate (they are artificial
creations), the mantra is a precise combination
of sounds created with some secret aim in mind
[Marques-Riviere (1972)1 See Akshara,
Numeral alphabet, Bhuta, Mahabhuta,
Trivarna, Vdchana. See also Chapter 20, for
similar practices in other cultures.
NUMERAL ALPHABET. This denotes any
system of representing numbers which uses
vocalised consonants of the Indian alphabet, to
which a numerical value is assigned, in a
predetermined, regular order. In keeping with
their diverse systems of recording numbers (in
numerals, in symbols or spoken), the Indians
knew and used different systems of this kind.
This is what is conveyed by the collective name
*varnasankhya, or systems of “letter-numbers”.
The inventor of the first numerical
alphabet in Indian history was the astronomer
* Aryabhata, who, c. 510 CE, had the idea
of using the thirty-three letters of the
Indian alphabet to represent all the numbers
from 1 to 10 18 . His aim in creating this system
was to express the constants of his
* astronomical canon, as well as the numerical
data of his diverse speculations on *yugas.
See Aryabhata (Numerical notations of),
D I C T IONARY OF INDIAN NUMERICAL SYMBOLS
484
Aryabhata’s numeration, Yuga
(astronomical speculation about).
After Aryabhata, many other numeral
alphabets were invented using Indian letters.
These vary both according to the numerical
value of the letters and the period and region,
and sometimes even the principle employed in
the numerical representations.
One such system is the katapayadi system,
which is still called *varnasamjna (or
“proceeding from syllables”); it was almost
certainly created by the astronomer *Haridatta
in the ninth century CE. and later adopted by
many astronomers, including Shankaranarayana
(c. 869 CE). It is a simplified version of
Aryabhata’s system; the successive vocalisations
of the consonants of the Indian alphabet are
suppressed. Each value which is superior or
equal to ten is replaced with a zero or one of the
first nine units. The author of the system thus
transformed the earlier system into an
alphabetical numeration which used the place-
value system and zero. See Katapayadi
numeration.
Amongst the diverse alphabetical
notations, it is also worth mentioning the
*aksharapa!li system, which is frequently used
in Jaina manuscripts. Such systems are still in
use today in various regions of India, from
Maharashtra, Bengal, Nepal and Orissa to
Tamil Nadu, Kerala and Karnataka. They are
also found amongst the Sinhalese, the
Burmese, the Khmers, the Thais and the
Javanese. They can also be found amongst the
Tibetans, who have long used their letters as
numerical signs, particularly when numbering
their registers and the pages of their
manuscripts. See Chapters 17 to 20 for similar
uses in other cultures.
NUMERAL. [S]. Value = 9. See Anka and Nine.
NUMERATION OF NUMERICAL SYMBOLS.
Name given here to the place-value system
written using Sanskrit numerical symbols, used
by Indian astronomers and mathematicians
since at least the fifth century CE. In Sanskrit,
this is often called *samkhya (or *sankhya). See
Numerical symbols (Principle of the
numeration of).
NUMERICAL NOTATION. Here is an
alphabetical list of terms relating to this
notion, which appear as headings in this
dictionary: *AksharapalH, *Andhra numerals,
*Anka, *Ankakramena, *Ankanam Varna to
Gatih, *Ankapalli, *Ankasthdna, *Arabic
numeration (Positional systems of Indian
origin), ‘Aryabhata’s numeration, ‘Brahmi
numerals, ‘Eastern Arabic numerals, ‘High
numbers, ‘Indian numerals, ‘Indian written
numeral systems (Classification of), ‘Indusian
numeration, ‘Katapayadi numeration,
‘Kharoshthi numeration, ‘Numeral alphabet,
‘Numeral 1, ‘Numeral 2, etc., ‘Numerical
symbols (Principle of the numeration of),
‘Sanskrit ‘Sthana, ‘Sthanakramad,
‘Varnasamjna and ‘Zero.
NUMERICAL SYMBOLS. These are words
which are given a numerical value depending
what they represent. They can be taken from
nature, the morphology of the human body,
representations of animal or plants, acts of
daily life, any types of tradition, philosophical,
literary or religious elements, attributes and
morphologies connected to the divinities of
the Hindu, Jaina, Vedic, Brahmanic, Buddhist,
etc. pantheons, legends, traditional
associations of ideas, mythologies or social
conventions of Indian culture. See Symbols.
See also all entries entitled Numerical
symbols or Symbolism of numbers.
NUMERICAL SYMBOLS (General alphabetic
list). These are Sanskrit numerical symbols which
are found in texts on mathematics or astronomy,
as well as in various Indian epigraphic
inscriptions (this list is not exhaustive):
*Abdhi (= 4), *Abhra (= 0), *Abja (= 1),
Abjadala (= 100), Abjagarbha (= 9), Achala
(= 7), *Adi (= 1), *Aditya (= 12), *Adri (= 7),
*Aga (= 7), Aghosha (= 13), *Agni (= 3), *Ahar
(= 15), *Ahi (= 8), Airavata (= 1), Aja (= 9),
*Akasha (= 0), *Akriti (= 22), *Akshara (= 1),
Akshauhini (= 11), Akshi (= 2), *Arnara (= 33),
Ambaka (= 2), *Ambara (= 0), *Ambhodha (=
4), Ambhodhi (= 4), Ambhonidhi (= 4),
*Ambodha (= 4), Ambodhi (= 4), Ambudhi
(= 4), *Amburdshi (= 4), *Anala (= 3), *Ananta
(= 0), *Anga (= 6), *Anguli (= 10), *Anguli
(= 20), Anika (= 8), *Anka (= 9), *Antariksha (=
0), *Anushtubh (= 8), *Aptya (= 3), Arhat
(= 24), Ari (= 6), *Arjundkara (= 1,000), *Arka
(= 12), *Arnava (= 4), Artha (= 5), *Asha (= 10),
Ashrama (= 4), *Asbti (= 16), *Ashva (= 7),
*Ashvin (= 2), *Ashvina (= 2), *Ashvinau (= 2),
*Atidhriti (= 19), Atijagati (= 13), * Atman (= 1),
*Atri (= 7), * Atrinayanaja (= 1), *Atyashti
(= 17), *Avani (= 1), *Avatara (= 10), Aya (= 4),
Aya (= 4), Ayana (= 2).
*Bahu (= 2), *Bana (= 5), Bandhu (= 4),
*Bha (= 27), *Bhdnu (= 12), *Bharga (= 11),
Bhdva (= 5), *Bhava (= 11), Bhaya (= 7), *Bhu
(= 1), *Bhubrit (= 7), *Bhudhara (= 7), *Bhumi
(= 1), *Bhupa (= 16), *Bhuta (= 5), Bhuti (= 8),
*Bhuvana (= 3), *Bhuvana (- 14), *Bindu (= 0),
*Brahmasya (= 4), Brihati (= 9).
* Chakra (= 12), *Chakshus (= 2), Chandah
(= 7), Chandas (= 7), *Chandra (= 1),
*Chaturdnanavadana (= 4), *Chhidra (= 9).
Dadhi (= 4), * Dab an a (= 3), *Danta (= 32),
*Dantin (= 8), *Darshana (= 6), *Dasra (= 2),
*Deva (= 33), *Dhara (= 1), *Dharani (= 1),
* Dhartarashtra (= 100), *Dhatri (= 1), Dhatu
(= 7), Dhi (= 7), *Dhriti (= 18), *Dhruva (= 1),
*Diggaja (= 8), Dik (= 8), *Dikpdla (= 8), Dina
(= 15), * Dish (= 4), *Dish (= 8), *Disha (= 10),
*Disha (= 4), * Dish a (= 10), *Divdkara (= 12),
Dosha (= 3), *Dravya (= 6), *Drishti (= 2),
*Durga (= 9), Durita (= 8), *Dvandva (= 2),
Dvdra (= 9), *Dvaya (= 2), *Dvija (= 2), *Dvipa
(= 8), *Dvipa (= 7), Dvirada (= 8), Dyumani
(= 12).
*Gagana (= 0), *Gaja (= 8), Gangamarga
(= 3), *Gati (= 4), *Gavyd (= 5), *Gayatri (= 24),
Ghasra (= 15), *Giri (= 7), *Go (= 1), *Go (= 9),
Gostana (= 4), *Graha (= 9), Grahana (= 2),
*Gulpha (= 2), *Guna (= 3), *Guna (= 6).
*Hara (= 11), *Haranayana (= 3),
*Haranetra (= 3), *Haribahu (= 4), *Hastin
(= 8), Maya (= 7), Himagu (= 1), Himakara (= 1),
Himamshu (= 1), *Hotri (= 3), *Hutdshana (= 3).
Ibha (= 8), Ikshana (= 2), Ila (= 1), *Indra
(= 14), * Indradrishti (= 1,000), *Indriya (= 5),
*Indu (= 1), *Irya (= 4), *lsha (=11), *fshadrish
(= 3), *Ishu (= 5), *lshvara (= 11).
*Jagat (= 3), *]agat (= 14), *Jagati (= 1),
*Jagatt (= 12), *]agati (- 48), *Jahnavivaktra
(= 1,000), *Jala (= 4), *Jaladharapatha (= 0),
*JaIadhi (= 4), *]alanidhi (= 4), Jaldshaya (= 4),
Jana (= 1), Jangha (= 2), Janu (= 2), Jati (= 22),
Jina (= 24), *Jvalana (= 3).
*Kakubh (= 10), *Kdla (= 3), Kald (= 16),
Kalamba (= 5), Kalatra (= 7), *Kama (= 13),
*Kara (= 2), Karaka (= 6), *Karaniya (= 5),
Karman (= 8), Karman (= 10), Kama (= 2),
* Karttikeyasya (= 6), Kashaya (= 4), *Kdya (= 6),
Kendra (= 4), *Keshava (= 9), *Kha (= 0),
Khanda (= 9), Khara (= 6), Khatvapada (= 4),
Koshtha (= 4), *Krishanu (= 3), *Krita (= 4),
*Kriti (= 20), Kritin (= 22), Kshapakara (= 1),
* Kshapeshvara (= 1), Kshara (= 5), *Kshauni
(= 1), *Kshema (= 1), *Kshiti (= 1), *Kshoni
(= 1), *Ku (= 1), Kucha (= 2), *Kumarasya (= 6),
* Kumdravadana (= 6), *Kuhjara= 6), (= 8),
Kutumba (= 2).
labdha (= 9), Labdhi (= 9), Labha (= 11),
Lakara (= 10), Lavana (= 5), Lekhya (= 6), *Loka
(= 3), *Loka (= 7), *Loka (= 14), *Lokapala
(= 8), *Lochana (= 2).
Mada (= 8), *Mahdbhuta (= 5), *Mahddeva
(= 11), *Mahdpdpa (= 5), *Mahayafha (= 5),
*Mahi (= 1), *Mahidhara (= 7), Mala (= 6),
*Mangala (= 8), Man math a (= 13), *Manu
(= 14), *Margana (= 5), *Mdrtanda (= 12),
*Masa (= 12), *Mdsardha (= 6), *Matanga
(= 8), *Matrika (= 7), *Mriganka (= 1), *Mukha
(= 4), Mulaprakriti (- 1), *Muni (= 7), *Murti (=
3), *Murti (= 8).
*Nabha (= 0), *Nabhas (= 0), Nadi (= 3),
Nadikula (= 2), *Naga (= 7), *Naga (= 8),
*Nakha (= 20), *Nakshatra (= 27), Nanda (=
9), Naraka (= 40), *Nasatya (= 2), Naya (= 2),
Nayaka (= 1), *Nayana (= 2), *Nctra (= 2),
*Netra (= 3), Nidhana (= 9), Nidhi (= 9),
*Nripa (= 16).
Oshtha (= 2).
Paddrtha (= 9), *Paksha (= 2), *Paksha
(= 15), Pallava (= 5), *Pandava (= 5), Pankti
(= 10), * Parabrahman (= 1), Parva (= 5), Pan>an
(= 5), *Parvata (= 7), *Pataka (= 5), *Patala
(= 7), *Pavaka (= 3), *Pavana (= 5), *Pdvana
(= 7), Payodhi (= 4), Payonidhi (= 4),
* Pinakanayana (= 3), *Pitamaha (= 1), *Prakriti
(= 21), Praleyamshu (= 1), * Prana (= 5), *Prithivi
(= 1), *Pura (= 3), *Purd (= 3), * Puranalakshana
(= 5), *Purna (= 0), Purushartha (= 4), Purushayus
(= 100), Purva (= 14), *Pushkara (= 7), Pushkarin
(= 8), *Putra (= 5).
* Rada (= 32), *Raga (= 6), Rajanikara (= 1),
*Rama (= 3), Rdmanandana (= 2), *Randhra
(= 0), *Randhra (= 9), *Rasa (= 6), *Rashi
(= 12), Rashmi (= 1), *Ratna (= 3), *Ratna (= 5),
*Ratna (= 9), *Ratna (= 14), *Ravanabhuja
(= 20), * Ravanashiras (= 10), *Ravi (= 1.2),
*Ravibana (= 1,000), *Ravichandra (= 2), Ripu
(= 6), *Rishi (= 7), *Ritu (= 6), *Rudra (= 11),
*Rudrasya (= 5), *Rupa (= 1).
*Sagara (= 4), *Sagara (= 7),
*Sahasramshu (= 12), Sahodarah (= 3),
Salilakara (= 4), *Samudra (= 4), *Samudra (=
7), Sankranti (= 12), *Sarpa (= 8), *Sayaka (=
5), Senanga (= 4), * Senaninetra (= 12),
*Shaddyatana (= 6), *Shaddarshana (= 6),
*Shadgunya (= 6), *Shaila (= 7), *Shakra (=
14), Shakrayajha (= 100), *Shakti (= 3),
* Shankarakshi (= 3), *Shanmukha (= 6),
*Shanmukhabahu (= 12), *Shara (= 5),
*Shashadhara (= 1), *Shashanka (= 1),
*Shashin (= 1), Shastra (= 5), Shastra (= 6),
*Sheshashirsha (= 1.000), *Shikhin (= 3),
*Shitamshu (= 1), *Shitarashmi (= 1), * Shiva
(= 11), *Shruti (= 4), *Shukranetra (= 1),
*Shula (= 3), *Shulin (= 11), *Shunya (= 0),
Shveta (= 1), Siddha (= 24), *Siddhi (= 8),
*Sindhu (= 4), Sindhura (= 8), *Soma (= 1),
*Sudhamshu (= 1), *Sura (= 33), *Surya (= 12),
*Suta (= 5), Svagara (= 21) *Svara (= 7).
485
NUMERICAL SYMBOLS
* Takshan (= 8 ), *Tana (= 49), Tanmatra (=
5 ), * Tanu (= 1), * Tartu (= 8), *Tapana (= 3),
*Tapana (= 12), (= 6), Tarkshadhvaj (= 9),
Tata (= 5), *Tattva (= 5), *Tattva (= 7), *Tattva
(= 25), *Tithi (= 15), * Trailokya (= 3), *Trayi
(= 3), Tridasha (= 33), Trigala (= 3), *Triguna
( = 3), *Trijagat (= 3), *Trikala (= 3), *Trikdya
(= 3), *Triloka (= 3), *Trimurti (= 3), *Trinetra
3 ) * Tripura (= 3), *Triratna (= 3), * Trishiras
(= 3), Trishtubh (= 11), *Trivarga (= 3), *Trivarna
(= 3), *Tryakshamukha (= 5), * Tryambaka (= 3),
*Turaga (= 7), *Turangama (= 7), *Turiya (= 4).
* Uchchaishravas (= 1), *Uda (= 27), *Udadhi
(= 4), *Udarchis (= 3), Upend ra (= 9), *Utkriti (=
26), * t/rvura (= 1 ).
*Vachana (= 3), *Vahni (= 3), *Vaishvanara
(= 3), ‘Vay/'n (= 7), Vanadhi (= 4), "Kara (= 7),
‘Varidhi (= 4), *H irinidhi (= 4), KarsAa (= 9),
*lfea (= 8), ‘Vasudha (= 1), ‘Vasundhara (= 1),
* Vayu (= 49), * Veda (= 3), * VWa (= 4), * Vidhu (=
1), Wrfya (= 14), ‘Vikriti (= 23), ‘Vindu (= 0),
Visbanidhi (= 4), ‘Vishaya (= 5), * Vishikha
(= 5), * Vishnupada (= 0), Vishtapa (= 3), Vishuvat
(= 2), ‘Vishva (= 13), ‘Vishvadeva
(= 13), Viyata (= 0), Vra/a (= 5), * tyanl (= 0),
tyaia»a (= 7), tyaya (= 12), ‘Vyoman (= 0),
tyaAa (= 4).
*Yama (= 2), Kama (= 8 ), *YamaIa (= 2),
* Yamau (= 2), Ka/i (= 7), *Yoni (= 4), *Yuga
(= 2), *y«ga (= 4), * Yugala (= 2), ‘Yugma (= 2).
To gain an idea of the symbolism of
these words, see Symbolism of words with
a numerical value and Symbolism of
numbers. The first of these two entries
gives an alphabetical list of English terms
which explain the various corresponding
associations of ideas, and the second entry
gives a list of the same associations of ideas,
set out this time in numerical order (one, two,
three, etc.). To understand the principle for
using word-symbols to represent numbers,
see Numerical symbols (Principle of the
numeration of).
Source: Biihler (1896), pp. 84ff; Burnell (1878); Datta
and Singh (1938), pp. 54-7; Fleet, in : Clin, VIII;
Jaquet, in : JA, XVI, 1835; Renou and Filliozat (1953),
P- 708-9; Sircar (1965), pp. 230-3; Woepcke (1863).
NUMERICAL SYMBOLS (Principle of the
numeration of). Procedure used to record
numbers by Indian scholars since at least as
early as the fifth century CE. This is simply a
series of Sanskrit word-symbols (which are
u sed as names of units), which are written in
conformity with the “principle of the
movement of numerals from the right to tl
( *ankdnam vamato gatih). See Sanskr
and Numerical symbols.
In other words, in this system, numerical
symbols have a variable value depending on
their position when numbers are written down.
The system possesses several different special
terms which symbolise zero and which thus
serve to mark the absence of units in any given
decimal order in this positional notation
(*shunya, *dkasha, *abhra, *ambara,
*antariksha, *bindu, *gagana, *jaladharapatha,
*kha, *nabha, *nabhas, etc.). An expression
such as:
agni. shunya. ashvi. vasu.
[literally : “fire (= 3). void (= 0). Horsemen
(= 2). Vasu (= 8)”] corresponds to the numbers:
3 + 0 x 10 + 2 x 10 2 + 8 xlO 3 = 8,203.
This method of expressing numbers uses the
place-value system and zero. What is remarkable
about it is that Indian scholars are the only ones
to have invented such a system. See Position of
numerals, and Zero.
NUMERICAL SYMBOLS (Sanskrit desig-
nation of). The generic term for words used as
numerical symbols is *samkhya, which literally
means “number’'. Also used to refer to the
system as a whole, which is the place-value
system expressed through numerical symbols.
NUMEROLOGY. See Numeral alphabet,
magic, mysticism and divination.
NYARBUDA. Name given to the number ten to
the power eight (= one hundred million). See
Names of numbers and High numbers.
Sources: *Vdjasaneyi Sam hit a (beginning of the
Common Era); *Taittiriya Samhita (beginning of the
Common Era); *Kdthaka Samhita (beginning of the
Common Era); *Parichavimsha Brahmana
(date uncertain); * Sankhydyana Shrauta Sutra
(date uncertain).
NYARBUDA. Name given to the number ten to
the power 11. See Names of numbers and High
numbers.
Source: *Ganitasarasamgraha by Mahahaviracharya
(850 CE).
O
OCEAN. Name given to the number ten to the
power four, ten to the power nine or ten to the
power fourteen. See Jaladhi, Samudra and High
numbers.
OCEAN. IS]. The entries entitled *sagara or
*samudra, which, as numerical symbols,
translate the idea of "sea" or “ocean”, can have
the value of either 4 or 7. The relation between
sagara and 4 can be explained through the
allusion to the “four oceans” (*chatursagara)
which, according to Hindu and Brahmanic
mythologies, surround *Jambudvipa, (India).
However, this explanation does not give the real
reason for the choice of the number four for the
oceans surrounding India. In reality, it is due to
the fact that the mystical symbol for “water”
( *jala ) is the number four. According to
Brahmanic doctrine of the five elements of the
manifestation (*bhuta), water (which is also
(called *apa) forms, along with earth (prithivf),
air {vayu) and fire (agnf), the ensemble of
elements which are said to participate directly in
the “material order of nature”. This order is
believed to be quaternary, and the diverse
phenomena of life boil down to the
manifestations of these four elements in the
determination of the essence of the forces of
nature as well as in the realisation of the latter in
its work of generation and destruction. In
traditional Indian thought (and even according
to a universal constant), the earth itself
corresponds symbolically to the number four,
because it is associated with a square due to its
four horizons (or cardinal points).
As for the relationship between *sdgara and
the number seven, this can be explained by
direct reference to the seven mythical oceans
(namely: The ocean of salt water, the ocean of
sugar cane juice, the ocean of wine, the ocean of
thinned butter, the ocean of whipped cheese, the
ocean of milk and the ocean of soft water),
which are meant to surround * Mount Meru. See
Sapta sagara.
Mount Meru is the mythical and sacred
mountain of Brahman mythology and Hindu
cosmology, which constitutes the meeting place
and dwelling of the gods. Situated at the centre
of the universe, this mountain is placed above
seven hells ( *patala ), and has seven faces, each
one looking at one of the seven "island-
continents”, themselves each in one of the seven
oceans, etc. In this symbolism, Mount Meru
represents the total fixedness and the absolute
centre around which the firmament and the
whole universe pivot in their eternal course.
This image is connected to one of the universal
symbolic representations of the *Pole star.
Mount Meru is said to be situated directly
underneath this star, and along exactly the same
axis. This symbolic correspondence comes from
the fact that the Pole star, the *Sudrishti, “That
which never moves”, is the last of the seven stars
of the Little Bear, which themselves are
considered by Indian tradition to be the
personification of the seven “great Sages” (in
other words the *saptarishi of Vedic times,
believed to be the authors of hymns and
invocations, as well as of the most important
texts of the *Veda). This is why the number
seven came to play a preponderant symbolic
role in the mythological representations
associated with Mount Meru.
It is this symbolism which determined the
number of cosmic oceans in the legends about
the creation of the universe, and gave words
expressing the idea of “ocean” a value of 7. In
its representations, India, (Jambudvipa) is
considered to be the “centre of the earth”, whilst
Mount Meru was regarded as the centre of the
universe. Ocean has two different numerical
values in order to mark the opposition between
the human character, essentially terrestrial, of
the oceans surrounding India, and the divine
character, essentially celestial, of the oceans
surrounding Mount Meru. In spite of the
apparent paradox, Indian scholars managed to
avoid any confusion. The words *samudra and
* sagara, which both mean “ocean”, were both
sometimes used as symbols for the number four.
But they were usually used (never
simultaneously) to express the number seven,
words such as *abdhi, *ambhonidhi, ambudhi,
*amburashi, *jaladhi, *jalanidhi, ala shay a,
*sindhu, * varidhi or *vdrinidhi being reserved
for the number four, and which more modestly
meant “sea”.
OCEAN. IS]. Value = 4. See Abdhi, Ambhonidhi,
Ambudhi, Amburashi, Amava, Jaladhi,
Jalanidhi, Sagara, Samudra, Sindhu, Udadhi,
Varidhi and Varinidhi. See also four, Jala.
OCEAN. [S]. Value = 7. See Sagara and
Samudra. See also Seven, Mount Meru.
OLD KHMER NUMERALS. Symbols derived
from *Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, Ganga, Valabhi, “Pali” and Vatteluttu
numerals. Used from the seventh century CE in
the ancient kingdom of Cambodia. The notation
used for dates in the * Shaka era were based on a
place- value system and had a zero (a dot or
small circle), whereas vernacular notation was
very rudimentary. See: Indian written
numerals systems (Classification of). See Fig.,
24.52, 61 to 69, 77, 78 and 80.
ONE. Ordinary Sanskrit name for this number:
*Eka. Here is a list of corresponding numerical
symbols: *Abja, *Adi, Airavata, *Akshara,
* Atman, *Atrinayanaja, *Avani, *Bhu, *Bhumi,
*Chandra, *Dhara, *Dharani, *Dhdtri, *Dhruva,
*Go, Himagu, Himakara, Himamshu, lid, *Indu,
*Jagati, Jana, Kshapakara, * Kshapeshvara,
*Kshauni, *Kshema, *Kshiti, *Kshoni,
Dic:x IONARY OF INDIAN NUMERICAL SYMBOLS
486
*Ku, *Mahi, *Mrigdnka, Mulapra-
kriti, Ndyaka, * Parabrahman , *Pitamaha,
Praleydmshu, * Prithivi, Rajanikara, Rashmi,
*Rupa, *Shashadhara, *Shashanka, *Shashin,
Shveta, *Shitdmshu, *Shitarashmi, *Shukranetra,
*Soma, *Sudhdmshu, *Tanu, * Uch-chaishravas ,
*Urvard, *Vasudha, *Vasundhara, *Vidhu,
These words have the following translation
or symbolic meaning: 1. The '“Moon".
( Abja , Atrinayanaja, Chandra, Indu, Jagati,
Kshapeshvara, Mriganka, Shashadhara,
Shashanka, Shashin, Shitamshu, Shitarashmi,
Soma, Sudhamshu, Vidhu). 2. The drink of
immortality (Soma). 3. The “Earth” (Avani, Bhu,
Bhumi, Dhara, Dharant, Dhatri, Go, Jagati,
Kshauni, Kshema, Kshiti, Kshoni, Ku, Mahi,
Prithivi, Urvara, Vasudha, Vasundhara). 4. The
“Ancestor”, the “First Father”, the “Great
Ancestor” ( Pitamaha ). 5. Individual soul,
supreme soul, ultimate Reality, the Self
(Atman). 6. The Brahman (Atman, Pitamaha,
Parabrahman). 7. The beginning (Adi). 8. The
body (Tanu). 9. The Pole star (Dhruva). 10. The
form (Rupa). 11. The “drop” (Indu). 12. The
“immense” (Prithivi). 13. The “Indestructible”
(Akshara). 14. The rabbit (Shashin,
Shashadhara). 15. The “Luminous”, in allusion
to the moon as a masculine entity (Chandra). 16.
The “cold Rays” of the moon (Shitamshu,
Shitarashmi). 17. The terrestrial world (Prithivi).
18. The eye of Shukra (Shukranetra). 19. The
“Bearer”, in allusion to the earth (Dharant). 20.
The primordial principle (Adi). 21. Rabbit figure
(Shashadhara). 22. The Cow (Go, Mahi). 23.
Curdled milk (Mahi). See Numerical symbols.
OPINION. [SI. Value = 6. See Darshana and Six.
ORDERS OF BEINGS (The five). See
Pahchaparamesthin .
ORIFICE. [SJ. Value = 9. See Chhidra, Randhra
and Nine.
ORIGINAL SERPENT (Myth of the). See
Infinity (Indian mythological representation
of) and Serpent (Symbolism of the).
ORISSI NUMERALS. See Oriya Numerals.
ORIYA NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Gupta, Nagari,
Kutila and Bengali. Now used mainly in the
state of Orissa. Also called Orissi numerals.
The symbols correspond to a mathematical
system that has place values and a zero (shaped
like a small circle). See Indian written numeral
systems (Classification of). See Fig. 24.12, 52
and 24.61 to 69.
OUROBOUROS. See Infinity (Indian
mythological representation of) and Serpent
(Symbolism of the).
P
PADMA (or PADUMA). This is the name for
the pink lotus. As well as the purity it
represents, to the Indian mind it symbolises
the highest divinity as well as innate reason.
PADMA. Name given to the number ten to the
power nine. See Names of numbers. See also
High numbers (The symbolic meaning of).
Source: *Kitab fi tahqiq i ma li’l hind by al Biruni
(c. 1030 CE).
PADMA. Name given to the number ten to the
power fourteen. See Names of numbers. For
an explanantion of this symbolism, see Padma
(or Paduma). See also High numbers (The
symbolic meaning of).
Source: * Ganitasarasamgraha by Mahaviracharya
(850 CE).
PADMA. Name given to the number ten to the
power twenty-nine. See Names of numbers.
See also High numbers (The symbolic
meaning of).
Source: *Rdmayana by Valmiki (early centuries CE).
PADUMA. Literally, “(pink) lotus”. Name
given to the number ten to the power twenty-
nine. See Names of numbers. See also High
numbers (The symbolic meaning of).
Source: * Lalitavistara Sutra (before 308 CE).
PADUMA. Name given to the number ten to
the power 119. See Names of numbers. See also
High numbers (The symbolic meaning of).
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
PAIR. [S]. Value = 2. See Dvaya and Two.
PAKOTI. Name given to the number ten to the
power fourteen. See Names of numbers and
High numbers.
Source: *Vydkarana (Pali grammar) by
Kachchayana (eleventh century CE).
PAKSHA. [Sj. Value = 2. “Wing”. This is due to
the symmetry of this organ. The word can also
mean one of the two halves of a month. Thus it
is sometimes also used to represent the number
fifteen. This double symbolism can be
explained by the division of the month (*masa)
into two periods of fifteen days called paksha,
each one corresponding to one phase of the
moon. The first, called “shining” (shudi), is
progressive, and the second, called “shadow”
(badi), is degressive. According to Hindu
mythology and cosmogony, these two periods
formed one whole being (before the churning
of the sea of milk); this being w'as decapitated
by Indra when he drank the *amrita (the nectar
of eternal life) that he had stolen. This created
the “Cut in twos” (Ashleshabava): two beings
named *Rahu and *Ketu, who personify the
ascending and descending nodes of the moon.
See Masa, Rdhu and Two.
PAKSHA. [SI. Value = 15. See Fifteen.
“PALf” NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Pallava, Chalukya,
Ganga and Valabhl. Formerly used in Magadha
(the ancient Hindu kingdon of present-day
Bihar, south of the Ganges) from the Mauryan
period. All the later numeral symbols of the
eastern and southeast Asia (Mon, Burmese,
Cham, Ancient Khmer, Thai-Khmer, Balinese,
etc.) derive from Pall numerals. The symbols
corresponded to a mathematical system that
was not based on place values and therefore did
not possess a zero. See: Indian written
numerals systems (Classification of). See Fig.
24.52 and 24.61 to 69.
PALLAVA NUMERALS. Symbols derived from
•Brahmi numerals and influenced by Shunga,
Shaka, Kushana and Andhra, arising at the
time of the Pallava dynasty (fourth to sixth
centuries CE). The symbols correspond to a
mathematical system that was not based on
place values and therefore did not possess a
zero. See: Indian written numeral systems
(Classification of). See Fig. 24.37, 24.61 to
24.69 and 24.70.
PANCHA. Ordinary Sanskrit term for the
number five, which appears in many words
which have a direct relationship with the idea
of this number. Examples:
*Pahchabana, *Pahchdbhijha, *Pahchabhuta,
*Pahchachakshus, *Pahchadisha, *Pahchagavya,
*Pahcha Indriyani, *Pahcha Jati, * Pahchaklesha,
*Pahchanana, *Pahchanantarya, * Pahcha-
parameshtin, * Pahcharashika, *Pahchatantra.
For words which have a more symbolic
relationship with this number, see Five and
Symbolism of numbers.
PANCHABANA. "Bow of five flowers”. This is
one of the attributes of *Kama, Hindu divinity
of Cosmic Desire and Carnal Love, who is
generally invoked in marriage ceremonies.
Kama is often represented as a young man
armed with a bow of sugar cane and five arrows
covered in or constituted by five flowers.
PANCHABHIJNA. Name given by the
Sinhalese to the “five supernatural powers” of
Buddha. The Buddhists of Sri Lanka only
recognise five of the six Abhijha, or
“supernatural powers”, which other Buddhist
philosophies believe in.
PANCHABHUTA. “Five elements”. Collective
name for the five elements of the manifestation
of Brahman and Hindu philosophies. See
Bhuta and Jala.
PANCHACHAKSHUS. “Five visions of Buddha”.
According to Buddhists, Buddha possesses the
five following types of visions: that of the body, of
the divine form, wisdom, doctrine and of his eye.
PANCHADASHA. Ordinary Sanskrit name for
the number fifteen. For words with a symbolic
relationship with this number, see Fifteen and
Symbolism of numbers.
PANCHADISHA. “Five horizons". These are the
four cardinal points plus the zenith. See Dish.
PANCHAGAVYA. “Five gifts of the Cow”.
See Gavyd.
PANCHA INDRIYANI. “Five faculties”. These
are the mental and physical faculties of
Buddhist philosophy, which are divided into
five groups. See Indriya.
PANCHA JATI. Name of the five fundamental
arithmetical rules of the reduction of fractions.
PANCHAKLESHA. “Five impurities”. According
to Hindu and Buddhist philosophies, these are
the five major obstacles which keep the faiihful
off the Way of Realisation. See Mahapapa.
PANCHANANA. Name of the five heads of
*Rudra. See Rudrasya.
PANCHANANTARYA. “Five mortal sins" of
Buddhism. These are the following sins: parricide;
matricide; the killing of an arhat (a saint issued
from karma)] causing division in the Buddhist
community (sangham)] and wounding a Buddha.
PANCHAPARAMESHTIN. Name of the five
orders of beings, considered to be the “five
treasures" (Pahcha Ratna) of *Jaina religion.
PANCHAPARASHIKA. [Arithmetic!. Sanskrit
name for the Rule of Five.
PANCHASHAT. Ordinary Sanskrit name for
the number fifty.
PANCHASIDDHANTIKA. “Five astronomical
canons”. See Varahamihira and Indian astrology.
PANCHATANTRA. “Five books”. Name of the
famous collection of moralistic tales and fables,
made up of five books. The fables of Aesop and
La Fontaine are more or less directly inspired
by this collection. See Pahcha.
487
PANCHAVIMSHA BRAHMAN A
pANCHAVIMSHA BRAHMANA. Text
derived from the Samaveda, a text of Vedic
literature. The contents were transmitted
orally since ancient times, but were constantly
re-worked and added to, and did not achieve
their finished form until relatively recently.
Date uncertain. See Veda. Here is a list of the
main names of numbers mentioned in the text
(see Datta and Singh (1938), p. 10):
*Eka (= 1), * Dasha (= 10), *Saia (= 10 2 ),
* Sahasra (= 10 1 ), *Ayuta (= 10 4 ), *Niyuta (=
10 s ). *Prayuta (= 10 6 ), *Arbuda (= 100.
*Nyarbuda (= 10 H ). *Samudra (= 10 9 ),
* Madhya (= 10 10 ), *Anta (= 10 u ), *P arardha
(=10 12 ), *Nikharva (= 10 n ), * Vddava (= 10 H ),
*Akshiti (= 10“). See Names of numbers and
High numbers.
PANCHAV1MSHATI. Ordinary Sanskrit name
for the number twenty-five. For words which
are symbolically related to this number, see
Twenty-five and Symbolism of numbers.
PANDAVA. IS]. Value = 5. “Son of Pandu”. This
refers to one of the five brothers, semi-
legendary heroes of the epic *Mahabharata
(namely: Yudishtira, Arjuna, Bhima, Nakula,
and Sahadeva), son of the king Pandu of
Hastinapura. See Five.
PAPER. See Patiganita.
PARA. See Numeral alphabet, magic,
mysticism and divination.
PARABRAHMAN. [S]. Value = 1. Literally,
“Supreme Brahman”. Expression synonymous
with *Paramatman, in terms of “Supreme
Soul”, and an epithet given to Mahapurusha
(supreme entity of the global spirit of
humanity), considered in Hindu philosophy to
be the Absolute Lord of the universe and thus
identified with the Brahman. See Atman,
Pitamaha and One.
PARADISE. [S], Value = 13. See Vishvadeva
and Thirteen.
PARADISE. [S]. Value = 14. See Bhuvana and
Fourteen.
PARAMABINDU. “Supreme Point”. This is the
supreme causal point, which, according to
Buddhist philosophy, is both inexistent and
identical to all the universe; it is also time con-
sidered as a point (*bindu) which lasts no
sequential time but gives the impression of having
a duration [see Frederic, Dictionnaire (1987)).
PARAMANU. “Supreme Atom”. This is the
smallest indivisible material particle, and has a
taste, odour and colour. This is different to our
notion of the “atom”, and is more like what we
c all a “molecule", the smallest particle which
constitutes part of a compound body. The
paramanu and the * paramatta raja (or “grain of
dust of the first atoms”) have long been the
smallest units of length and weight in India.
These are found notably in the Legend of
Buddha, told in the * Lalitavistara Sutra, where
the paramanu corresponds to 0.000000287 mm
and the paramanu raja to 0.000000614 g.
PARAMANU RAJA. “Grain of dust of the
first atoms”. Name of the smallest Indian
unit of weight. At the time of the writing of
the * Lalitavistara Sutra (before 308 CE),
it corresponded to 0.000000614g. See
Paramanu.
PARAMATMAN. “Supreme Soul". Epithet
given to the * Brahman. See Parabrahman.
PARAMESHVARA. Indian astronomer c. 1431
CE. His works notably include the text entitled
Drigganita , in which there is abundant use of
the place-value system using Sanskrit
numerical symbols [see Sarma (1963)]. See
Numerical symbols, Numeration of
numerical symbols and Indian mathematics
(History of).
PARARDHA. From para, “beyond”, and ardha
“half”. This is the spiritual half of the path
which leads to death, identical to devayana, the
“way of the gods”, which, according to the
* Vedas, is one of the two possibilities offered to
human souls after death (this path being said
to lead to the deliverance from *samsara or
cycles of rebirth). The symbolism which has
led to these words having such high numerical
values as ten to the power twelve or ten to
the power seventeen comes from an
association of ideas, not only with the
immeasurable immensity of the sky, but also
with the eternity which it represents. For
more details, see High numbers (Symbolic
meaning of).
PARARDHA. Literally “half of the beyond".
Name given to the number ten to the power
twelve (= billion). See Names of numbers. For
an explanation of this symbolism, see Parardha
(first entry) and High numbers (Symbolic
meaning of).
Sources: * Vdjasaneyi Samhita, *Taittiriya Samhita and
*Kdthaka Samhita (from the start of the first millen-
nium CE); * Pahchavimsha Brahmana (date uncertain).
PARARDHA. Literally “half of the beyond".
Name given to the number ten to the power
seventeen. See Names of numbers. For an
explanation of this symbolism, see Parardha
(first entry) and High numbers (The symbolic
meaning of).
Sources: *Kitab fi tahqiq i ma li'l hind by al-Biruni
(c. 1030 CE); *IJIdvati by Bhaskaracharya (1150
CE); *Ganitakaumudi by Narayana (1350 CE):
*Trishatika by Shridharaeharya (date unknown).
PARASHURAMA (Calendar). See Kollam.
pARAVARA. Name given to the number ten to
the power fourteen. See Names of numbers
and High numbers.
Source; *Ganitakaumudi by Narayana (1350 CE).
PARJKARMA. Word used in arithmetic to
denote “fundamental operations" carried out
on whole numbers. See Kalasavarna.
PART. [SI- Value = 6. See Anga and Six.
PARTHIAN (Calendar). Calendar beginning
in the year 248 BCE. Formerly used in the
northwest of the Indian sub-continent. To find
a corresponding date in the Common Era,
subtract 248 from a date expressed in the
Parthian calendar. See Indian calendars.
PARVATA. [S]. Value = 7. “Mountain". Clearly
an allusion to the “Mountain of the gods"
(* devaparvata), one of the names for * Mount
Meru, which is said to be the home of the gods.
This numerical symbolism is due to the
preponderance of the number seven in the
mythological representations of Mount Meru.
See Adri and Seven.
PARVATI. See Mount Meru.
PASHYANT1. See Numeral alphabet, magic,
mysticism and divination.
PASSION. [SI. Value = 6. See Rdga and Six.
PATAKA. [Si. Value = 5. “Great sin”. See
Mahapapa and Five.
pAtAlA. [S]. Value = 7. “Inferior world”. This
refers to one of the seven "hells” of Hindu and
*Jaina mythology (namely: Atala, Vitala,
Nitala, Gabhastimat, Mahatala, Sutala and
Patala). These inferior worlds are said to be
situated one on top of the other underneath
*Mount Meru. They are the dwelling place of
the *naga, who are ruled by *Muchalinda, a
chthonian genie in the form of a cobra,
depicted as having seven heads. See Seven.
pAtAlA KUMARA. “Princess of the
Underworlds”. Name given to the daughter of
Himalaya, sister of Vishnu and wife of Shiva.
See Parvati.
PATI. Literally “Board”, “tablet”. Term used for
the calculating board or tablet, upon which
Indian mathematicians carried out their
calculations. See Patiganita and Indian
methods of calculation.
PATiGANITA (or GANITAPAt!). In its most
general sense, this word is used today to mean
“abstract mathematics". In the past, however, it
referred to “arithmetic” and to the “practice of
calculation”, and appeared in the titles of
many works relating to this discipline, for
example: Pdtisara by Munishvara (1658);
Ganitapatikaumudi by *Narayana (1356),
which deals notably with magic squares; and
Ganitatilaka by *Shripati (1039), the sub-
heading of which is Patiganita. See: Datta and
Singh (1938); Kapadia (1935).
Moreover, in his Brahmasphutasiddhanta
(628), *Brahmagupta describes the ensemble
of basic arithmetical operations with the word
patiganita. He writes: “Those that know
the twenty logistic operations separately
and individually, [these being] addition,
multiplication, etc., as well as the eight
[methods] of determination, including [in
particular measurement by] shadow, is a
[true] mathematician." See: BrSpSi; Datta and
Singh (1938).
To Brahmagupta’s mind, the eight
fundamental operations of the Indian
mathematicians were the same as the first eight
operations of patiganita (namely: addition,
subtraction, multiplication, division, the
squaring or cubing of a number, the extraction
of the square or cube root), to which the five
fundamental rules of the reduction of fractions
were added: the *trairashika or “Rule of Three”,
etc. This shows the high level that had been
reached by the Indian mathematicians in their
calculating techniques at the beginning of the
seventh century CE. The methods of calculation
which originated in India are known to us today
not only because of the information provided
by Arabic and European authors, but also by
Indian authors themselves. See Square roots
(How Aryabhata calculated his).
See: Allard (1981); Datta and Singh (1938);
Iyer (1954); Kaye (1908); Waeschke (1878).
In some rural regions of India, these
processes have been taught through the
centuries with hardly any modifications, and
calculations are still carried out on the
pati (small board) [see Datta and Singh
(1938)]. The word patiganita (or ganitapdtf) is
composed of *ganita, which means
“calculation, arithmetic, science of
calculation”, and *pati, synonymous with Patta
in the sense of “board” or “tablet". See: AMM,
XXXV, p. 526; Datta and Singh, pp. 7-8 and
123. This etymology dates back to the time
when Indian mathematicians carried out their
calculations on either a board or a tablet.
Today, the most natural support for carrying
out mathematical operations on is paper. Paper
was invented in China, although the
circumstances are not fully known. There are
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
488
texts that attribute the invention of a type of
paper made from the pulp created by removing
the fibre from rags and fishing nets to Cai-Lun
in the year 105 BCE. However, the ideogram
used to write the word paper in Chinese
contains the sign for silk. It seems that paper
made from silk preceded paper made from
vegetable fibres, the latter quickly replacing the
former type because it was cheaper and more
resistant. Cai-Lun and other paper makers then
went on to use the pulp of vegetable matter,
particularly the bark of the mulberry tree. It
was probably in the tenth century that they
began to use bamboo and, around the
fourteenth century, straw. It would be a long
time after the Chinese discovery before the
West would know about paper. The production
of paper began in Samarkand in 751 after the
Chinese were taken prisoner by the Arabs at
the battle of Talas. Paper was then made by
Chinese workers in Baghdad (from 793) and
Damascus, which for centuries remained the
principal supplier to Europe. From there,
methods of fabrication spread to Egypt (c. 900)
and Morocco, from where the Arabic invaders
introduced it to Spain [see Galiana, (1968)].
Paper was introduced to India by the
Persians, who learned the methods of
manufacture from the Arabs. It was not until
the fourteenth century, however, that the
Indians learned the secrets of paper-making.
In other words, Indians almost never used
paper to carry out their calculations, until very
late on in their history. The Arabs and
Persians never used paper for this purpose
until the twelfth or thirteenth century,
because it was such a rare and expensive
commodity. The Indians could have used the
same material as they used for their
manuscripts, carving or writing on palm-
leaves or tree-bark (see Indian styles of
writing, The materials of). However, carrying
out calculations was a completely different
practice to writing manuscripts: working out
sums was “rough work”, whilst manuscripts
were written to last, on durable material and
in indelible ink. They used something much
more economic than palm-leaves or tree-bark
for their calculations: they used chalk and
slate, just as most people in the Western
world used at school until very recently (or
chalk and a blackboard). The mathematician
*Bhaskaracharya (whose favourite instrument
was the pati, or “board”, which he wrote upon
with a piece of chalk) refers to the use of these
materials in his texts, notably in his *Lilavati,
where he writes the following:
khatikaya rckhd ucchddya . . .. “After having
drawn the lines [of the numerals for the
calculations] on the pati with chalk . .
[See: Datta and Singh (1938), p. 129;
Dvivedi (1935), p. 41.]
In other words, the Indian mathematicians
began, at some point to use if not slate, then at
least a wooden board painted black, and chalk
to write their numerals on and cross them out,
and a rag to rub them out.
Just as the Arabs and Persians adopted the
Indian numerals and methods of calculation,
so they began to use the support upon which
the Indians carried out their mathematical
operations. They gave it the Arabic name takht
or luha (especially in northern Africa) which
means “table" or “board" (whether it is made
of wood, leather, metal, earth, clay or even
slate). As for “arithmetic", this was described
by the expression ‘/7m al hisab al takht
(“science of calculation on the board”). This
support had the advantage of overcoming all
the difficulties created when calculations w r ere
carried out on boards covered in dust. See
Indian methods of calculation.
PAVAKA. [S]. Value = 3. “Fire”. See Agni and
Three.
PAVANA. “Purification”, and by extension, “He
who purifies”. This is another name for *Vayu,
the ancient Brahmanic god of the wind. He is
often represented on a mount in the form of an
antelope or a deer and holding a fan, an arrow
and a standard, respectively symbols of the air
{vayu), of speed and of the wind [see Frederic,
Dictionnaire (1987)].
PAVANA. [SI. Value = 5. “Purification”. This
symbolism is due to this word being associated
with one of the attributes of *Vayu, god of the
wind, because the wind itself in Indian
cosmologies is considered to be the “cosmic
breath”. Vayu is king of the subtle and
intermediary domain between the sky and the
earth who penetrates, breaks up and purifies.
Vayu is also known by the name Anila, which
means “breath of life”. Thus, according to the
Hindus, he is the cosmic energy that penetrates
and conserves the body and is manifested most
clearly in the form of breath in creatures. Vayu
is also the *prana, the “breath” in terms of
“vital respiration”. Hinduism distinguishes
between five types of breath: prajha, the very
essence of breath, the pure vital force; vyana,
the regulator of the circulation of the blood;
samana, which regulates the process of
absorption and assimilation of food and
maintains the balance of the body by looking
after the processes of feeding; apana, which
looks after secretion; and udana, which acts on
the upper part of the organism and facilitates
spiritual development by creating a link
between the physical part and the spiritual part
of the being. Thus pdvana = 5. See Pdvana
(previous entry), Prana and Five.
The use of this numerical symbol can be
found in Bhaskaracharya [see SiShi, I, 27] and
in Bhattotpala’s Commentary on Brihatsamhita
(chapter II). [See Datta and Singh (1938), p. 55].
PAVANA. [SJ. Value = 7. “Purification", “He who
purifies”. This is one of the attributes of *Vayu,
god of the wind (see previous article). To
understand the reason for this symbolism, it is
necessary to be familiar with the relevant episode
in Brahmanic mythology. One day Vayu revolted
against the deva, or "gods”, who live on *Mount
Meru. He decided to destroy the mountain, and
started a powerful hurricane. But the mountain
was protected by the wing of Garuda, the bird-
helper of Vishnu, which meant that the assaults
of the wind had no effect. One day, however,
when Garuda was absent, Vayu cut off the peak
of Mount Meru and threw it into the ocean. This
is how Lanka was bom, the island of Sri Lanka.
This mythological tale explains how the wind
came to have this value. The mythical mountain,
*Mount Meru, the living and meeting place of
the gods, and centre of the universe, is said to be
situated above the seven *patdla (or “inferior
worlds”), and has seven faces, each one turned
towards one of the seven *dvipa (or “island-
continents”) and one of the seven *sagara (or
“oceans”); when Vayu attacked the mountain, he
created seven strong winds, one for each face of
Mount Meru. Thus: pdvana = 7. See Seven.
PERFECT. A synonym for a large quantity. See
High numbers (Symbolic meaning of).
PERFECT. [S]. Value = 0. See Puma and Zero.
PHENOMENAL WORLD. [S]. Value = 3. See
Jagat, Loka, Three, Triloka.
PHILOSOPHICAL POINT OF VIEW. [S].
Value = 6. See Darshana and Six.
PHILOSOPHY OF VACUITY. See Shunya,
Shiinyald.
PHILOSOPHY OF ZERO. See Symbolism of
zero, Shunya, Shunyatd, and Zero.
PINAKANAYANA. [SJ. Value = 3. This is one of
Shiva’s names, the third divinity of the Hindu
trinity, god of destruction and dissolution. He is
often represented with a third eye on his
forehead (which symbolises perfect Knowledge).
Moreover, his emblem is the *trishula, or
“trident", symbols of the three aspects of
the manifestation (creation, preservation,
destruction). See Haranetra and Three.
PINK LOTUS. As name of the numbers ten to
power nine, ten to power fourteen and ten to
power twenty-nine. See: Padma, High Numbers
(Symbolic Meaning of).
PINK LOTUS. As name of the numbers ten to
power twenty-nine, ten to power 119. See:
Padma, High Numbers (Symbolic Meaning of).
PITAMAHA. [S]. Value = 1. “Great ancestor”,
“grandfather”, “first father”. This is an allusion
to the god Brahma, first divinity of the trinity
of Hinduism; he is the "Director of the sky”, the
“Master of the horizons", the "One” amongst
the diversity. See One.
PLACE-VALUE SYSTEM. The most common
Sanskrit term for this is * sthana, which literally
means “place”. See Sthana, Anka,
Ankakramena, Ankasthana, Sthanakramad
and Indian written numeral systems
(Classification of).
PLANET. [S]. Value = 9. See Graha and Nine.
PLANETS. See Graha, Saptagraha and
Navagraha.
PLENITUDE. [SJ. Value = 0. See Puma and Zero.
POETRY. See Indian metric, Poetry and
writing of numbers, Naga, Serpent
(Symbolism of the) and Poetry, zero and
positional numeration.
POETRY AND WRITING OF NUMBERS. Like
all the scholars of India, astronomers and
mathematicians of this civilisation usually wrote
in Sanskrit, often writing their numerical tables
and texts in verse. These scholars loved to play
with and speculate with numbers, and their
enjoyment can be seen in the form of their
wording which, if not lyrical, is at least in verse.
Thus numbers came to be written using words
which were connected to them symbolically, and
one such word could be chosen from an almost
limitless selection of synonyms so that it would
fit the rules of Sanskrit versification and give the
desired effect. The transcription of a numerical
table or of the most arid mathematical formula
would often resemble an epic poem. Their
language lent itself admirably to the rules of
versification, thus giving poetry a significant
role in Indian culture and Sanskrit literature. See
Sanskrit and Numerical symbols.
POETRY, ZERO AND POSITIONAL
NUMERATION. See Zero and Sanskrit poetry.
POINT. [SJ. Value = 3. See Shula and Three.
POSITION. [SJ. Value = 4. See Iryd and Four.
POSITION OF NUMERALS. See Sthana,
Sthanakramad, Ankasthana and Ankakramena.
489
POWER
POWER. [S]. Value = 14. See lndra and
Fourteen.
POWERFUL- [S]. Value = 14. See Shakra and
Fourteen.
POWERS OF TEN. See Ten, Hundred,
Thousand, Ten thousand, Million, Ten million,
Hundred million, Thousand million, Ten
thousand million, Hundred thousand million,
Billion, Ten billion, Hundred billion, Trillion,
Ten trillion, Hundred trillion, Quadrillion,
Quintillion, Names of numbers, High numbers
and Infinity.
PRAKRIT. “Unrefined", “Basic”. Generic name
commonly used by Indians to refer to the
numerous Indo-European dialects of the “Indo-
Aryan” category.
PRAKRITI. “Nature, material”. According to
Indian philosophy, this is the original material
that the universe was made from. It is the
principal transcendental material, which is
associated with terrestrial elements, as
opposed to the principal spirit (which is
represented by the skies).
PRAKRITI. [SJ. Value = 21. In Sanskrit poetry,
this is the metre of four times twenty-one
syllables per verse. See Indian metric.
PRALAYA. Name of the total destruction of the
universe in Hindu and Brahman cosmogonies.
See Day of Brahma, Kalpa, Katiyuga and Yuga.
PRANA. [S]. Value = 5. “Breath”. In Hindu
philosophy, this describes the five breaths
which are said to govern the vital functions of
the human being (prajria, apana, vyana, uddtia
and samana). This term not only applies to
respiratory rhythms (like the prdnayama
physical exercises, which are meant to control
breathing and form part of the techniques of
hathayoga), but also to “subtle breathing”
identified with intelligence and wisdom
I prajria ) [see Frederic, Dictionnaire (1987)]. See
Pavana and Five.
PRAYUTA. Name given to the number ten to
the power five (a hundred thousand). See
Names of numbers and High numbers.
Source: Kdlhaka Samhita (beginning of the
Common Era).
PRAYUTA. Name given to the number ten to
the power six (= million). See Names of
numbers and High numbers.
Sources: 'Vajasaneyt Samhita, *Taittiriya Samhita
and *Kathaka Samhita (from the start of the first
millennium CE); * Pahchavimsha Brdhmana (date
uncertain); * Aryabhatiya (510 CE). *I.ildvati by
Bhaskaracharya (1150 CE); *Ganitakaumudi by
Narayana (1350 CE); *Trishatika by
Shridharacharya (date uncertain); *Kitab Ji tahqiq i
ma li'l hind by al-Biruni (c. 1030 CE); *Sankhyayana
Shrauta Sutra (date unknown).
PRECEPT. [SJ. Value = 6. See Six.
PRIMORDIAL PRINCIPLE. [S[. Value = 1. See
Adi. One.
PRIMORDIAL PROPERTY. [S], Value = 3. See
Guna and Triguna.
PRIMORDIAL PROPERTY. [S], Value = 6. See
Guna and Shaddyatana.
PRINCIPLE OF THE ENUNCIATION OF
NUMBERS. See Ankanam vamato gatih and
Sanskrit.
PRINCIPLE OF POSITION. The Sanskrit term
usually designating it is *sthana, literally:
“place". See Sthdna.
PRITHIVt [SJ. Value = 1. “Immense", “Earth",
“terrestrial world". This symbolism primarily
refers to the unique nature of the earth,
considered to be the spouse of the sky. However,
this is also and above all an allusion to the fact
that the earth, as principal transcendental
material ( *prakriti ), as opposed to the principal
spirit (represented by the skies), is regarded as
the mother of all things. See One.
PROGENITOR OF THE HUMAN RACE. [SJ.
Value = 14. See Manu and Fourteen.
PROTO-INDIAN NUMERALS. Symbols used
from about 2500 to 1500 BCE by people of the
Indus civilisation (Mohenjo-daro, Harappa,
etc.) who preceded the Aryan settlement of the
Indian sub-continent. It is not known how
these very different symbols could have
evolved into early Brahml numerals (nor if
indeed there is a connection between them).
Only the signs for the nine units have been
identified so far; a full understanding of
proto-Indian numerals must await further
archaeological evidence. See Fig. 1.14.
PUNDARIKA. Literally “(white) lotus”. Name
given to the number ten to the power twenty-
seven. See Names of numbers. For an
explanation of this symbolism, see: High
numbers (Symbolic meaning of). Source:
* Lalitavistara Sutra (before 308 CE).
PUNDARIKA. Literally “(white) lotus”.
Name given to the number ten to the power
112. See Names of numbers. For an
explanation of this symbolism, see: High
numbers (Symbolic meaning of).
Source: *Vydkarana (Pali grammar) by
Kachchayana (eleventh century CE).
PUNJABI NUMERALS. Symbols derived from
‘Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Gupta and Sharada.
Currently used in the Punjab (Northwest India).
The symbols correspond to a mathematical
system that has place values and a zero (shaped
like a small circle). See: Indian written numeral
systems (Classification of). See Fig. 24.5, 52,
and 24.61 to 69.
PURA. [SJ. Value = 3. “City”. Allusion to the
* tripura , the “three cities" of the *Asura (or
“anti-gods”), flying iron fortresses from which
they directed the war they waged against the
*deva. See Three.
PURA. [SJ. Value = 3. “State”. Allusion to the
three *tripura, the “three states of
consciousness" according to Hinduism (awake,
asleep and dreaming). See Three.
PURANA. Literally: “Ancients". Traditional
Sanskrit texts, dealing with highly diverse
subjects, such as the creation of the world,
mythology, legends, the genealogy of mythical
sovereigns, castes, etc. These texts were written
for ordinary people and those of “low caste”.
Analysis has shown that they are made up of
documents written at various times and are
from many different sources, and were
compiled, revised, added to and corrected in an
interval of time oscillating between the sixth
and the twelfth century, some even being dated
as nineteenth century. Thus the documentation
that they contain should be treated with
caution, as, from a purely historical point of
view at least, they are of no interest. See Indian
documentation (Pitfalls of).
PURANA AND POSITIONAL NUMERATION.
See Indian documentation (Pitfalls of).
PURANAJLAKSHANA. [SJ. Value = 5. (Late
usage). Allusion to the texts of the *Purana,
which tell of the Pahchalakshana, which, in
Hindu philosophy, correspond to the “five
characteristics" which are said to have defined
history: creation ( sarga ), periodical creations
( pratisarga ), divine geneaologies ( vamsha ), the
era of a *manu ( manvantara ) and the genealogies
of human sovereigns ( vamshanucharita ) [see
Frederic, Dictionnaire (1987)]. See Five.
PURIFICATION. [SJ. Value = 5. See Pavana
and Five.
PURIFICATION. [SJ. Value = 7. See Pavana
and Seven.
PURNA. [SJ. Value = 0. Literally “full, fullness,
fulfilled, perfect, finished”. To a Western reader,
this symbolism might seem paradoxical: how
can a word that means “full” represent zero, the
void? The allusion is to *Vishnu, the second
great divinity of the Hindu and Brahman trinity,
whose essential role is to preserve, and cause
the evolution of, creation (‘Brahma being the
creator, ‘Vishnu the conserver and ‘Shiva the
destroyer). Vishnu is considered to be the
internal cause of existence and the guardian of
*dharma. Each time the world goes wrong, he
hastens (incarnating himself in the form of
* avatdra ) to show humanity new ways in which
to develop. He is often represented as a
handsome young man with four arms, holding
a conch in the first hand, a bow in the second, a
club in the third and a lotus flower in the fourth.
The conch represents riches, fortune and
beauty, which are the attributes of Vishnu as the
principal conserver of the manifestation,
because the sound and the pearl are conserved
within the shell. As for the ‘lotus, it symbolises
the highest divinity, innate reason and mental
and spiritual perfection. It also symbolises the
“third eye”, that of perfect Knowledge; however,
it is also the superior illumination and the
divine corolla, the totality of revelation and
illumination, as well as intelligence, wisdom
and the victory of the mind over the senses. See
High numbers (Symbolic meaning of).
Moreover, like the thousand-petalled lotus,
Vishnu possesses a thousand attributes and
qualities ( *sahasrandma ). He represents the
innumerable ( thousand here being treated in its
figurative sense). See Thousand. Thus Vishnu
is associated with the idea of wholeness,
integrity, completeness, absoluteness and
perfection. The “foot of Vishnu” ( *vishnupada ),
is the “sky”, the “zenith", the “place of the
blessed" and the meeting place of the gods; it
is, in Hindu cosmology, the summit of ‘Mount
Meru, the mythical mountain situated at the
centre of the universe, the source of the
celestial Gangd (the sacred Ganges). This
makes it easier to understand how “full" came
to mean infinity, eternity, and by extension
completion and perfection. It is upon ‘Ananta,
the serpent with a thousand heads floating on
the primordial waters of the “ocean of
unconsciousness”, that Vishnu lies to rest
during the time separating two creations of the
universe. Ananta represents infinity, and has
also often represented zero as a numerical
symbol. Thus it is clear how purna came to
mean zero. See Ananta, Jaladharapatha,
Shunya, Zero. See also Infinity, Infinity
(Indian mythological representation of) and
Serpent (Symbolism of the).
PUSHKARA. [SJ. Value = 7. This is a surname
attributed to Krishna and Shiva, as well as to
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
490
Dyaus (the Sky) considered to be a “reservoir of
water". The allusion here is to Pushkara, one of
the seven mythical continents that surround
*Mount Meru. See Dvipa, Sapta, Sagara,
Ocean and Seven.
PUTRA. [S], Value = 5. “Son”. In this
symbolism, the word in question is a synonym
of *Pandava, which means the “sons of Pandu".
See Five.
PUTUMANASOMAYAJIN. Indian astronomer
of the eighteenth century. His works notably
include a text entitled Karanapaddhati , in
which there is frequent use of the place-value
system written in the Sanskrit numerical
symbols [see Sastri (1929)1. See Numerical
symbols. Numeration of numerical symbols,
and Indian mathematics (History of).
such as love, nostalgia, sadness, etc., combine
with lines and colours to provoke diverse
sensations within the spectator. In the
symbolism in question, the allusion is to the
janaka rdga, the six “eastern raga", who are male,
associated with their six ragini (or female raga),
and with the six “sons” of the latter ones, each of
these groups in turn being associated with the
'shadayatana or “six 'guna" of Buddhist
philosophy (in other words the six sense organs:
eye, nose, ear, tongue, body and mind) [see
Frederic, Dictionnaire (1987)]. See Rasa, Six and
Naga.
RAHU. Demon who, according to ancient
Indian mythology and cosmology, caused
eclipses by “devouring” the sun or the moon,
due to a privilege conferred on him by
*Brahma. See Paksha.
Q
QUADRILLION (= ten to the power eighteen).
See Shankha and Names of numbers.
QUALITY. [SJ. Value = 3, See Guna, Triguna
and Three.
QUALITY. [S]. Value = 6. See Guna,
Shadayatana and Six.
QUINTILLION (= ten to the power twenty-
one). See Kotippakoti, Kumud, Mahdkshiti and
Utsanga. See also Names of numbers.
QUOTIENT. [Arithmetic]. See Labdha.
QUTAN XIDA. Chinese astronomer of Indian
origin. Qutan Xida is none other than the
Chinese rendering of the Indian name
*Gautama Siddhanta.
R
RABBIT. [S]. Value = 1. See One, Shashin,
Shashadhara.
RADA. [S]. Value = 32. “Tooth”. See Danta and
Thirty-two.
RADIANCE. (SJ. Value = 9. See Go and Nine.
RAGA. [S]. Value = 6. “Attraction, colour,
passion, musical mode”. This Sanskrit term
describes the moments of emotion provoked by
a piece of music (the modes and rhythms
causing diverse sensations in the listener) or by
a visual work of art. The instants of emotion,
which can be provoked by the perception of an
exterior agent such as the rain, the wind, a
storm, etc., or even by an interior sentiment
RAJAMRIGANKA. See Bhoja.
Rajasthan! numerals. Symbols
derived from *Brahmi numerals and
influenced by Shunga, Shaka, Kushana,
Andhra, Gupta, Nagari and Maharashtri.
Currently used in the state of Rajasthan in the
west of the sub-continent (bordering on
Pakistan, Punjab, Haryana, Uttar Pradesh,
Madhya Pradesh and the Gujurat). Rajasthani
numerals are a variant of Marwari numerals.
The symbols correspond to a mathematical
system that has place values and a zero
(shaped like a small circle). See: Indian
written numeral systems (Classification
of). See Fig. 24.52 and 24.61 to 69.
RAMA. [S]. Value = 3. Allusion to the three
Rama of Indian tradition and philosophy: the
first, also called Parashu-Rama, or “Rama of the
axe”, is the sixth incarnation of Vishnu, who
came to crush the tyranny of the Kshatriyas, the
caste of warriors; the second, called Rama-
chandra, seventh incarnation of Vishnu, came
to develop sattva in humankind, in other words
uprightness, equilibrium, serenity and
peacefulness; and the third, called simply
Rama, was the famous hero of the epic poem
*Ramayana (see Frederic, Dictionnaire (1987)].
RAMAYANA. “The march of Rama”. This is an
epic Indian poem, written down in Sanskrit by
the poet Valmiki. It is derived from very
ancient legends, but did not find its definitive
form until the early centuries of the Common
Era. Here is a list of names of the high numbers
mentioned in this text (from a passage where,
in order to express the number of monkeys that
made up Sugriva’s army, the author gives the
following names successively, which increase
each time on a scale of one hundred thousand):
*Koti (= 10 7 ), *Shanka (= 10 12 ), *Vrinda (=
10 l7 ), * Mahavrinda (= 10 22 ), *Padma
(= 10 29 ), *Mahdpadma (= lO 34 ’, *Kharva (= 10 39 ).
See Names of numbers and High numbers.
[See Weber in: JSO, XV, pp. 132-40;
Woepcke (1863)J.
RANDHRA. [S]. Value = 0. “Hole”. Numerical
word-symbol used rarely and not until a relatively
recent date. The origin of this association of ideas
clearly comes from the lack of consideration
attached to the anal orifice. See Zero.
RANDHRA. [Si. Value = 9. “Hole”. This is an
allusion to the nine orifices of the human body
(the two eye sockets, the two ears, the two
nostrils, the mouth, the navel and the anal
orifice). See Chhidra and Nine.
RASA. [SJ. Value = 6. “Sensation". In its most
general meaning, this word denotes the
sensation(s) that a Shadayatana can experience,
in other words the “six senses or sense organs"
of Indian philosophy (which are: the eye, the
nose, the ear, the tongue, the body and the
mind). However, the explanation for this
symbolism is much more subtle than that. It
can only be understood with reference to the
Indian aesthetic canons, where this word
describes the emotional state of the spectator,
listener or reader, in terms of the essence of the
evocative power of the musical, pictorial, poetic,
theatrical, (etc.) art. This aesthetic distinguishes
between nine different types of rasa, including
the least agreeable, namely: shringara (love or
erotic passion); hashya (comedy and humour);
karund (compassion); vira (heroic sentiment);
adbhuta (amazement); shanta (peace and
serenity); raudra (anger and rage); bhayanaka
(fear or anguish); and vibhatsa (disgust or
repulsion). The first sue are the ones which
enable enjoyment, and this is what rasa refers to
in this symbolism: the idea of “savouring”. Thus
rasa = 6. See Shadayatana and Six.
RASHI. “Rule”. Often used in arithmetic to
denote the “Rule of Three".
RASHI. [SJ. Value = 12. “Zodiac". This, of
course, refers to the twelve signs of the Indian
zodiac: Mesha (Aries); Vrishabha (Taurus);
Mithuna (Gemini); Karka (Cancer); Simha
(Leo); Kanyd (Virgo); Tula (Libra); Vrishchika
(Scorpio); Dhanus (Sagittarius); Makara
(Capricorn); Kumbha (Aquarius); Mina
(Pisces). See Twelve.
RASHIVIDYA. Name given to arithmetic in the
Chdndogya Upanishad. Literally: “Knowledge of
the rules”.
RATNA. [S]. Value = 3. “Jewel”. This is
probably an allusion to the *triratna, the “three
jewels” of Buddhism, namely: the Community
(sangha), the Buddhist Law ( *dharma ) and
Buddha. These “jewels” are represented by a
trident. See Dharma, Shiila and Three.
Note: this symbol is found very rarely
representing this value, except for in the
*Ganitasarasamgraha by Mahaviracharya [see
Datta and Singh (1938), p. 551.
RATNA. [SJ. Value = 5. “Jewel”. This is the most
frequent value that this word is used for as a
numerical symbol. It is probably an allusion to
the *pahchaparameshtin, the “five orders of
beings" considered to be the “five treasures" of
*Jaina religion: the *siddha, human beings who
are omniscient and who became immortal after
being freed from the bonds of karma and
*samsara\ the arhat, sages liberated from the
bonds of karma, but still subject to the laws of
*samsara\ the acharya or “great masters”; the
upadhya or “masters”; and the ascetics (sadhu)
[see Frederic, Dictionnaire (1987)1. See Five.
RATNA. [S]. Value = 9. “Jewel”. This allusion
could be to the *Navaratna, “Nine Jewels", the
collective name given to the nine famous poets
who wrote in Sanskrit who lived in the court of
the king Vikramaditya. See Nine.
RATNA. [S]. Value = 14. “Jewel”. There is no
concrete explanation for this symbolism.
However, it could have some connection to the
*saptaratna or “seven jewels” of Buddhism,
which constitute the seven attributes of the
current Buddha (“Golden wheel”; Chintamani, or
miraculous pearl said to grant all wishes; “White
horse”; “Noble woman”; “Elephant” carrying the
sacred Scriptures; “Minister of Finances”; and
“Head of war”); these are attributes that would
have been associated symbolically with the
*saptabuddha, or seven Buddhas of the past
(Vipashyin, Shikhin, Vishvabhu, Krakuchhanda,
Kanakamuni and Kashyapa), including the
current Buddha (Shakyamuni Siddhartha
Gautama); thus, by symbolic addition: ratna = 7
+ 7 = 14. See Fourteen.
RATNASANU. One of the names for *Mount
Meru. See Adri, Dvipa, Purna, Patdla, Sagara,
Pushkara, Pavana and Vayu.
RAVANA. Name of the king-demon Lanka
who, according to the legends of *Rdmayana,
usurped the throne of his half-brother Kuvera
and stole his flying palace (pushpaka ).
RAVANABHUJA. [S]. Value = 20. “Arms of
*Ravana”. Allusion to the twenty arms of this
king-demon. See Twenty.
RAVANASHIRAS. [S]. Value = 10. “Heads of
*Ravana”. This king-demon is said to have had
ten heads. See Ten.
491
RAVI
RAVI. [SJ. Value = 12. This is another name for
'Surya, the divinity of the sun. See Twelve.
RAVIBANA. [S]. Value = 1,000. “Beams of
Ravi”. This refers to one of the attributes of
*Ravi (= ‘Surya), the divinity of the sun, and
expresses the * sahasrakirana or “Thousand
Rays" of the sun. See Thousand.
RAVICHANDRA. IS]. Value = 2. The couple
uniting Ravi and Chandra (named Ravi after
‘Surya, the sun whose other attribute is *Ravi,
and ‘Soma, the moon, the masculine entity
also called *Chandrci). See Two.
REALITY. (S). Value = 5. See Taltva and Five.
REALITY. [SJ. Value = 7. See Tattva and Seven.
REALITY. [S]. Value = 25. See Tattva and
Twenty-five.
REMAINDER. [Arithmetic]. See Shesha.
RISHI. [S]. Value = 7. “Sage”. Allusion to the
*Saptarishi, the seven great mythical Sages of
Vedic times (Gotama, Bharadvaja, Vishvamitra,
Jamadagni, Vasishtha, Kashyapa and *Atri),
created by ‘Brahma and said to be the authors
of the hymns and invocations of the Rigveda and
most of the other *Vedas. They are said to form
the seven stars of the Little Bear. See Seven.
RITU. [S]. Value = 6. “Season”. Allusion to the
six seasons, each lasting two “months” in the
Hindu calendar: spring ( vasanta)-, the hot
season (grishma ); the rain season (varsha)\
autumn (sharada)-, winter ( hemanta ) and the
cold season ( shishira ). See Six.
RUDRA-SHIVA (Attributes of). [SI. Value =
11. See Bharga, Bhava, Hara, Isha, Ishvara,
Mahadeva, Rudra, Shiva, Shulin and Eleven.
RUDRA. [S[. Value = 11. "Rumbling”,
"Violent”, “Lord of tears". This is the name of
the ancient Vedic divinity of the tempest who,
according to the * Vedas, was the personification
of the vital breaths, which came from
‘Brahma’s forehead, of which there were
eleven. Thus: Rudra = 11. See Eleven.
RUDRASYA. [S], Value = 5. “Faces of ‘Rudra”.
This god is said to have had five heads. He is
also lord of the "five elements”, “the five sense
organs”, the five “human races” and the five
points of the compass (if the zenith is
included). See Five.
RULE OF THREE. [Arithmetic). See Rashi,
Trairashika and Vyastatrairashika.
RULE OF FIVE. [Arithmetic). See
Pahchapardshika,
RULE OF SEVEN. [Arithmetic]. See Saptarashika.
RULE OF NINE. [Arithmetic]. See Navarashika.
RULE OF ELEVEN. [Arithmetic). See
Ekadasharashika.
RUPA. [S]. Value = 1. “Form”, “Appearance”.
This word is synonymous here with “body” as a
symbol for the number one. See Tana and One.
s
SAGARA. [S]. Value = 4. “Sea, Ocean”. This
symbolism can be explained by an allusion to
the four “oceans" (* chatursagara) which
surround the four “island-continents”
(* chaturdvipa) which, according to Hindu
cosmology, surround Jambudvipa (India). See
Four and Ocean.
SAGARA. [S). Value = 7. “Sea, Ocean”. This
symbolism can be explained by an allusion to
the seven “oceans” ( *sapta Sagara ) which,
according to Hindu cosmology and Brahmanic
mythology, surround *Mount Meru. See Four
and Ocean.
SAGE. [SJ. Value = 7. See Atri, Rishi, Saptarishi,
Muni and Seven.
SAHASRA. Ordinary Sanskit name for the
number * thousand, the consecutive multiples
of which are formed by placing the word
sahasra to the right of the name of the
corresponding unit: dvasahasra (two
thousand), trisahasra (three thousand),
chatursahasra (four thousand), panchasahasra
(five thousand), etc. This name appears in
many words which have a direct relationship
with the idea of this number.
Examples: *Sahasrabhuja, * Sahasrakirana,
* Sahasraksha, *Sahasrdmshu, *Sahasranama,
*Sahasrapadma, * Sahasrarjuna.
For words which have a more symbolic
relationship with this number, see Thousand
and Symbolism of numbers.
SAHASRABHUJA. “Thousand arms”. This is
one of the names of the Sun-god *Surya (in
allusion to his rays). In the schools of Buddhism
of the North, this term refers to an ancient
divinity whose thousand arms represented his
multiple powers and omniscience.
SAHASRAKIRANA. “Thousand rays”. One of
the names of the Sun-god *Surya.
SAHASRAKSHA. “Thousand eyes”. One of the
attributes of *Indra and *Vishnu. See
Indradrishti and Sahasra .
SAHASRAMSHU. [SJ. Value = 12. “Thousand
(of the) Shining” (from sahasra, “thousand”,
and amshu, “shining”). This is a metaphorical
name for the Sun (the “thousand rays” of its
“shining”), and the symbolism has nothing to
do with the idea of a thousand, but with the
name of the Sun-god as a numerical symbol
equal to twelve. See Surya and Twelve.
SAHASRANAMA. “Thousand names”. One of
the attributes of *Vishnu and *Shiva.
SAHASRAPADMA. “Lotus of a thousand
petals”. See Lotus and High numbers
(Symbolic meaning of).
SAHASRARJUNA. “Arjuna's thousand".
Name for the thousand arms of
Arjunakartavirya, mythical sovereign of the
*Mahabharata. See Arjunakara.
SALILA. Name given to the number ten to the
power eleven. See Names of numbers and
High numbers. Source: *Samkhyayana Shrauta
Sutra (date uncertain).
SAMAPTALAMBHA. Name given to the
number ten to the power thirty-seven. See
Names of numbers and High numbers.
Source: *I.alitavistara Sutra (before 308 CE).
SAMIKARANA. Term used to denote an
“equation”. Literally “to make equal” (from
sama “equal”, and kara "to make”). Synonyms:
samikara, sdmikriyd, etc.
SAMKALITA. Sanskrit term denoting
addition. Literally: “put together”. Synonyms:
samkalana (literally: “act of reuniting”);
mishrana (“act of mixing”); sammelana;
prakshepana; samyojana, etc.
SAMKHYA (SANKHYA). “Number”. Term
often used to describe the system of writing
numbers using numerical symbols. See
Numerical symbols and Numeration of
numerical symbols.
SAMKHYA (SANKHYA). Literally “calculator”.
This term describes the adept of the mystical
philosophy of * samkhya.
SAMKHYA (SANKHYA). Literally “number”.
This denotes one of the six orthodox systems of
Indian philosophies. See Darshana and Tatt\>a.
SAMKHYA (SANKHYA). Literally “number”.
Word used to denote "expert-calculator” and,
by extension, the arithmetician and
mathematician. See Darshana and
Samkhyana.
SAMKHYANA (SANKHYANA). “Science of
numbers", and by extension “arithmetic” and
“astronomy”. Word used in this sense in
Buddhist and Jaina literature. This science was
considered to be one of the fundamental
conditions for the development of a Jaina
priest. For the Buddhists, it was also considered
(although somewhat later) to be the first and
most noble of arts.
SAMKHYEYA (SANKHYEYA). “Number”, in
the operative and arithmetical sense of the word.
SAMSARA. Cycle of rebirth. See Gati, Kama
and Yoni.
SAMSKRITA. “Complete”, "perfect”,
“definitive”. Term used to denote the Sanskrit
language. See Sanskrit.
SAMUDRA. Literally “ocean”. Name given to
the number ten to the power nine. See Names
of numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Sources: * Vajasaneyi Samhita, *Taittiriya Samhita and
*Kalhaka Samhita (from the start of the first millen-
nium CE); * Pahchavimsha Brahmana (date
uncertain).
SAMUDRA. Literally “ocean”. Name given to
the number ten to the power ten. See Names of
numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Source: *Sankhyayana Shrauta Sutra (date uncertain).
SAMUDRA. Literally “ocean”. Name given to
the number ten to the power fourteen. See
Names of numbers. For an explanation of this
symbolism, see High numbers (Symbolic
meaning of).
Source: Kitab ji tahqiq i ma li'l hind by al-Biruni
(c. 1030 CE).
SAMUDRA. [S]. Value = 4. “Ocean". This is
because of the four oceans that are said to
surround *Jambudvipa (India). See Sagara,
Four and Ocean.
SAMUDRA. [S]. Value = 7. “Ocean”. This is
because of the seven oceans that are said to
surround * Mount Meru. See Sagara, Seven
and Ocean.
SAMVAT (Calendar). See Vikrama.
SANKHYA, etc. See Samkhya, etc.
SANKHYANA. See Samkhyana.
SANKHYAYANA SHRAUTA SUTRA. Philoso-
phical Sanskrit text (date uncertain). Here is a
list of the principal names of numbers
mentioned in the text (see Datta and Singh
(1938), p. 10]:
*Eka (= 1), * Dasha (= 10), *Sata (= 10 2 ),
*Sahasra (= 10 3 ), *Ayuta (= lO 4 ), *Niyuta
(= 10 5 ), *Prayuta (= 10 6 ), *Arbuda (= 10 7 ),
*Nyarbuda (= 10 8 ), *Nikharva (= 10 9 ),
*Samudra (= 10 10 ), *Salila (= 10 u ), *Antya (=
10 12 ), *Ananta (= 10 13 ). See Names of numbers
and High numbers.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
492
SANKHYEYA. See Samkhyeya.
SANSKRIT. In India and Southeast Asia,
Sanskrit has played, and still does play today, a
role comparable with Greek and Latin in
Western Europe. This language is capable of
translating, through meditation, the mystical
transcendental truths said to have been
revealed to the *Rishi in Vedic times. See
Akshara, AVM, Trivarna, Vdchana and
Mysticism of letters. Moreover, the name of
the Sanskrit language itself is quite significant,
because the word *samskrita (“Sanskrit”) means
“complete”, “perfect” and “definitive”. The
people who know this Sanskrit are said to speak
the divine language and are thus gifted with
divine knowledge. Bearing in mind the power
accorded to the spoken word (and
consequentially its written expression), Sanskrit
is considered to be the “language of the gods”.
In fact, this language is extremely
elaborate, almost artificial. It is capable of
describing multiple levels of meditations,
states of consciousness and physical, spiritual
and even intellectual processes. The inflection
of nouns is richly articulated and there are
numerous personal forms of the verb, even
though the syntax is rudimentary. The
vocabulary is very rich and highly diversified
according to the means for w'hich it is intended
[see Renou (1930); see also Filliozat (1992)].
This show's how, over the centuries, Sanskrit
has lent itself admirably to the rules of prosody
and versification. This explains why poetry has
always played such an important role in Indian
culture and Sanskrit literature. It is clear w'hy
Indian astronomers favoured the use of Sanskrit
numerical symbols, based on a complex
symbolism which was extraordinarily fertile and
sophisticated, possessing as it did an almost
limitless choice of synonyms. See Poetry and
writing of numbers and Numerical symbols.
SAPTA (SAPTAN). Ordinary Sanskrit name
for the number seven, which forms part of the
composition of many words directly related to
the idea of this number. Examples:
*Saptabuddha, *Saptagraha, *Saptamdtrika,
*Saptapadi, *Saptarashika, * Saptarishi,
*Saptarishikdla, *Saptasindhava. For words
w'hich have a more symbolic connection
with this number, see Seven and Symbolism
of numbers.
SAPTA. Literally “seven”. Term used
symbolically in the texts of the Atharvaveda as a
synonym for each of the following ideas: “sage”,
“ocean”, “mountain”, “island-continent”, etc.
The allusion here is to the “Seven Sages” of
Vedic times (* saptarishi), to the seven cosmic
oceans ( *sapta sdgara), to the seven peaks of
Mount Meru, or to the seven “island
continents” (sapta dvipa ) of Indian mythology
and cosmology. See Saptarishi, Adri, Giri,
Sdgara, Dvipa, Mount Meru and Ocean.
For an example, see Atharvaveda, I, 1, 1;
Datta and Singh (1938), p. 17.
SAPTABUDDHA. Name of the seven Buddhas.
See Sapta and Ratna (= 14).
SAPTADASHA. Ordinary Sanskrit name for
the number seventeen. For words which have a
symbolic link with this number, see Seventeen
and Symbolism of numbers.
SAPTA DViPA. “Seven islands”. In Hindu
cosmology and Brahmanic mythology, this is
the name given to the seven island-continents
said to surround *Mount Meru. See Sapta. For
an explanantion of the symbolism and the
choice of this number, see Ocean.
SAPTAGRAHA. Literally “seven planets”.
These are the following: *Surya (the Sun);
* Chandra (the Moon); *Angaraka (Mars);
*Budha (Mercury); *Brihaspati (Jupiter);
*Shukra (Venus); and *Shani (Saturn). See
Graha and Paksha.
SAPTAMATRIKA. Name for the seven “divine
Mothers”. See Mdtrika.
SAPTAN. Ordinary name for the number
seven. See Sapta.
SAPTAPADI. “Seven paces”. Name of a Hindu
rite which forms part of the nuptial
ceremonies, where the bride and groom must
take seven paces around the sacred fire in order
to consummate the union.
SAPTARASHIKA. (Arithmetic]. Sanskrit
name for the Rule of Seven.
SAPTARATNA. Name of the “Seven Jewels of
Buddhism”. See Ratna (= 14).
SAPTARISHI. “Seven Sages”. These are the
seven *Rishi of Vedic times, w r ho are said to
have resided in the seven stars of the Little
Bear. See Atri and Mount Meru.
SAPTARISHIKALA. “Time of the seven
*Rishi". Name of an Indian calendar. See
Saptarishi, Kdla and Laukikasamvat.
SAPTA SAGARA. “Seven oceans”. These are the
seven oceans which are said to surround
*Mount Meru in Hindu cosmology and
Brahmanic mythology: the ocean of salt water,
the ocean of sugar cane juice, the ocean of wine,
the ocean of thinned butter, the ocean of
whipped cheese, the ocean of milk and the ocean
of soft w'ater). See Sdgara. For an explanation of
the choice of this number, see Ocean.
SAPTASINDHAVA. “Seven rivers”. This is one
of the seven sacred rivers of ancient
Brahmanism (Gangd, Yamuna, Sarsvati, Satlej,
Parushni , Marurudvridha and Arjikiya).
SAPTATI. Ordinary Sanskrit name for the
number seventy.
SAPTAVIMSHATI. Ordinary Sanskrit name for
the number twenty-seven. For w'ords which have
a symbolic relationship with this number, see
Twenty-seven and Symbolism of numbers.
SARITAPATI. Name given to the number ten
to the power fourteen (= hundred billion). See
Names of numbers and High numbers.
Source: *Trishatikd by Shridharacharya (date
uncertain).
SAROJA. Name given to the number ten to
the power nine. See Names of numbers and
High numbers.
Source: *Ganiiakaumudi by Narayana (1350 CE).
SARPA. (Sf Value = 8. “Serpent”. See Naga,
Eight and Serpent (Symbolism of).
SARVABALA. Name formed with the Sanskrit
adjective sarva, which signifies "everything”. It is
given to the number ten to the pow'er forty-five.
See Names of numbers. For an explanation of
this symbolism, see High numbers (Symbolic
meaning of).
Source: * l.alitavistara Sutra (before 308 CE).
SARVADHANA. (Arithmetic]. Term denoting
the “total”, or the “whole”.
SARVAJNA. Name formed with the Sanskrit
adjective sarva, which means “everything”.
Given to the number ten to the pow'er forty-
nine. See Names of numbers. For an
explanation of this symbolism, see High
numbers (Symbolic meaning of).
SATA. Ancient Sanskrit form of the name for
hundred. See Shata and Names of numbers.
Use of this word is notably found in
* Vajasaneyi Samhitd, *Taittiriya Samhitd and
*Kathaka Samhitd (from the start of the first
millennium CE); and in * Pahchavimsha
Brdhmana (date uncertain) and *Sankhydyana
Shrauta Sutra (date uncertain).
SATYAYUGA. Synonym of *Kritayuga. See Yuga.
SAYAKA. [SI. Value = 5. “Arrow”. Allusion to
the Pahchasayaka, the “five arrows" of *Kama.
See Bana, Panchabana, Shara and Five.
SEASON. [S] . Value = 6. See Ritu and Six.
SELEUCID (Calendar). This calendar began in
the year 311 BCE, and was used in the
northwest of the Indian subcontinent. To find
the corresponding date in the Common Era,
subtract 311 from a date expressed in the
Seleucid calendar. See Indian calendars.
SELF (THE). |S]. Value = 1. See Abja and One.
SENANINETRA. [S). Value = 12. “Eyes of
Senani”. This is one of the names of
*Karttikeya, who is often depicted as having six
heads. Thus Sendninetra = 6 x 2 = 12 eyes. See
Karttikeyasya and Twelve.
SENSATION. (SJ. Value = 6. See Rasa and Six.
SENSE. IS]. Value = 5. See Vishaya and Five.
SERPENT (Cult of the). See Serpent
(Symbolism of the). See also Infinity (Indian
mythological representation of).
SERPENT (Symbolism of the). In India and all
its neighbouring regions, since the dawn of
Indian civilisation, the Serpent has been an
object of veneration worshipped by the most
diverse of religions. At the beginning of the rain
season in Rajasthan, Bengal and Tamil Nadu,
the serpent is worshipped through offerings of
milk and food. In popular religion, the cobra is
very highly considered and these snakes are to
be found adorning stones called Gramadevata,
or “divinities of the village”, which are placed
under the banyans. Frederic (1987) explains
that the serpents, in most local religions, are
genies of the ground, chthonian spirits who
possess the earth and its treasures. The cobras
are the most significant type of snake in Indian
mythology; they are deified and have their own
personality. They are often associated with the
cult of *Shiva, and in some pictures of Shiva, he
has a cobra wound one of his left arms. In these
representations, cobras are actually *naga,
chthonian divinities with the body of a serpent,
considered to be the water spirits in all folklore
of Asia, especially in the Far East where they are
depicted as dragons.
In fact, in traditional Indian iconography,
the *ndga are usually represented as having the
head of a human with a cobra’s hood. They live
in the *pdtala, the underworlds, and guard the
treasure which is under the earth. They are said
to live with the females, the nagini (renowned
for their beauty) and devote themselves to
poetry. They are considered to be excellent
poets, and are even called the princes of poetry:
493
SERPENT
first they mastered numbers, which led them
naturally to becoming masters of the art of
poetic metric. They are also princes of
arithmetic because, according to legend, there
ar e a thousand of them. In other words, due
to their considerable fertility, the naga
represent the incalculable. Just as metric
involves the regulation of rhythm, so they are
also sometimes associated with the rhythm of
the seasons and the weather cycles.
Coming back to the cobra, this is a long
snake which can measure between one metre
and one and a half metres. Because of this, the
Hindus classified them amongst the demons
called *mahoraga (or “large serpents"). It is the
“royal” cobra, however, (which can be up to two
metres in length) that was the logical choice of
leader of the tribe. This snake, as king of
the naga , was given several different
names: *Vasuki, *Muchalinda, Muchilinda,
Muchalinga, Takshasa, *Shesha, etc., and there
are many corresponding myths. See Vasuki.
According to a Buddhist legend, the king
Muchilinda protected the Buddha, who was in
deep meditation, from the rain and floods, by
making his coils into a high seat and sheltering
him with the hoods of his seven heads. The
name which is used most frequently, however, is
*Shesha. He is sometimes depicted as a snake
with seven heads, but he is usually represented
as a serpent with a thousand heads. This is why
the term *Sheshashirsha (literally “head of
Shesha”) often means “thousand” when it
is used as a numerical word-symbol.
Etymologically, the word shesha means
“vestige”, “that which remains”. Shesha is also
called Adi Shesha (from *Adi, “beginning”). This
is because Shesha is also and most significantly
the “original serpent”, born out of the union of
Kashyapa and Kadru (Immortality). And
because he married Anantashirsha (the “head of
*Ananta”), Shesha, according to Indian
cosmology and mythology, became the son of
immortality, the vestige of destroyed universes
and the seed of all future creations all at once.
The king of the naga thus represents
primordial nature, the limitless length of
eternity and the boundless limits of infinity.
Thus Shesha is none other than Ananta: that
immense serpent that floats on the primordial
waters of original chaos and the “ocean of
unconsciousness”, and *Vishnu lies on his coils
when he rests in between two creations of the
world, during the birth of *Brahma who is
born out of his navel (see Fig. D. 1 in the entry
entitled * Ananta). Ananta is also the great
prince of darkness. Each time he opens his
mouth, he causes an earthquake. At the end of
each *kalpa (cosmic cycle of 4,320,000,000
years), Ananta spits and causes the fire which
destroys all creation in the universe. He is also
*Ahirbudhnya (or Ahi Budhnya), the famous
serpent of the depths of the ocean who,
according to Vedic mythology, is born out of
dark waters. Thus, as well as being the genie of
the ground and the chthonian spirit who owns
the earth and its treasures, the serpent is also a
“spirit of the waters” (*aptya), who lives in the
“inferior worlds” (* pa tala).
Some myths clearly indicate this
ambivalence surrounding the nature of the
reptile, for example the legend which tells the
story of Kaliya, the king of the naga of the
Yamuna river; this is a serpent with four heads
of monstrous proportions, who defeated by
*Krishna, who was then only five years old,
went to hide in the depths of the ocean. In this
myth, the four heads of the king of the naga is
significant, because when this serpent goes by
the name of Muchalinda, it often has seven
heads, or a thousand heads like *Ananta. The
choice of these numerical attributions is not
simply a question of chance. In fact, in these
allegories, the seven heads of Muchalinda
represent the subterranean kingdom of the
naga, each one being associated with one of the
seven hells which constitute the “inferior
worlds”. These Hells are situated just below
*Mount Meru, the centre of the universe, which
itself has seven faces, each one facing one of the
seven oceans ( *sapta sagara) and one of the
“island-continents” ( *sapta dvipa). Muchalinda
was the “original serpent” who created
primordial nature. *Mount Meru, the sacred
and mythical mountain of Indian religions,
which is thus associated with the number seven,
receives its light from the *Pole star, the last of
the seven stars of the Little Bear, situated on
exactly the same line as this “axis of the world”.
On the other hand, the four heads of Kaliya
represent the essentially terrestrial nature of the
serpent, which crawls along the ground. In
Indian mystical thought, earth corresponds
symbolically to the number four, which is linked
to the square, which in turn is associated with
the four cardinal points. On the other hand, the
thousand heads of Shesha-Ananta symbolise
both the incalculable multitude and an eternal
length of time. As for the battle mentioned
above between * Krishna and the king of the
naga, this is the mystical expression of the rivalry
between man and serpent. This man-snake
duality is expressed in a very symbolic manner
in Vedic literature (notably in the Chhdndogya
Upanishad), where Krishna, the “Black”, before
his deification, is a simple scholar or *asura (an
“anti-god”). After his victory over the snake, he
becomes one of the divinities of the Hindu
pantheon: he becomes the eighth “incarnation"
(*avatara) of Vishnu, even before becoming the
“beneficent protector of humanity".
This duality is also expressed numerically,
because Krishna’s position as an incarnation of
Vishnu is equal to eight, which is exactly the
mystical value of the naga. The naga is not only
considered to be a genie of the ground, a
chthonian spirit who owns the earth and its
treasure, but also and above all an aquatic symbol;
it is a “spirit of the waters” living in the
underworlds. The symbolic value of the earth is 4.
In Indian mystical thought, water (see *Jala) also
has the value 4, thus the ambivalence surrounding
the serpent is expressed by the relation: naga =
earth + water = 4 + 4 = 8. This value is confirmed
by the fact that the naga reproduce in couples and
always develop in the company of the nagini (their
females); this gives the number eight as the result
of the symbolic multiplication of two (the naga
and his nagini) by 4 (the earth or water). This is
why the name of this species became a word-
symbol for the numerical value of 8 (see the entry
entitled *Naga).
As well as its terrestrial character, the
serpent symbolises primordial nature: “The
underworlds and the oceans, the primordial
water and the deep earth form one materia
prima, a primordial substance, which is that of
the serpent. He is spirit of the first water and
spirit of all waters, be they below, on the
surface of or above the earth. Thus the serpent
is associated with the cold, sticky and
subterranean night of the origins of life: All the
serpents of creation together form a unique
primordial mass, an incalculable primordial
thing, which is constantly in the process of
deteriorating, disappearing and being reborn.”
[Keyserling, quoted in Chevalier and
Gheerbrant (1982)]. Thus the serpent
symbolises life. The primordial thing is life in its
latent form. Keyserling says that the
Chaldaeans only had one word to express both
“life” and "serpent”. The symbolism of the
serpent is linked to the very idea of life; in
Arabic, serpent is hayyah and life is hay at.
[Guenon, quoted in Chevalier and Gheerbrant
(1982)]. The serpent is one of the most
important archetypes of the human soul
[Bachelard, quoted in Chevalier and
Gheerbrant (1982)]. The same images are
found in Indian cosmological and mythological
representations. Thus in tantric doctrine, the
Kundalini, literally the “Serpent” of Shiva,
source of all spiritual and sexual energies
(energies = *shakti) is said to be found coiled
up at the base of the vertebral column, on the
chakra of the state of sleep. And when he wakes
up, “the serpent hisses and becomes tense, and
the successive ascent of the chakra begins: this
is the arousal of the libido, the renewed
manifestation of life" [Frederic, Dictionnaire
(1987)]. Moreover, from a macroscopic point
of view, the Kundalini is the equivalent of the
serpent * Ananta, who grasps in his coils the
very base of the axis of the universe. He is
associated with Vishnu and Shiva, and
symbolises cyclical development and
reabsorption, but, as guardian of the nadir, he
is the bearer of the world, and ensures its
continuity and stability. But Ananta is first and
foremost the serpent of infinity, immensity and
eternity. All these meanings are in fact various
applications, depending on the field in
question, of the myth of the original Serpent,
which represents primordial indifferentiation.
The serpent is considered to be both the
beginning and the end of all creation. It is not
by chance that the Sanskrit language uses the
word Shesha, the “remainder”, to denote the
serpent Ananta; to the Indians, the naga with a
thousand heads represents the “vestige” of
worlds which have disappeared as well as the
seed of worlds yet to appear. This explains the
importance which so many cosmologies and
mythologies place on the eschatological
symbolism of the serpent.
In summary, the snake has always been
associated with ideas of the sky, celestial bodies,
the universe, of the night of origins, materia
prima, the axis of the world, primordial
substance, the vital principle, life, eternal life
and sexual or spiritual energy. It is also
connected to ideas of the vestige of past
creations and the seed of future creations,
cyclical development and reabsorption,
longevity, an innumerable quantity, abundance,
fertility, immensity, wholeness, absolute
stability, endless undulating movement, etc.
In other words, since time immemorial,
and amongst all the races of the earth, the
serpent, as well as being a symbol of the earth
and water, personifies the notion of infinity
and eternity. See Infinity, Infinity (Indian
concepts of) and Infinity (Indian
mythological representation of).
SERPENT OF INFINITY AND ETERNITY.
See Ananta, Sheshashirsha, Infinity (Indian
mythological representation of) and Serpent
(Symbolism of the).
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
494
SERPENT OF THE DEEP. [S]. Value = 8. See
Ahi, Eight and Serpent (Symbolism of the).
SERPENT WITH ATHOUSAND HEADS. [S].
Value = 1,000. See Sheshashirsha and
Thousand. See also Infinity (Indian
mythological representation of) and Serpent
(Symbolism of the).
SERPENT. [S]. Value = 8. See Naga, Ahi, Sarpa
and Eight.
SEVEN. The ordinary Sanskrit words for the
number seven are *sapta and *saptan. Here is
a list of corresponding numerical symbols:
*Abdhi, Achala, *Adri, *Aga, *Ashva, *Atri,
Bhaya, *Bhubhrit, * Bhudhara , Chandas, Dhatu,
Dhi, *Dvipa, *Giri, Hay a, Kalatra, *Loka,
*Mahidhara, *Mdtrika, *Muni, *Naga,
*Parvata, *Patdla, * Pavana, *Pushkara, *Rishi,
*Sagara, *Sagara, *Samudra, *Shaila, *Svara,
*Tattva, *Turaga, *Turangama, *Vajin, *Vara,
*Vyasana and Yati. These words have the
following symbolic meaning or translation:
1. “Purification” and by extension
“Purifier” (Pavana). 2. The horses ( Ashva ,
Turaga, Turangama, Vajin). 3. The island-
continents (Dvipa). 4. The seas or oceans
(Sagara, Samudra ). 5. The divine mothers
(. Matrika ). 6. The worlds ( Loka ). 7. The
inferior worlds ( Patala ). 8. The mountains or
hills ( Adri , Aga, Bhubhrit, Bhudhara, Giri,
Mahidhara, Naga, Parvata, S hail a). 9. The
syllables (Svara). 10. The musical notes
(Svara). 11. The “Sages” of Vedic times (Muni,
Rishi). 12. The last of the seven Rishi (Atri).
13. The days of the week (Vara). 14. “That
which does not move” (Naga). 15. The seventh
“island-continent” (Pushkara). 16. The fears
(Bhaya) (only in *Jaina religion). 17. The
winds (Pavana).
See Numerical symbols.
SEVENTEEN. Ordinary Sanskrit name:
*saptadasha. Corresponding numerical
symbol: *atyashti.
SEVENTY. See Saptati.
SEVERUS SEBOKT. Syrian bishop of the
seventh century CE. His w'orks notably include
a manuscript dated 662 CE, where he talks of
the system of nine numerals and Indian
methods of calculation.
SHAD (SHASH, SHAT). Ordinary Sanskrit
name for the number six, this word forms part
of the composition of many other words
which are directly related to the idea of
this number.
Examples: *Shddanga, *Shaddyatana,
*Shaddarshana, *Shadgunya, *Shatkasampatti.
For w r ords which are symbolically related to this
number, see Six and Symbolism of numbers.
SHADANGA. “Six parts”. This is the name for
the six aesthetic rules of painting, which are
described in a commentary on the Kamasutra
by Yashodhara (these six rules being as follows:
rupabheda, “shape”; pramanarn, “size”; bhava,
“sentiment"; lavana, “grace"; sadrishyam,
“comparison”; and varnikabahanga, “colour”).
SHADAYATANA. [SI. Value = 6. “Six *guna".
These are the “six bases”, or “six categories”.
These are the six senses, objects or sense organs
of Buddhist philosophy (namely: the eye, the
nose, the ear, the tongue, the body and the
mind). See Six.
SHADDARSHANA. [S]. Value = 6. “Six visions”,
“six contemplations”, “six philosophical points
of view”. These are the six principal systems of
Hindu philosophy. See Darshana and Six.
SHADDASHA. Ordinary Sanskrit name for
the number sixteen. For words which are
symbolically connected to this number see
Sixteen and Symbolism of numbers.
SHADGUNYA. [S]. Value = 6. “six *guna". This
is a synonym of *shaddyatana. See Six.
SHADVIMSHATI. Ordinary Sanskrit name for
the number twenty-six. For words which are
symbolically associated with this number see
Twenty-six and Symbolism of numbers.
SHAILA. [SJ. Value = 7. “Mountain”. This
concept is related to the myth of *Mount
Meru, where the numbers seven plays a
significant role. See Adri and Seven.
SHAKA (Calendar). This is the most widely
used calendar in Hindu India, as well as in the
parts of Southeast Asia influenced by India. It is
also known as Shakakdla, Shakardja or
Shakasamvat. It began in the year 78 of the
Common Era. According to certain traditions,
this calendar was begun in the first century CE
by a Satrap (Kshatrapa) king called Shalivahana
(or Nahapana), who then reigned over the city
of *Ujjain. To find a corresponding date in the
Common Era, add seventy-eight to a date
expressed in Shaka years. See Indian calendars.
SHAKA NUMERALS. Symbols derived from
‘Brahmi numerals and influenced by Shunga
numerals, arising at the time of the Shunga
dynasty (second to first centuries BCE). The
symbols corresponded to a mathematical
system that was not based on place values and
therefore did not possess a zero. See: Indian
written numeral systems (Classification of).
See Fig. 24.52 and 24.61 to 69.
SHAKASAMVAT (Calendar). See Shaka.
SHAKRA. [S]. Value = 14. “Powerful”. Allusion
to the “strength" of the god ‘Indra, amongst
whose attributes is Shakradevendra, “Powerful
Indra". This explains the symbolism in
question, becuase Indra = 14. See Fourteen.
SHAKTI. [SI. Value = 3. “Energy”. In
Brahmanism and Hinduism, this word denotes
feminine energy or the active principle of all
divinity. The allusion here is to the shakti of the
most important divinities, namely those of the
triad formed by * Brahma, * Vishnu and
*Shiva. See Three.
For an example of the use of this word-
symbol, see: El, XIX, p. 166.
SHANKARACHARYA. Hindu philosopher of
the late ninth century. His works notably
include Shdrirakamimdmsdbhashya (great
commentary on the Vedanlasutra), where there
is a reference to the place-value system of the
Indian numerals.
SHANKARAKSHI. [S]. Value = 3. Synonym of
*Haranetra, “eyes of ‘Shiva”. See Three.
SHANKARANARAYANA. Indian astronomer
c. 869 CE. His w'orks notably include a text
entitled Laghubhaskariyavivarana in which the
place-value system of Sanskrit numerical
symbols is used frequently. He also uses the
katapayadi method invented by Haridatta
[see Billard (1971), p. 8]. See Numerical
symbols, Katapayadi numeration and Indian
mathematics (History of).
SHANKHA. Word which expresses the sea
conch. It is a symbol of riches and of certain
Hindu and Buddhist divinities (such as
‘Vishnu). It is a name given to the number ten
to the power twelve. See Names of numbers.
For an explanation of this symbolism, see High
numbers (Symbolic meaning of).
Source: *Rdmayana by Valrmki (early centuries CE).
SHANKHA. Word which expresses the sea
conch. It is given to the number ten to the power
thirteen (ten billion). See Names of numbers.
For an explanation of this symbolism, see High
numbers (Symbolic meaning of).
Source: *Kitah fi tahqiq i ma Hi hind by al-Biruni
(c. 1030 CE).
SHANKHA. Word meaning sea conch. It is
given to the number ten to the power eighteen.
See Names of numbers. For an explanation of
this symbolism, see High numbers (Symbolic
meaning of).
Source: *Ganitasdrasamgraha by Mahaviracharya
(850 CE).
SHANKU. Literally: “Diamond”. Name given
to the number ten to the power thirteen (ten
billion). See Names of numbers. For an
explanation of this symbolism, see High
numbers (Symbolic meaning of).
Sources: *Lildvati by Bhaskaracharya (1150 CE);
*Ganilakaumudi by Narayana (1350 CE),
*Trishatika by Shridharacharya (date uncertain).
SHANMUKHA. [S]. Value = 6. Synonym of
*Kumdrasya, “Faces of *Kumara (=
Shanmukha)”. This is an allusion to the six
heads of ‘Karttikeya. See Karttikeyasya and Six.
SHANMUKHAJBAHU. [SJ. Value = 12. “Arms
of ‘Shanmukha (= ‘Kumara = ‘Karttikeya)”.
Karttikeya is said to have had twelve arms. See
Karttikeyasya and Twelve.
SHARA. [SJ. Value = 5. “Arrow". This is one of
the attributes of ‘Kama, Hindu divinity of
Cosmic Desire and Carnal Love, who is
generally invoked during wedding ceremonies,
and whose action is said to determine the laws
of *samsara for human beings. The symbolism
in question is due to the fact that Kama is often
represented as a young man armed with a bow
made of sugar cane shooting five arrows
(* panchabdna) which are either flowers or
adorned with flowers. See Arrow and Five.
SHARADA NUMERALS. Symbols derived
from ‘Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, and Gupta.
Used in Kashmir and the Punjab from the
ninth to the fifteenth centuries CE. The
symbols correspond to a mathematical system
that has place values and a zero (shaped like a
dot). The more or less direct ancestor of
Takari, Dogri, Kulul, Sirmauri, Kochi, Landa,
Maltani, Khudawadi, Sindhi, Punjabi,
Gurumukhi, etc. numerals. For historic
symbols, see Fig. 24.40; for current symbols, see
Fig. 24.14; for derived notations, see Fig. 24.52.
For the corresponding graphical development,
see Fig. 24.61 to 69. See: Indian written
numeral systems (Classification of).
SHASH. Ordinary Sanskrit word for the
number six. See Shad.
SHASHADHARA. [SJ. Value = 1. “Which
represents a rabbit". This is connected with an
attribute of the moon. According to legend, a
rabbit, who offered its own flesh to relieve the
poor, was rewarded by having its own image
impressed on the face of the moon. This
explains the symbolism in question, because
“Moon” = 1. See Abja and One.
495
SHASHANKA
SHASHANKA. [SJ. Value = 1. “Moon”. See
A bja and One.
SHASHIN. IS]. Value = 1. “To the Rabbit”. This
is the rabbit which, according to legend, was
drawn on the face of the moon. See
Shashadhara, Abja and One.
SHASHTI. Ordinary Sanskrit name for the
number sixty.
SHAT. Ordinary Sanskrit name for the number
six. See Shad.
SHATA. Ordinary Sanskrit name for the number
one hundred. Its multiples are formed by placing
it to the right of the names of the corresponding
units: dvashtat (two hundred), trishata (three
hundred), chatushata (four hundred), etc. This
name forms part of the composition of several
words which are symbolically associated with the
idea of this number.
Examples: * Shatapathabrahmana, *Shat-
arvdriya, *Shatarupa, * Shatottaraganana,
* Shatottaraguna, *Shatottarasamjna. For words
which have a more symbolic link with this number,
see Hundred and Symbolism of numbers. See
also Sata for an ancient form of this number.
SHATAKOTI. Literally: a hundred *koti”. This
is the name given to the number ten to the
power nine. See Names of numbers and High
numbers.
Source: *Ganitasarasamgraha by Mahaviracharya
(850 CE).
SHATAPATHABRAHMANA. “Brahmana of
the Hundred ways”. This is the title of an
important work of Vedic literature, divided
into a hundred adhydya (“recitations”).
SHATARUDRIYA. Name of a Sanskrit hymn
which is part of the *Taittiriya Samhita
( Yajurveda ), addressed to *Rudra considered
from a hundred different perspectives.
SHATARUPA. "Of a hundred forms”. One of
the names for the first woman, daughter and
wife of *Brahma, who is said to have been
gifted with a “hundred bodies”. See Rupa.
SHATKASAMPATTI. Literally “six great
victories" (from shatka, “made up of six”, and
sampatti, "to obtain, achieve, succeed”). In
Hinduism, this refers to the “Six Great Victories"
of Tattvabodha of Shankara, which constitutes
the first of the four conditions that an adept of
the philosophy of the Vedanta must fulfil.
SHATOTTARAGANANA. “Centesimal
arithmetic". There is a reference to this in
* Lalitavistara Sutra [see Datta and Singh
(1938), p. 10).
SHATOTTARAGUNA. “Hundred, primordial
property”. Sanskrit name for the centesimal
base. Reference to this is found in the
* Lalitavistara Sutra.
SHATOTTARASAMJNA. “Names of multiples
of a hundred”. This term applies to names of
numbers in Sanskrit numeration in the
centesimal base. There is a reference to this in
* Lalitavistara Sutra [see Datta and Singh
(1938), p. 10]. The equivalent of this w'ord in
terms of the decimal base is
*Dashagunasamjna. See Shatottaraganana,
Shatottaraguna and Dashagundsamjha.
SHESHA. "Vestige”, “that which remains” or
"he who remains". In Brahman and Hindu
mythologies, this is the name of *Ananta, the
king of the *naga and serpent of the infinity,
eternity and immensity of space. See Serpent
(Symbolism of the).
SHESHA. [Arithmetic]. “Vestige". Term
describing the remainder in division.
SHESHASHiRSHA. [S], Value = 1,000.
Literally “heads of *Shesha”. Shesha is the king
of the *naga who lives in the inferior worlds
(*patala) and who is considered to be the
“Vestige" of destroyed universes as well as the
seed of all future creation. This symbolism
comes from the fact that Shesha is represented
as a serpent with a thousand heads, the
number thousand here meaning “multitude”
and the “incalculable”. See Ananta, Thousand,
High numbers (Symbolic meaning of) and
Serpent (Symbolism of the).
SHIKHIN. [S]. Value = 3. “Ablaze”. This is one
of the names for *Agni, the god of sacrificial
fire, whose name is equal to the number three.
See Three.
SHIRSHAPRAHELIKA. From shirsha, “head”,
and prahelika, “awkward question, enigma”.
This term is used in the texts of *Jaina
cosmology to denote a period of time which
corresponds approximately to ten to the power
196. See Anuyogadvdra Sutra, Names of
numbers, High numbers and Infinity.
SHITAMSHU. [S]. Value = 1. “Of the cold rays".
This is a synonym of “moon", the opposite of the
warm rays of the sun. See Abja and One.
SHITARASHMI. [S]. Value = 1. “Of the cold
rays”. This is a synonym of “Moon”, the opposite
of the warm rays of the sun. See Abja and One.
SHIVA. [S]. Value = 11. One of the three main
divinities of the Brahmanic pantheon
(*Brahma, *Vishnu, *Shiva). There is no
mention of Shiva in the *Veda, and it would
seem that Shiva did not become a god until
relatively recently. The symbolism in question
comes from the fact that Shiva is none other
than an incarnation of *Rudra, ancient Vedic
divinity of tempests and cosmic anger. As
Rudra symbolises the number 11 (because
of the eleven vital breaths, born from
Brahma’s forehead, of which he was the
personification), the name of Shiva also came
to represent this number. See Rudra-Shiva
and Eleven.
SHRiDHARACHARYA. Indian mathematician.
The date of his birth is not known. His works
notably include Trishatika. Here is a list of
the principal names of numbers mentioned in
this work:
*Eka (= 1), * Dasha (= 10), *Shata (= 10 2 ),
*Sahasra (= 10 3 ), *Ayuta (= 10 4 ), * Laksha (= 10 5 ),
*Prayuta (= 10 6 ), *Koti (= 10 7 ), *Arbuda (= 10 8 ),
*Abja (= 10 9 ), *Kharva (= 10 10 ), *Nikharva (=
10 u ), *Mahasaroja (= 10 12 ), *Shanku (= 10 13 ),
*Saritapati, (= 10 14 ), *Antya (= 10 15 ), *Madhya
(= 10 16 ), *Parardha (= 10 17 ).
Ref: TsT, R. 2-3 [see Datta and Singh
(1938), p. 13].
See Names of numbers, High numbers
and Indian mathematics (History of).
SHRIPATI. Indian astronomer c. 1039 CE. His
works notably include a text entitled
Siddhantashekhara, in which the place-value
system of the Sanskrit numerical system is used
frequently [see Billard (1987), p. 10]. See
Numerical symbols, Numeration of
numerical symbols, and Indian mathematics
(History of).
SHRUTI. [S]. Value = 4. “Recital”. Name given
to the ancient Brahmanic and Vedic religious
texts, which are said to have been revealed by a
divinity to one of the seven “Sages” ( *rishi ),
poets and soothsayers of Vedic times
(*Saptarishi). As this allusion primarily
concerns the *Veda, and there are four of them,
shruti = 4. See Four.
SHUKRANETRA. [S]. Value = 1. The “Eye of
Shukra”. According to legend, this divinity had
one eye destroyed by *Vishnu, thus the
symbolism in question. See One.
SHULA. [S]. Value = 3. “Point". Allusion to the
three points of *Shivas Trident ( *trishula ),
which symbolise the three aspects of the
manifestation (creation, preservation,
destruction), as well as the three primordial
principles ( *triguna ), and the three states of
consciousness ( *tripura ). See Guna and Three.
SHULIN. [S]. Value = 11. This is one of the
attributes of *Rudra, who is invoked as “lord of
the animals” in the Shulagava, Brahmanic
sacrifices of two-year-old calves with the aim of
obtaining prosperity. Thus Shulin = Rudra = 11.
See Rudra-Shiva and Eleven.
SHUNGA NUMERALS. Symbols derived from
*Brahmi numerals, arising during the Shunga
dynasty (second century BCE). The symbols
corresponded to a mathematical system that
was not based on place-values and therefore
did not possess a zero. See: Indian written
numeral systems (Classification of). See Fig.
24.30, 24.52 and 24.61 to 69.
SHUNYA. Literally “void”. This is the principal
Sanskrit term for “zero”. However, the Sanskrit
language (the excellent literary instrument of
mathematicians, astronomers and all Indian
scholars) has many synonyms for expressing
this concept ( *abhra , *akasha, *ambara,
*ananta, *antariksha, *bindu, *gagana,
*jaladharapatha, *kha, *nabha, *nabhas,
*purna, *vishnupada, *vindu, *vyoman, etc.).
The words *kha, *gagana, etc., are used for
“sky”, “firmament”, and the words *ambara,
*abhra, *nabhas, etc., signify “space”,
“atmosphere”, etc. The word *dkasha means
the fifth “element”, “ether”, the immensity of
space, as well as the essence of all that is
uncreated and eternal. There is also the word
*bindu, which means “dot" or “point". At least
since the beginning of the Common Era, shunya
means not only void, space, atmosphere or
ether, but also nothing, nothingness,
negligible, insignificant, etc. In other words,
the Indian concept of zero far surpassed the
heterogeneous notions of vacuity, nihilism,
nothingness, insignificance, absence and non-
being of Greek and Latin philosophies. See
Shunyatd, Numerical symbols, Zero, Zero
(Graeco-Latin concepts of), Zero (Indian
concepts of) and Indian atomism.
SHUNYA-BINDU. Literally: “void-dot”. Name
of the graphical representation of zero in the
shape of a dot. See Shuya, Bindu, Numeral 0
(in the shape of a dot) and Zero.
SHUNYA-CHAKRA. Literally: “void-circle".
Name of the graphical representation of zero in
the shape of a little circle. See Shunya, Numeral
0 (in the shape of a little circle) and Zero.
SHUNYA-KHA. Literally: “void-space”. Name
given to the function of zero in numerical
representations: it is the empty space which marks
the absence of units of a given order in positional
numeration. See Kha, Shunya and Zero.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
496
SHUNYA-SAMKHYA. Literally: 'void-
number’’. Name given to zero as a numerical
symbol. It is also the “zero quantity"
considered to be a whole number in itself. See
Samkhya, Shunya and Zero.
SHUNYATA. In Sanskrit, the privileged term
for the designation of zero is * shunya , which
literally means “void”. But this word existed
long before the discovery of the place-value
system. Since Antiquity, this word has
constituted the central element of a mystical
and religious philosophy, developed as a way
of thinking and behaving, namely the
philosophy of “vacuity" or shunyala. See
Shunya. This doctrine is a fundamental
concept of Buddhist philosophy and is called
the “Middle Way" ( Madhyamaka ), which
teaches that every thing in the world
( samskrita ) is empty ("shunya), impermanent
(anitya), impersonal ( anatman ), painful
(dukha) and without original nature. Thus this
vision, which does not distinguish between the
reality and non-reality of things, reduces these
things to complete insubstantiality.
This philosophy is summed up in the
following answer that the Buddha is said to
have given to his disciple Shariputra, who
wrongly identified the void ("shunya) with
form ( "rupa ): "That is not right,” said the
Buddha, “in the shunya there is no form, no
sensation, there are no ideas, no volitions, and
no consciousness. In the shunya, there are no
eyes, no ears, no nose, no tongue, no body, no
mind. In the shunya, there is no colour, no
noise, no smell, no taste, no contact and no
elements. In the shunya, there is no ignorance,
no knowledge, or even the end of ignorance. In
the shunya, there is no aging or death. In the
shunya, there is no knowledge, or even the
acquisition of knowledge.
“Buddhists did not always use shunya in
this sense: in the ancient Buddhism of
Hinaydna (known as the “Lesser Vehicle"), this
notion only applied to the person, whereas in
Mahdydna Buddhism (of the schools of the
North and known as the “Greater Vehicle”), the
idea of vacuity stretched to include all things.
To explain the difference between these two
concepts, the Buddhists of the schools of the
North make the following comparison: in the
ancient vision, things were regarded as if they
were empty shells, whereas in the Mahdydna
the very existence of the empty shells is denied.
This concept of the whole of existence being a
void should not lead to the conclusion that this
is an attitude of nihilism. Far ffom meaning
that things do not exist, this philosophy
expresses that things are merely illusions.
Through criticising the knowledge of things as
being a pure illusion (maya), it actually means
that absolute truth is independent of the being
and the non-being, because it is the shunyala or
“vacuity”. The shunyala has a real existence; it
is composed of'dkasha, or “ether”, the last and
most subtle of the “five elements”
("pahchabhuta or “ether) of Hindu and
Buddhist philosophies, which is considered to
be the essence of all that is uncreated and
eternal, and the element which penetrates
everything. The "akasha has no substance, yet
it is considered to be the condition of all
corporeal extension and the receptacle of all
matter which manifests itself in the form of one
of the other four elements (earth, water, fire,
air). According to this philosophy, the shunyala
is the ether-universe, the only “true universe".
This is why the being and the non-being are
considered to be insignificant and even illusory
compared to the shunyata, which thus excludes
any possible mixing with material things, and
which, as an unchanging and eternal element, is
impossible to describe. In * Mddhyamika
Buddhism (the followers are still called
"Shunyavadin or “vacuitists”), the void has been
identified with the absence of self and salvation.
Both are meant to achieve redemption, which is
only possible in the shunyata. In other words, in
order to be granted deliverance, vacuity must be
achieved; for this, the mind must be purified of
all affirmation and all negation at once.
This ontology is inextricably linked to the
mysticism of universal vacuity, and represented
the great philosophical revolution of Buddhism
amongst the schools of the North, implemented
by Nagarjuna, the patriarch of this sect. The
Madhyamakashdstra, the fundamental text
which is traditionally attributed to Nagarjuna,
was translated into Chinese in the year 409 CE,
when he had already achieved almost god-
like status, and was renowned far beyond
the frontiers of India. [See Bareau (1966),
pp. 143ff.-; Frederic (1987); Percheron (1956);
Renou and Filliozat (1953)]. This proves that
the fundamental concepts of this mysticism
were already fully established at the beginning
of the Common Era.
These concepts were pushed to such an
extent that twenty-five types of shunya were
identified. Amongst these figured: the void of
non-existence, of non-being, of the unformed,
of the unborn, of the non-product, of the
uncreated or the non-present; the void of the
non-substance, of the unthought, of
immateriality or insubstantiality; the void of
non-value, of the absent, of the insignificant, of
little value, of no value, of nothing, etc. This
means that in the shunyatdvada, the
philosophical notions of vacuity, nihilism,
nullity, non-existence, insignificance and
absence were conceived of very early and
unified according to a perfect homogeneity
under the unique label of shunyata (“vacuity”).
In this domain at least, India was very advanced
in comparison with corresponding Graeco-
Latin ideas. See Zero (Graeco-Latin concepts
of), Zero and Zero (Indian concepts of).
SHUNYATAVADA. Name of the Buddhist
doctrine of vacuity. See Shunya and Shunyata.
SHUNYAVADIN. “Vacuitist”. Name given to
the followers of the Buddhist philosophy of
vacuity. See Shunyata and Mddhyamika.
SIDDHA. In Hindu and Jaina philosophies, this
is the name given to human beings that become
immortals after having obtained liberation.
SIDDHAM NUMERALS. Symbols derived
ffom ‘Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, Gupta and
Nagari. Used in Ancient Nepal (sixth to eighth
centuries CE). The symbols corresponded to a
mathematical system that was not based on
place-values and therefore did not possess a
zero. Ancestor of Nepali, Tibetan, Mongolian,
etc. numerals. Siddham also influenced the
shapes of Kutila numerals, whence came
Bengali, Oriya, Kaithi, Maithili, Manipuri, etc.
numerals. See Fig. 24.41. For systems derived
ffom Siddham, see Fig. 24.52. For graphical
development, see Figs. 24.61 to 69. See: Indian
written numeral systems (Classification of).
SIDDHANTA. [Astronomy]. Generic name of
the astronomical texts which describe such
things as the calculation for an eclipse of the
Moon or the Sun, and the procedures, methods
and instruments of observation. Diverse
parameters and data are supplied, as well as the
procedure for trigonometric operations, etc.
See Indian astronomy (History of) and Yuga
(Astronomical speculations on).
SIDDHANTADARPANA.
See Nilakanthasomayajin.
SIDDHANTASHEKHARA. See Shripati.
SIDDHANTASHIROMANI. See Bhakaracharya.
SIDDHANTATATTVAVIVEKA. See Kamalakara.
SIDDHI. [S[. Value = 8. “Supernatural power".
This is an allusion to the ashtasiddhi, the eight
major siddhi, or eight supernatural powers which
the siddha and the purnayogin (perfect adepts of
the techniques of yoga) are gifted with. See Eight.
SIGNS IN THE FORM OF “S” OR “8".
See Numeral 8, Serpent (Symbolism of
the) and Infinity (Indian mythological
representation of).
SIGNS OF NUMERATION. See Fig. 24.61
to 69. See Indian numerals, which gives
the complete list of signs of numeration, as
well as Numerical notation, which gives a
list of the main systems of numeration used
in India since Antiquity. See also Indian
written numeral systems (Classification
of), which recapitulates on all the numerical
notations of the Indian sub-continent,
ffom both a mathematical and a palaeographic
point of view.
SIMHASAMVAT (Calendar). Calendar
beginning in the year 1113 CE. Add 1113 to a
date in this calendar to find the corresponding
year in the Common Era. Formerly used in
Gujarat. It was probably abandoned during the
thirteenth century. See Indian calendars.
SIMPLE YUGA (Non-speculative). See Yuga
(Astronomical speculation on).
SINDHI NUMERALS. Symbols derived
from ‘Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, Gupta and
Sharada. Currently used in the region of Sindh
(whose name derives from the river now
called the Indus). The symbols correspond
to a mathematical system that has place
values and a zero (shaped like a small circle).
See: Indian written numeral systems
(Classification of). See Fig. 24.6, 24.52 and
24.61 to 69.
SINDHU. [S[. Value = 4. “Sea”. See Sagara,
Four and Ocean.
SINE (Function). This is referred to as
ardhajya, which literally means: “demi-chord".
This is the name used since ‘Aryabhata (c. 510
CE) by Indian astronomers to denote this
function of trigonometry.
SINHALA (SINHALESE) NUMERALS.
Symbols derived from ‘Brahmi numerals and
influenced by Shunga, Shaka, Kushana,
Andhra, Pallava, Chalukya, Ganga, Valabhi
and Bhattiprolu numerals. Currently used
mainly in Sri Lanka (Ceylon), in the Maldives
and in other islands to the north of the
Maldives. (Note that in the north and
northwest of Sri Lanka, ‘Tamil numerals are
used by the Tamil inhabitants.). The symbols
correspond to a mathematical system that is
not based on place values and therefore does
497
SIX
not possess a zero. See: Indian written
numeral systems (Classification of). See Fig.
24.22, 24.52 and 24.61 to 69.
SIX. Ordinary Sanskrit names for this number:
‘shad, ‘shash, * shat . Here is a list of
corresponding numerical symbols: ‘Anga, Ari,
‘Darshana, ‘Dravya, *Guna, Kardka,
‘Kirttikeyisya, ‘ Kiya , Kharn, ‘Kumirisya,
‘Kumiravadana, Lekhya, Mala, ‘ Misirdha ,
‘Riga, ‘Rasa, Ripu, ‘Rita, 'Shidiyatana,
•Shaddarshana, 'Shadgunya, * Shanmukha ,
Shistra, Tarka.
These words have the following symbolic
meanings or translations: 1. The
philosophical points of view (Darshana) 2.
The six philosophical points of view
(Shaddarshana), 3. The bodies (Kaya), 4. The
colours (Riga), 5. The musical modes (Riga),
6. The weapons (Shistra), 7. The limbs (Anga),
8. The * Vedinga (Anga), 9. The merits, the
qualities, the primordial properties (Guna),
10. The six primordial properties, the six
bases, the six categories (Shadiyatana,
Shadgunya), 11. The seasons (Misirdha, Ritu),
12. The substances (Dravya), 13. The faces of
Karttikeya-Kumara (Kirttikeyisya, Kumirisya,
Kumiravadana, Shanmukha), 14. The
sensations, in the sense of “flavours” (Rasa).
See Numerical symbols.
SIX AESTHETIC RULES OF PAINTING. See
Shidanga.
SIXTEEN. Ordinary Sanskrit name: ‘shaddasha.
Here is a list of corresponding numerical
symbols: * Ashti , ‘Bhupa, Kali and ‘Nripa. These
words refer to or are related to the following: 1. A
particular element of Indian metric (Ashti) 2. The
sixteen kings of the legend of the * Mahibhirata
(Bhupa and Nripa) 3. The “fingers of the Moon”
(Kali). See Numerical symbols.
SIXTY. See Shashti.
SMALLEST UNIT OF LENGTH. See
Paramanu.
SMALLEST UNIT OF WEIGHT. See Paramanu
raja. See also ‘Indian weights and measures.
SOGANDHIKA. Name given to the number
ten to the power ninety-one. See Names of
numbers and High numbers.
Source: ‘Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
SOMA. [S], Value = 1. Name of an intoxicating
drink, used in Vedic times for religious
ceremonies and sacrifices: “It is a drink made
from a climbing plant, with which an offering is
made to the gods and which is drunk by
Brahmanic priests. This drink plays an
important role in the Rigi’eda. It is considered to
be capable of conferring supernatural powers
and is worshipped as though it were a god. The
Hindus also call it the wine of immortality
( * amrita ). It is the symbol of the transition from
ordinary sensory pleasures to divine happiness
(inanda). K. Friedrichs, etc. ‘In Indian thought,
Soma also represents the source of all life and
symbolises fertility; thus it is the sperm, the
receptacle of the seeds of cyclic rebirth. In this
respect, the soma is connected to the symbolism
of the moon. This is why the soma is also the
lunar star, a masculine entity compared with a
full goblet of the drink of immortality. Thus:
soma = 1. See Abja and One.
Source: ‘Lalitavistara Sutra (before 308 CE).
SPECULATIVE YUGA. See Yuga
(Astronomical speculation on).
SQUARE ROOT. [Arithmetic]. See
Varganmula. See also Square roots (How
Aryabhata calculated his).
SQUARE ROOTS (How Aryabhata calculated
his). In the chapter of Ganitapida in
Aryabhatiya devoted to arithmetic and
methods of calculation, the astronomer
‘Aryabhata (c. 510 CE) described, amongst
other operations, the rule for the extraction of
square roots [see Arya, Ganita, line 4]:
Always divide the even column by twice the
square root. Then, after subtracting the square of
the even column, put the quotient in the next place.
This will give you the square root.
The rule, thus formulated, is a typical
example of Aryabhata's extremely concise style,
only giving the essential information in his
definitions, operations or concepts, any other
information being deemed useless for reasons
only known by the man himself. Here is
the extract again, with the necessary
information added for easy understanding:
[After subtracting the largest possible
square from the figure found in the last uneven
column, then having written the square root of
the subtracted number in the line of the square
root] always divide the [figure in the] even column
[written on the right] by twice the square root.
Then, after subtracting the square [of the
quotient] from the [figure found in the] even column
[written on the right], place the quotient in the next
place [to the right of the figure which is already
written down in the line of the square root]. This
will give you the square root [desired]. [But if there
are figures remaining on the right, repeat the
process until there are no more of these figures],
[See: Datta and Singh (1938), pp. 169-75;
Clark (1930), pp. 23ff; Shukla and Sarma (1976),
pp. 36-7; Singh, in BCMS, 18, (1927)].
Here is the reproduction (with no theoretical
justification) of the first of these rules, in order to
calculate the square root of the number
55, 225, according to the information given
notably by Bhaskara (in 629) in his Commentary
on the Aryabhatiya: First, the number in
question is written in the following manner,
marking each uneven place with a vertical line
and each even place with a horizontal line:
I - I - I
5 5 2 2 5
Then a horizontal line is drawn (to the right of
the number in question), in order to write
down the successive numbers of the square
root:
5 5 2 2 5
line of the square root
By beginning the operation with the highest
figure of the uneven column, the biggest
square it contains is 4, thus the square root is
equal to 2. Therefore a 4 is placed in a line
underneath and a 2 on the line of square roots:
I - I - I 2
5 5 2 2 5
4 line of the square root
Then a line is drawn below the 4, which is
subtracted from the preceding 5; the result is 1,
and this figure is placed under the line in the
even position of this first section, without
forgetting to return the 4 to the extreme left of
this lower line:
I - I - I 2
5 5 2 2 5
4 line of the square root
4) 1
Next the figure in the even column written
immediately to the right (the 5) is considered,
and is placed below the bottom line, to the
right of the 1:
I - I - I 2
5 5 2 2 5
4 line of the square root
4) 1 5
Now the number 15 which has been obtained is
divided by twice the square root that was
previously found (2), in other words by 4; as the
quotient found is 3, thus 3 is written on the line of
square roots, to the right of the 2 that is already
there, without forgetting to record the same
figure on the extreme right of the line of the 15:
I - I - I 2 3
5 5 2 2 5
4 line of the square root
4) 1 5 (3
The product of the numbers 4 and 3 (placed to
the left and right of the line of 15) is 12, and
this is placed on the line below 15:
I - I - I 2 3
5 5 2 2 5
4 line of the square root
4) 1 5 (3
1 2
Then 12 is subtracted from the above 15, and
the result is placed on the line below, after
drawing a line below the number 12:
I - I - I 2 3
5 5 2 2 5
4 line of the square root
4) 1 5 (3
1 2
3
Then the 2 from the following uneven column
is placed next to the 3:
I - I - I 2 3
5 5 2 2 5
4 line of the square root
4) 1 5 (3
1 2
3 2
And a 9 (the square of the quotient 3 found
above, indicated to the right of the line of 15) is
placed in the line below the 32:
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
498
2 3
5 5 2
2 5
4
line of the square root
4) 1 5
(3
1 2
3
2
9
A line is drawn and the 9 is subtracted from the
32, then the result is placed below this line:
1 - 1
- 1 2 3
5 5 2
2 5
4
line of the square root
4) 1 5
(3
1 2
3
2
2
9
3
Now the 2 is
taken from the even column and
placed to the
right of the positions of 23:
1 - 1
- 1 2 3
5 5 2
2 5
4
line of the square root
4) 1 5
(3
1 2
3
2
2
9
3 2
Then the number 232 which has been thus
obtained is divided by 46, which is double the
square root found (23), and as the quotient is 5,
the numbers 46 and 5 are written as follows
(the divisor 46 on the left and the quotient 5 on
the right), by placing a 5 on the line of square
roots to the right of the 3:
I - I - I 2 3 5
5 5 2 2 5
4 line of the square root
4) 1
5
(3
1
2
3
2
9
46)
2
3 2
And as the product of 46 times 5 is 230, this
number is placed below 232:
I - I - I 2 3 5
5 5 2 2 5
4 line of the square root
4) 1 5 (3
1 2
3 2
9
46) 2 3 2 (5
2 3 0
Another line is drawn, and the following
subtraction is carried out:
I - I - I 2 3 5
5 5 2 2 5
4 line of the square root
4) 1 5 (3
2 2
3 2
9
46) 2 3 2 (5
2 3 0
2
The last figure (5) is lowered and placed to the
right of the 2:
I - I - I 2 3 5
5 5 2 2 5
4 line of the square root
4) 1 5 (3
1 2
3 2
9
46) 2 3 2 (5
2 3 0
2 5
The last quotient is equal to 5, and the square
of this number is taken (25) and subtracted
from this last number. As the result is equal to
zero, the operation is finished. It is clear that
the operation has worked, because the square
root of 55 225 is equal to 235.
2 3 5
5
5
2
2
5
4
line of the square root
4)
1
5
(3
1
2
3
2
9
46)
2
3
2
(5
2
3
0
2
5
2
5
0
Thus it is clear that this procedure is not
algebraic (contrary to Kaye’s allegations, who
gave the unwarranted affirmation that
Aryabhata’s method was identical to that of
Theon of Alexandria), and it is also clear that
it is impossible to use Aryabhata’s method if
the numbers in question are not expressed in
writing using distinct numerals as a base for
the calculations. In other words, the
operations described by Aryabhata involve
placing the numbers involved in the
calculation in two or three blocks of
numbers, according to whether it is the
square root or the cube root that is being
extracted. It can be proved mathematically
that these operations could not be carried out
using a written numeration that was not
based upon the place-value system and did
not have a zero.
STHANA. Sanskrit term meaning “place".
Generally used by Indian scholars to express
the place-value system. See Sthanakramad,
Ankakramena and Ankasthana.
Source: *Lokavibhaga (458 CE).
STHANAKRAMAD. Sanskrit term which
literally means: “in the order of the position”.
Often used by Indian scholars in ancient
times (fifth - seventh century CE) to indicate
that a series of numbers or numerical word-
symbols were written according to the
place-value system. An example of this is
found in the *Jaina cosmological text, the
*Lokavibhdga (“Parts of the universe”), which
is the oldest known Indian text to contain an
example of the place-value system written in
numerical symbols. [See Anon. (1962), chap.
IV, line 56, p. 79].
SUBANDHU. Indian poet from the beginning
of the seventh century CE. His works notably
include a love story entitled Vdsavadattd, where
there are precise references to zero written as a
dot ('shunya-bindu). See Zero.
SUBSTANCE. [S]. Value = 6. See Dunya and Sue.
SUBTRACTION. [Arithmetic]. See Vyavakalita.
SUDDHA SVARA. These are the seven notes of
the sa-grdmma (Sa, Ri, Ga, Ma, Pa, Dha, Ni),
the first scale in Indian music. The notes are
represented by short syllables, each one
corresponding to the initial of the Sanskrit
name of the note (Ni = Nishada ; Ga =
Gandhara, etc.) [see Frederic, Dictionnaire,
(1987)].
SUDHAMSHU. [S], Value = 1. “Moon”. See
Abja and One.
SUDRISHTI. “That which is seen clearly”.
Name given to the Pole star, “the Star which
never moves”. See Dhruva, Grahddhdra and
Mount Meru.
SUMERU. One of the names of *Mount Meru.
See Adri, Dvipa, Purna, Patala, Sdgara,
Pushkara, Parana and Vayu.
SUN. As a concept associated with the number
thousand. See Sahasrakirana, Sahasramshu
and High numbers.
SUN. As a mystical value equal to 7. See Mount
Meru.
SUN. [S]. Value = 12. See Bhanu, Divakara,
Dyumani, Martanda, Shasramshu, Surya and
Twelve.
SUN RAYS. [S[. Value = 12. See Shasramshu
and Twelve.
SUN-MOON (The couple). [S[. Value = 2. See
Ravichandra and Two.
SUPERNATURAL POWER. [S]. Value = 8. See
Pahchahhijha, Siddhi and Eight.
SURA. [S[. Value = 33. “Gods”. See Deva and
Thirty-three.
SURYA. [S[. Value = 12. Name of the Brahmanic
sun god. This symbolism is explained by the
“course” of the sun during the twelve months of
the year. See Rdshi and Twelve.
SUTA. [S[. Value = 5. “Son”. See Putra and Five.
SVARA. [S[. Value = 7. “Note”, “syllable”. This
is probably an allusion to the *suddha svara,
the seven notes of the first scale in Indian
music. See Seven.
499
SYMBOLISM OF NUMBERS
SYLLABLE. [S]. Value = 7. See Svara and Seven.
SYMBOLISM OF NUMBERS. Here is a list of
associations of ideas contained in Indian
numerical symbolism, given in arithmetical
order (list not exhaustive):
Number One. Concept often directly or
symbolically related to: the god ‘Surya; the god
*Ganesha; a type of deep concentration
( *ekagratd ); the sacred Syllable of the Hindus
( *ekakshara ); a certain monotheist doctrine
( *ekantika ); the study of the unique reality; the
contemplation of Everything {*ekatva)\ the
‘moon; the drink of immortality ( *soma)\ the
♦earth; the * Ancestor; the ‘Great Ancestor; the
‘First Father; the ‘beginning; the ‘body; the
‘Self; ‘Ultimate reality; the superior soul;
the ‘individual soul; the ‘Brahman; the
“‘form"; the “‘drop"; the “‘immense"; the
“‘indestructible”; the ‘rabbit; the
“‘Luminous”; the ‘Pole star; the “Cold rays”;
the ‘eye of Shukra; the ‘terrestrial world; the
"‘Bearer”; the ‘primordial principle; the
‘rabbit figure; the ‘Cow; the sour milk, etc. See
Eka and One.
Number Two. Concept often directly or
symbolically related to: ‘duality; the idea of
couple, ‘pair, twins or contrast; the
‘symmetrical organs; ‘wings; the ‘hand; the
‘arms; the ‘eyes; ‘vision; the ‘ankles; the
primordial couple; the couple; ‘Sun-Moon; the
twin gods; the conception of the world;
‘contemplation; revelation; the ‘Horsemen;
the epithet "Twice born”; the third age of a
*mahayuga ( *Yuga)\ etc. See Dva and Two.
Number Three. Concept often directly or
symbolically related to: the ‘three “classes” of
beings; the ‘Triple science; the first three
*Vedas\ the ‘eyes; the three eyes of ‘Shiva; the
“‘three worlds”; the god Shiva; the god Vishnu;
the god Krishna; the ritual dress of Buddhist
monks {*trichivara)\ the “three primary forces”;
the “three eras”; the “‘three bodies”; the
‘three forms”; the “three baskets”; the ‘three
city-fortresses; the ‘three states of
consciousness; the “‘three jewels”; the triple
town-fortress ( *tripura)\ the town-fortress with
the triple rampart ( tripura ); a demon with three
heads (*trishiras); Shiva’s Trident; the principal
castes of Brahmanism; the “‘three aims”; the
‘three letters”; the god *Agni; “fire”; the
*god of sacrificial fires; the “three rivers;
the ‘phenomenal world; the “‘aphorism”;
‘Feminine Energies; the “‘merits”; the
‘qualities”; the Spirit of the waters; the ‘Eye;
the ‘points; the ‘times; the “‘three heads”; the
‘three ‘Rama; etc. See Traya, Vajra and Three.
Number Four. Concept often directly or
symbolically related to: the "‘four oceans”; the
“‘four stages”; the “‘four island-continents”;
the four “‘great kings”; the “four ‘months";
the “four ‘faces”; the “four aims"; the “four
‘ages”; the “four ‘ways of rebirth"; water; sea;
‘ocean; “‘horizons"; the ‘cardinal points; the
‘arms of Vishnu; the ‘positions; the ‘vulva;
the ‘births; the “‘Fourth” (as an epithet of
Brahma); the conditions of existence; the
*Vedas; the ‘faces of Brahma; the four ages of a
*mahayuga\ “‘faces”; etc. See Chatur and Four.
Number Five. This number is considered to
be sacred and magic in India and all Indianised
civilisations of Southeast Asia. It is often directly
or symbolically related to: the ‘Bow with five
flowers; the “‘five supernatural powers”; the
‘five elements of the manifestation; the “five
visions of Buddha”; the “five ‘horizons"; the
‘gifts of the Cow; the “five ‘faculties”; the “five
‘impurities"; the five ‘heads of Rudra (= Shiva);
the “five ‘mortal sins”; the five ‘orders of beings;
the “five treasures" of Jaina religion
( *panchaparameshtin'y, the “sons of Pandu”; the
‘arrows; the characteristics; ‘Purification; the
“‘Great Elements"; the ‘great sacrifices; the
‘main observances; the ‘fundamental principles;
the ‘realities; the ‘truths; the “‘true natures”;
the ‘Jewels; the ‘breaths; the ‘senses; the
‘winds; the sense organs; the ‘faces of Rudra;
etc. See Pancha; and Five.
Number Six. Concept often directly or
symbolically related to: the “six ‘parts; the “six
‘bases”; the “six categories”; the six
‘philosophical points of view; the six aesthetic
rules {*shddanga)\ the ‘bodies; the ‘colours;
the ‘musical modes; the weapons; the limbs;
the ‘merits; the ‘qualities; the ‘primordial
properties; the ‘substances; the ‘seasons; the
*Vedanga\ the ‘faces of Karttikeya (=
‘Kumara); the ‘sensations; the flavours; etc.
See Shad and Six.
Number Seven. Concept often directly or
symbolically related to: the seven Buddhas
( *saptabuddha)\ the seven ‘planets; the “seven
paces” ( *saptapddi)\ the “seven ‘Jewels”
(saptagraha); the “seven ‘sages"; the *Rishi;
“‘Purification”; the ‘horses; the “seven ‘divine
mothers"; the “seven rivers” ( *saptasindhava);
the seven ‘horses of Surya; the ‘island-
continents; the ‘seas; the ‘oceans; the “worlds”;
the seven ‘inferior worlds; the seven ‘hells; the
‘mountain; the seven ‘syllables; the seven
‘musical notes; the last of the seven Rishi
( *Atrf ); the seven ‘days of the ‘week; “That
which never moves”; ‘blue lotus flower; the
seven ‘winds; etc. See Sapta and Seven.
Number Eight. Concept often directly or
symbolically related to: the “eight parts
(i ashtasansa )”; the “eight ‘horizons”; the “eight
‘forms”; the “eight ‘limbs" of prostrating oneself
(*ashtanga); the ‘serpent; the ‘serpent of the
deep; the “eight liberations” (* ashtavimoksha)]
the ‘elephant; the eight “things which augur
well"; the “eight ‘elephants”; the ‘guardians of
the horizons; the ‘guardians of the points of the
compass; the "‘jewel”; the “shapes”; the eight
divinities ( *Vasu ); the spheres of existence; the
‘supernatural powers; the acts; the “body”
( *tanu ); etc. See Ashta, Serpent (Symbolism of
the) and Eight.
Number Nine. Concept often directly or
symbolically related to: the nine planets
( *navagraha)\ the “nine ‘Jewels”; the Feast of
nine days; the “nine precious stones"
( *navaratna ); the ‘Brahman; the ninth month
of the year *chaitradi\ the numeral of the place-
value system {*anka)\ the “Unborn”; the
“‘Inaccessible”; the “ ‘Divine Mother”; the
divinity ‘Durga; the ‘holes; the ‘orifices; the
‘radiance; the “*Cow”; etc. See Nava and Nine.
Number Ten. Concept often directly or
symbolically related to: the digits; the Feast of
the tenth day; the ten powers of a Buddha
( *dashabala)\ the ‘descents; the "ten ‘earths”;
the “ten paradises" (*dashabhumf); the “ten
stages” of the Buddha ( *dashabhumi ); the
‘horizons; the ‘heads of Ravana; the ten
‘major incarnations of Vishnu ( *dashdvatara)\
etc. See Dasha, Ten, and Durga.
Number Eleven. Concept often
symbolically associated with: the god ‘Rudra
(= ‘Shiva), who is often referred to by one of
his attributes instead of by name (“Supreme
Divinity”, “‘Great God”, “‘Lord of the
universe", “‘Lord of tears”, “Rumbling", Lord
of the animals”, “Violent”, etc.). See Ekadasha
and Eleven.
Number Twelve. Concept often symbolically
associated with: the “brilliant”; the sun; the
Sun-god; the “solar fire”; the ‘sun rays; the
“‘months”; the ‘zodiac; the ‘arms of Karttikeya;
the “‘wheel”; the ‘eyes of Senani; the sons of
Aditi; etc. See Dvadasha and Twelve.
Number Thirteen. Concept often
symbolically associated with the ‘god of carnal
love and of cosmic desire ( *Kama) and with the
‘universe formed by thirteen worlds. See
Trayodasha and Thirteen.
Number Fourteen. Concept often
symbolically associated with: the god ‘Indra, who
is often referred to by one of his attributes instead
of by name (“‘Courage”, “Strength”, “ ‘Power”,
“‘Powerful”, etc.); the “‘human” (in the sense of
the progenitor of the human race); the worlds; the
fourteen universes ( *bhuvana ); the “‘Jewels”; etc.
See Chaturdasha and Fourteen.
Number Fifteen. Concept often
symbolically associated with: “‘wing”; “‘day”;
etc. See Pahchadasha and Fifteen.
Number Sixteen. Concept often
symbolically associated with: the sixteen
‘kings of the legend of the *Mahdbhdrata\ the
“fingers of the moon” (kald). See Shaddasha
and Sixteen.
Number Twenty. Concept often directly or
symbolically associated with: the digits; the
‘nails; the ‘arms of Ravana; etc. See Vimshati
and Twenty.
Number Twenty-five. Concept often
symbolically associated with: the
‘fundamental principles; the “‘true natures”;
the ‘truths; the ‘realities; etc. See
Pahchavimshati and Twenty-five.
Number Twenty-seven. Concept often
directly related to: the “stars”; “‘lunar
mansions"; the ‘constellations; etc. See
Saptavimshati and Twenty-seven.
Number Thirty-two. Concept often
directly related to: the teeth. See Dvatrimshati
and Thirty-two.
Number Thirty-three. Concept symbolically
associated with: the “‘gods”; the “immortals".
See Trayastrimsha and Thirty-three.
Number Forty-nine. Concept often
symbolically associated with the ‘winds. See
Navachatvdrimshati and Forty-nine.
Number thousand. Concept often
interpreted in the sense of the multitude or the
incalculable, often associated with: the attributes
of many Hindu and Brahmanic divinities (the
“Thousand arms", the “‘Thousand rays” or the
“Thousand of the Brilliant" all denote the Sun-
god ‘Surya; the “Thousand names” denotes the
gods ‘Vishnu and ‘Shiva; the “Thousand eyes"
refers to the gods Vishnu and Indra; etc.); or
mythological figures (such as the demon Arjuna,
who is referred to by the name “Thousand arms
of Arjuna”). This number is also associated with:
the Mouths of the Ganges {*jdhnavivaktra)\ the
Arrows of Ravi (= Surya); ‘Ananta (the serpent
with a thousand heads); the ‘lotus with a
thousand petals; etc. See Sahasra and Thousand.
SYMBOLISM OF NUMBERS (Concept of a
large quantity). Here is an alphabetical list of
English words which have a connection with
Indian high numbers, and which can be found
as entries in this dictionary: ‘Arithmetical
speculations, ‘Astronomical speculations,
‘Billion, ‘Blue lotus, Conch, ‘Cosmic cycles,
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
500
‘Day of Brahma, ‘Diamond, ‘Dot, ‘Earth,
‘High numbers, ‘High numbers (Symbolic
meaning of), ‘Hundred billion, ‘Hundred
million, Hundred quadrillion, Hundred
quintillion, Hundred thousand, ‘Hundred
thousand million, ‘Hundred trillion,
Incalculable, ‘Indeterminate, ‘Infinity,
*Kalpa, *Kalpa (Arithmetical-cosmogonical
speculations on), ‘Lotus, ‘Million, ‘Moon,
‘Names of numbers, ‘Ocean, ‘Pink lotus,
Pink-white lotus, ‘Powers of ten, ‘Quadrillion,
‘Quintillion, ‘Serpent with a thousand heads,
‘Serpent of infinity and eternity. Sky, ‘Ten
billion, ‘Ten million, Ten quadrillion, Ten
quintillion, ‘Ten thousand, ‘Ten thousand
million, ‘Ten trillion, ‘Thousand (in the sense
of “multitude”), ‘Thousand, ‘Thousand
million, ‘Trillion, Unlimited, ‘White lotus,
‘Zero. See High numbers, which gives a list of
the principal corresponding Sanskrit words, as
well as all the necessary explanations.
SYMBOLISM OF NUMBERS (Concept of
Infinity). Here is an alphabetical list of English
words which are connected to the Indian idea of
infinity, and which can be found as entries in
this dictionary: ‘Arithmetical speculations,
Arithmetical-cosmogonical speculations, ‘Blue
lotus. Conch, ‘Cosmic cycles, ‘Cosmogonic
speculations, ‘Day of Brahma, ‘Diamond,
‘Dot, ‘Earth, ‘Eternity, ‘High numbers, ‘High
numbers (Symbolic meaning of), Incalculable,
Indefinite, ‘Infinitely big, ‘Infinity, ‘Infinity
(Indian concepts of), *Kalpa, *Kalpa
(Arithmetical-cosmogonical speculations on),
‘Lotus, ‘Moon, ‘Names of numbers, ‘Ocean,
‘Pink lotus. Pink-white lotus, ‘Serpent of
infinity and eternity, ‘Serpent (Symbolism of
the), ‘Serpent with a thousand heads, Sky,
‘Thousand, Unlimited, ‘White lotus.
SYMBOLISM OF NUMBERS (Concept of
Zero). Here is an alphabetical list of words
which are connected to Indian notions of
vacuity, the void and zero, and which appear as
entries in this dictionary: Sanskrit terms: * Abhra ,
* Akasha , ‘Ambara, * Atlanta , * Antariksha ,
*Bindu, "Gagana, * Jaladharapalha , *Kha,
*Khachheda, * Khahara , * Nabha , *Nabhas,
*Puma, *Randhra, *Shunya, *Shunya-bindu,
* Shunya-chakra , *Shunya-kha, *Shunya-samkhya,
*Shunyata, *Shunyavadin, *Vindu, *Vishnupada,
*Vyant, * Vyoman . English terms: ‘Absence,
‘Atmosphere, ‘Canopy of heaven, ‘Dot, ‘Ether,
‘Firmament, ‘Hole, ‘Indian atomism,
‘Infinitely small, ‘Infinity, ‘Insignificance,
‘Low numbers. Negligible, ‘Nihilism, ‘Non-
being, ‘Non-existence, ‘Non-present,
‘Non-substantiality, ‘Nothing, ‘Nothingness,
‘Numeral 0, Sky, Space, Uncreated, Unformed,
Unproduced, ‘Unthought, ‘Vacuity, ‘Void,
‘Zero, ‘Zero (Graeco-Latin concepts of), ‘Zero
(Indian concepts of), ‘Zero and Sanskrit poetry.
See also Durga.
SYMBOLISM OF WORDS WITH A
NUMERICAL VALUE. Here is an alphabetical
list of English words which correspond to the
associations of ideas contained in Sanskrit
numerical symbols, which appear as entries in
this dictionary (the list is not exhaustive):
‘Ablaze (= 3), ‘Ancestor (= 1), ‘Ankle (= 2),
‘Aphorism (= 3), ‘Arms (= 2), ‘Arms of
Arjuna (= 1,000), ‘Arms of Karttikeya (= 12),
‘Arms of Ravana (= 20), ‘Arms of Vishnu (=
4), ‘Arrow (= 5), ‘Arrows of Ravi (= 1,000),
‘Atmosphere (= 0). ‘Bearer (= 1), ‘Beginning
(= 1), ‘Birth (= 4), ‘Blind king (= 100), ‘Body
(= 1), ‘Body (= 6), ‘Body (= 8), ‘Brahma (= 1),
‘Breath (= 5), ‘Brilliant (= 12).
‘Canopy of heaven (= 0), ‘Cardinal point (= 4),
‘Characteristic (= 5), ‘City-fortress (= 3),
‘Colour (= 6), Condition of existence (= 4),
‘Constellation (= 27), ‘Contemplation (= 6),
‘Courage (= 14), ‘Cow (= 1), ‘Cow (= 9).
‘Day (= 15), ‘Day of the week (= 7),
‘Delectation (= 6), ‘Demonstration (= 6),
‘Descent (= 10), Digit (= 10), Digit (= 20),
‘Divine mother (= 7), ‘Dot (= 0), ‘Drop (= 1).
‘Earth (= 1), ‘Earth (= 9), ‘Element (= 5),
‘Elephant (= 8), Energy (= 3), ‘Ether (= 0),
‘Eye (= 2), ‘Eye (= 3), ‘Eye of Shukra (= 1),
‘Eyes (= 2), ‘Eyes of Senani (= 12), ‘Eyes of
Shiva (= 3), ‘Eyes of Indra (= 1,000).
‘Face (= 4), ‘Faces of Brahma (= 4), ‘Faces of
Karttikeya (= 6), ‘Faces of Kumara (= 6), ‘Faces
of Rudra (= 5), ‘Faculty (= 5), ‘Fire (= 3), ‘Fire
(= 12), ‘Firmament (= 0), ‘First father (= 1),
‘Form (= 1), ‘Form (= 3), ‘Form (= 8), ‘Four
cardinal points (= 4), ‘Fourth (= 4),
‘Fundamental principle (= 5), ‘Fundamental
principle (= 7), ‘Fundamental principle (= 25).
‘Ganges (= 1,000), ‘Gift of the Cow (= 5), ‘God
of carnal love (= 13), ‘God of cosmic desire (=
13), ‘God of sacrificial fires (= 3), ‘Gods (= 33),
‘Great Ancestor (= 1), ‘Great god (= 11), ‘Great
element (= 5), ‘Great sacrifice (= 5), ‘Great sin
(= 5), ‘Guardian of the horizons (= 8),
‘Guardian of the points of the compass (= 8).
‘Hand (= 2), He who has three heads (= 3),
‘Heads of Ravana (= 20), ‘Hell (= 7), ‘Hole (= 0),
‘Horizon (= 4, ‘Horizon (= 8), ‘Horizon (= 10),
‘Horse (= 7), ‘Horsemen (= 2), ‘Human (= 14).
‘Immense (= 1), ‘Inaccessible (= 9),
‘Incarnation (= 10), Indestructible (= 1),
‘Individual soul (= 1), ‘Indra (= 14), ‘Inferior
world (= 7), ‘Infinity (= 0), ‘Island-continent
(= 4), ‘Island-continent (= 7).
‘Jewel (= 8), ‘Jewel (= 5), ‘Jewel (= 9), ‘Jewel
(= 14).
‘King (= 16). Limb (= 6), ‘Lord of the universe
(= 11), ‘Luminous (= 1), ‘Lunar mansion (=
27).
‘Main observance (= 5), ‘Merit (= 6), ‘Merit
(= 3), ‘Month (= .12), ‘Moon (= 1), ‘Mountain
(= 7), ‘Mouths of Jahnavi (= 1,000), ‘Musical
mode (= 6), ‘Musical note (= 7).
‘Nail (= 20), ‘Numeral (= 9).
‘Ocean (= 4), ‘Ocean (= 7), ‘Opinion (= 6),
‘Orifice (= 9).
‘Pair (= 2), ‘Paradise (= 13), ‘Paradise (= 14),
‘Part (= 6), ‘Passion, ‘Phenomenal world (=
3), ‘Philosophical point of view (= 6), ‘Planet
(= 9), ‘Point (= 3), ‘Position (= 4), ‘Power (=
14), ‘Powerful (= 14), ‘Precept (= 6),
Primordial couple (= 2), ‘Primordial principle
(= 1), ‘Primordial property (= 3), ‘Primordial
property (= 6), ‘Progenitor of the human race
(= 14), ‘Purification (= 7).
‘Quality (= 3), ‘Quality (= 6).
‘Rabbit (= 1), Rabbit figure (= 1), ‘Radiance
(= 9), ‘Reality (= 5), ‘Reality (= 7), ‘Reality (=
25), ‘Rudra-Shiva (= 11), Rumbler (= 11).
‘Sage (= 7), ‘Season (= 6), ‘Self (= 1),
‘Sensation (= 6), ‘Sense (= 5), ‘Sense organs
(= 5), ‘Serpent (= 8), ‘Serpent of the deep
(= 8), ‘Serpent with a thousand heads
(= 1,000), Sky (= 0), Son (= 5), Sons of Adit!
(= 12), ‘Sons of Pandu (= 5), Sour milk (= 1),
Space (= 0), Spirit of the waters (= 3), Star
(= 27), State (= 3), State of the manifestation
(= 5), Strength (= 14), ‘Substance (= 6),
‘Sun (= 12), ‘Sun (= 1,000), ‘Sun-god (= 12),
‘Sun-Moon (= 2), ‘Sun rays (= 12),
‘Supernatural power (= 8), Supreme Divinity
(= 11), Supreme soul (= 1), ‘Syllable (= 7),
‘Symmetrical organs (= 2).
‘Taste (= 6), ‘Terrestrial world (= 1), That
which augurs well (= 8), That which must be
done (= 5), That which belongs to all humans
(= 3), ‘Thousand (= 12), ‘Thousand rays (=
12), ‘Three aims (= 3), ‘Three bodies (- 3),
‘Three city-fortresses (= 3), ‘Three classes of
beings (= 3), ‘Three eyes (= 3), ‘Three forms (=
3), ‘Three fundamental properties (= 3),
‘Three heads (= 3), ‘Three jewels (= 3), ‘Three
letters (= 3), ‘Three sacred syllables (= 3),
‘Three states (= 3), ‘Three times (= 3), ‘Three
universes (= 3), ‘Three worlds (= 3), ‘Time (=
3), ‘Tone (= 49), Tooth (= 32), ‘Triple science
(= 3), ‘True nature (= 7), ‘True nature (= 25),
‘Truth (= 5), ‘Truth (= 7), ‘Truth (= 25), Twice
born (= 2), Twin gods (= 2), Twins, pairs or
couples (= 2).
‘Ultimate reality (= 1), ‘Universe (= 13).
‘Veda (= 3), ‘Veda (= 4), ‘Vedanga (= 6),
‘Violent (= 11), ‘Vision (= 6), ‘Voice (= 3),
‘Void (= 0), ‘Vulva (= 4).
Water (= 4), ‘Week (= 7), ‘Wheel (= 12),
‘Wind (= 5), ‘Wind (= 7), ‘Wind (= 49),
‘Wing (= 2), ‘Wing = 15), ‘Word (= 3),
‘World (= 3), ‘World (= 7), ‘World (= 14).
*Yuga (= 2), *Yuga (= 4).
‘Zenith (= 0), ‘Zodiac (= 12).
See Symbols, Numerical symbols, One, Two,
Three, Four, Five, Six, Seven, Eight, Nine,
Ten, Eleven, . . . Zero and Names of numbers.
SYMBOLISATION OF THE CONCEPT OF
INFINITY. See Infinity, Infinity (Indian
concepts of), Infinity (Mythological
representation of) and Serpent (Symbolism
of the).
SYMBOLISATION OF THE CONCEPT OF
ZERO. See Zero, Dot and Circle.
SYMBOLS. In the Brahmanic religion,
and other religions of the Indian
sub-continent, symbols have always been
very important. They are either visible and
understood by everyone and resume a
number of concepts which are difficult to
write down (stupa, for example), or they
are invisible because they have a sense which
the profane cannot see (such as the bija, the
yantra, the *mudra, etc.).
The symbols are represented by numerous
categories of beings (such as animals), objects
or even plants. As with Mahayana Buddhism,
each divinity of Brahmanism possesses a
carrier-animal which symbolises the god
himself: Garuda for ‘Vishnu, Nandin for
‘Shiva, etc.: they also have a bija (a letter-
symbol for the corresponding sound to invoke
them), * mantras (or sacred formulas), yantras
(geometrical diagrams with symbolic meaning)
and various “signs" or distinctive marks
which allow the faithful to identify the
representations of the gods immediately.
The combination of signs is also symbolic,
and different from a sole, isolated symbol (like
‘vajra and ganthd). Some symbols are raw
materials like the ' linga of Shiva or the
501
SYMMETRICAL ORGANS
shalagrama of Vishnu; others are constructions
(such as stupas, chaityas, temples and various
sculptures).
As for the plant kingdom, many trees
(pipal, banyan, etc.) plants (tulast) and seeds
(, rudraksha ) constitute symbols to Hindus,
Buddhists and followers of the Jaina religion. In
India, all things are potentially symbolic, not
only in philosophy and religion, but also in
literature, art and music. The most significant
symbols are the attributes of the divinities. The
Trident ( •trishula) belongs to Shiva, but like the
serpent {"naga) or the elephant, it has other
meanings. See Serpent (Symbolism of the).
The club ( danda , gada) is the sign of the
guardians of the gate ( dvarapdla ), but also a
symbol of solar energy. The lance ( shakti ) and
other weapons: dagger (kshurikd), axe
( parashu ), bow and arrow ( dhanus , *bana),
shield (khetaka), sword ( khadga ), etc., are used
to show the power of divinities.
*Lotus flowers are most important to
Buddhism, but are also highly symbolic of the
pure nature of Hindu divinities.
Other very common symbolic attributes
include: musical instruments (the vina of
Sarasvati, the damaru of Shiva-Nataraja); the
conch ( *shankhd)\ the bell (ganthd ); everyday
objects (the mirror of Maya, darpana)', the cord
I pasha ) that joins the soul to matter; the book
( pushtaka ) which represents all the *Vedas\ etc.
The sun ( chakra ) and the moon ( kulika ), the
symbols of constellations, all have specific
meanings which are either obvious or hidden
(esoteric or tantric). There is a lot of symbolism
connected to the human body: nudity suggests
detachment from contingencies; colour of skin
means anger and fury or peace and joy. Hair (in
a bun) symbolises Yogin; dishevelled hair
represents the mobility of Maya; frizzy, untidy
hair means rage.
The number of arms and legs that a
divinity possesses is also highly symbolic: the
more arms, the more active the god is. When a
god only has two arms, this represents
angelic", peaceful qualities. If a god has no
attributes whatsoever, this represents
neutrality, like the *Brahman. Jewels and
ornaments also have precise meanings, which
ran vary according to era, beliefs and
philosophies. [The information in this entry is
taken from Frederic, Dictionnaire de la
civilisation indienne (1987)].
SYMMETRICAL ORGANS. As symbols for
the number two. See Baku, Gulpha, Nayana,
Netra, Paksha and Two.
T
TAITTIRtYA SAMHITA. Text derived from
the Yajurvcda “black”, which figures amongst
the texts of Vedic literature. It is the result of
oral transmission since ancient times, and did
not appear in its definitive form until the
beginning of the Common Era. See Veda.
Here is a list of the principal names of
numbers mentioned in the text: *Eka (= 1),
* Dash a (= 10), *Sata (= 10 2 ), *Sahasra (= 10 3 ),
*Ayuta (= 10 1 ), *Niyuta (= 10 5 ), *Prayuta (=
10 6 ), *Arbuda (= 10 7 ), *Nyarbuda (= 10 8 ),
*Samudra (= 10 9 ), *Madhya (= 10 10 ), *Anta (=
10 u ), *Parardha (=10 12 ). [See Names of
numbers and High numbers. See: *Taittiriya
Samhitd, IV, 40. 11. 4; VII, 2. 20. 1; Datta and
Singh (1938), p. 9; Weber, in: JSO, XV, p. 132-
40],
TAKARI NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Gupta and Sharada
numerals. Currently used in Kashmir
alongside the so-called “Hindi" numerals of
eastern Arabs. Also called Tankri numerals.
The symbols correspond to a mathematical
system that has place values and a zero
(shaped like a small circle). See Indian written
numeral systems (Classification of). See Fig.
24.13, 52 and 24.61 to 69.
TAKSHAN. [S]. Value = 8. “Serpent”. See
Naga, Eight and Serpent (Symbolism of the).
TAKSHASA. Name of the king of the *naga.
See Serpent (Symbolism).
TALLAKSHANA. Name given to the number
ten to the power fifty-three. According to the
legend of Buddha, this number is the highest
in the first of the ten numerations of high
numbers defined by the Buddha child during
a contest in which he competed against the
great mathematician Arjuna. Tallakshana
contains the word lakshana, which literally
means “character”, “mark", “distinguishing
feature”. In Buddhism, this word often
expresses the “hundred and eight distinctive
signs of perfection" which distinguish a
Buddha from other human beings (108 being
considered a magic and sacred number which
symbolises perfection). See Names of
numbers and High numbers (Symbolic
meaning of).
Source: *I.alitavistara Sutra (before 308 CE).
TAMIL NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Pallava, Chalukya,
Ganga, Valabhi, Bhattiprolu and Grantha
numerals. Currently in use by the Dravidian
population of the state of Tamil nadu (Southeast
India). The symbols correspond to a
mathematical system that is not based on place
values and therefore does not possess a zero. For
contemporary symbols, see Fig. 24.17; for
historical symbols, see Fig 24, 49. See Indian
written numeral systems (Classification of).
See also Fig. 24.52 and 24.61 to 69.
TANA. [SI- Value = 49. “Tone”. In Indian
music, this refers to the combinations of seven
octaves of seven notes.
TANKRi NUMERALS. See Takari Numerals.
TANU. [SI. Value = 1. “Body”. This symbolism
comes from astrology, where “house I" is that
which refers to the person, and by extension
the body ( tanu ) of the person, whose
horoscope is being prepared. See One.
TANU. [SJ. Value = 8. “Body”. This is an
allusion to the *dshtanga, the “eight limbs” of
the human body that are involved in the act of
prostrating oneself. See Ashtanga and Eight.
TAPANA. [SJ. Value = 3. "Fire”. See Agni, Three
and Fire.
TAPANA. [SJ. Value = 12. The word means
“fire”, but here it is taken in the sense of “solar
fire" and thus of the Sun-god *Surya. See Surya
and Twelve.
TASTE. [SJ. Value = 6. See Rasa and Six.
TATTVA. [SJ. Value = 5. “Reality, truth, true
nature, fundamental principle”. Allusion to the
five “fundamental principles” identified by
Indian philosophers and considered to be the
basis for thought. See Five.
TATTVA. [SJ. Value = 7. “Reality, truth, true
nature, fundamental principle”. Allusion to the
seven “fundamental principles” identified by
Jaina philosophy and considered to be the basis
of the system for thought. See Seven. This
symbol is very rarely used to represent this
value, except for in the Ganitasarasamgraha by
the Jaina mathematician *Mahaviracharya [see
Datta and Singh (1938), p. 56].
TATTVA. [SJ. Value = 25. “Reality, truth, true
nature, fundamental principle”. Allusion to the
twenty-five “fundamental principles" identified
by the orthodox philosophy of *Sdmkhya:
avyakta (the “non -manifest”); buddhi
(intelligence); ahamkara (Ego, the
consciousness of the Me); the tanmatra (or
“original substances”, five subtle elements from
which the basic elements are said to derive); the
mahabhuta (the five elements of the
manifestation); the buddhindriya (the five
"sense organs”); the karmendriya (the five
organs of activity, namely: the tongue, the
hands, the legs, the organs of evacuation, and
the reproductive organs); manas (the “Ability
for reflection”; and purusha (the Self, the
Absolute, pure consciousness) See Twenty-five.
TELINGA NUMERALS. See Telugu numerals.
TELUGU NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Pallava, Chalukya,
Ganga, Valabhi and Bhattiprolu numerals.
Currently in use amongst the Dravidian
population of Andhra Pradesh (formerly
Telingana). Also called Telinga numerals. The
symbols correspond to a mathematical system
that has place values and a zero (shaped like a
small circle). For contemporary symbols, see Fig.
24.20; for historical symbols, see Fig. 24, 47. See:
Indian written numeral systems (Classification
of). See Fig. 24.13, 52 and 24.61 to 69.
TEN. Ordinary name in Sanskrit: »dasha. List
of corresponding numerical symbols: *Anguli,
*Asha, *Avatara, *Dish, *Dishd, *Kakubh,
Karman, Lakdra, Pankti, * Ravanshiras.
These terms translate or designate
symbolically: 1. Descendances and incarnations
(Avatara); 2. Fingers ( Anguli ); 3. Horizons (Dish,
Disha, Asha, Kakubh ); 4. The heads of Ravana
(Ravanshiras). See Numerical symbols.
TEN BILLION ( = ten to power thirteen; in US
expressed as “ten trillion"). See Ananta, Kankara,
Khamba, Makakharva, Nikharva, Shankha,
Shangku. See also Names of numbers.
TEN MILLION ( = ten to power seven). See
Arbuda. Koti. See also Names of numbers.
TEN THOUSAND ( = ten to power four). See
Ayuta, Dashashasra. See also Names of
numbers.
TEN THOUSAND MILLION ( = ten to power
ten; in US expressed as “ten billion”). See
Arbuda, Kharva, Madhya, Samudra. See also
Names of numbers.
TEN TRILLION (in British sense of ten to
power nineteen; otherwise called “ten
quadrillion”). See Attata, Mahdshankha
Vivaha. See also Names of numbers.
TERRESTRIAL WORLD. [SJ. Value = 1. See
One, Prithivi.
THAI (THAI-KHMER) NUMERALS. Symbols
derived from *Brahmi numerals and
influenced by Shunga, Shaka, Kushana,
Andhra, Pallava, Chalukya, Ganga, Valabhi,
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
502
“Pali” and Vatteluttu numerals. Currently used
in Thailand, Laos and Cambodia (Kampuchea).
The symbols correspond to a mathematical
system that has place values and a zero (shaped
like a small circle). See Indian written numeral
systems (Classification of). See Fig. 24.24, 52
and 24.61 to 69.
THAKURI (Calendar). Calendar beginning in
the year 595 CE. To find the corresponding
date in the Common Era, add 595 to a date
expressed in the Thakuri calendar. Formerly
used in Nepal. See Indian calendars.
THIRTEEN. Ordinary Sanskrit name:
*trayodasha. Here is a list of the corresponding
numerical symbols: Aghosha, Atijagati, *Kama,
Manmatha, *Vishva, *Vishvadeva.
These words have the following translation
or symbolic meaning: 1. The god of carnal love
and of cosmic desire (Kama). 2. The universe
comprised of thirteen worlds ( Vishva ,
Vishvadeva).
See Numerical symbols.
THIRTY. Ordinary Sanskrit name: *trimshat.
THIRTY-TWO. Ordinary Sanskrit name:
*dvatrimshati. The corresponding numerical
symbols are: *Danta and *Rada. These words
both mean “teeth”. See Numerical symbols.
THIRTY-THREE. Ordinary Sanskrit word:
*trdyastrimsha . The corresponding numerical
symbols are: * Amara, *Deva, *Sura, Tridasha.
These words have the following meaning: 1.
The “gods" ( Amara , Deva, Sura ) 2. The
“immortals", in allusion to the gods (Amara).
See Numerical symbols.
THOUSAND. Ordinary Sanskrit name:
*Sahasra. Corresponding numerical symbols:
*Arjunakara, *Indradrishti, *Jdhnavivaktra,
*Ravibana, * Sheshashirsha.
These terms name or refer to: 1. The mouth
of the Ganges or Jahnavi (Jdhnavivaktra ). 2. The
arms of Arjuna (Arjunakara). 3. The arrows of
Ravi (Ravibana). 4. The thousand-headed
serpent ( Sheshashirsha ). 5. The eyes of Indra
(Indradrishti). See Numerical Symbols.
THOUSAND. In the sense of “many, a
multitude of. . .”. See Jahnavivakta. See also
High Numbers (Symbolic Meaning of).
THOUSAND. In the sense of infinity and
eternity. See Sheshashirsha.
THOUSAND. [S]. Value = 12. See
Sahasramshu, Twelve.
THOUSAND MILLION. ( = ten to power
nine, known in US as “one billion”). See Abja,
Ayuta, Nahut, Nikharva, Padma, Samudra,
Saroja, Shatakoti, Vddava, Vrinda. See also
Names of numbers.
THOUSAND RAYS. [SJ. Value = 12. See
Sahasramshu. Twelve.
THREE. The ordinary Sanskrit names for this
number are: *traya, *trai and *tri. Here is a list
of corresponding word-symbols:
*Agni, *Anala, *Aptya, *Bhuvana, *Dahana,
Dosha, Gangamarga, *Guna, *Haranayana,
*Haranetra, *Hotri, *Hutashana, *Ishadrish,
*Jagat, *Jvalana, *Kala, *Krishanu, *Loka,
*Murti, Nadi, *Netra, *Pavaka, *Pinakanayana,
*Pura, *Rama, *Ratna, Sahodara, *Shakti,
*Shankarakshi, *Shikhin, *Shula, *Tapana,
*Trailokya, *Trayi, Trigata, *Triguna, * Trijagat,
* Trikala, *Trikaya, *Triloka, *Trimurti,
*Trinetra, * Tripura, * Tripura, *Triratna,
* Trishiras, *Trivarga, *Trivarna, *Tryambaka,
*Udarchis, *Vachana, *Vahni, *Vaishvanara,
*Veda, Vishtapa.
These words have the following translation
or symbolic meaning: 1. The god of fire ( Agni ).
2. “Fire”, in allusion to the god of sacrificial fire
(Agni, Anala, Dahana, Hotri, Hutashana,
Jvalana, Krishanu, Pavaka, Shikhin, Tapana,
Udarchis, Vahni, Vaishvanara). 3. “That which
belongs to all humans” (Vaishvanara). 4. Ablaze
(Shikhin). 5. The worlds, the universe (Bhuvana,
Loka). 6. The three worlds (Triloka). 7. The
phenomenal worlds (Jagat). 8. The three
phenomenal world (Trijagat). 9. The “three
letters”, in allusion to the three sacred syllables
(Trivarna). 10. The “aphorism” (Vdchana). 11.
Feminine energies (Shakti). 12. The City-
Fortresses (Pura). 13. The Three City, Fortresses
(Tripura). 14. The “States”, in allusion to the
States of consciousness (Purd). 15. The Three
states of consciousness (Tripura). 16. The
“forms” (Murti). 17. The three forms (Trimurti).
18. The Jewels (Ratna). 19. The three Jewels
(Triratna). 20. The “qualities”, the “primordial
properties” (Guna). 21. The “three primordial
properties" (Triguna). 22. The Eye, in allusion to
the “three eyes” (Netra). 23. The three eyes
(Trinetra, Tryambaka). 24. The points (Shula).
25. Time, in allusion to the “three times” (Kala).
26. The three times (Trikala). 27. The triple
science (Trayf). 28. The three aims (Trivarga).
29. The three classes of beings (Trailokya). 30.
The three bodies (Trikaya). 31. The three states
(Tripura). 32. The spirit of the waters (Aptya).
33. The eyes of Shiva (Haranetra), 34. The god
Shiva (Pinakanayana). 35. “The one with three
heads” (Trishiras). 36. The three Ramas (Rama).
See Numerical symbols.
THREE AIMS. [SJ. Value = 3. See Trivarga and
Three.
THREE BODIES. [S]. Value = 3. See Trikaya
and Three.
THREE CITY-FORTRESSES. IS). Value = 3.
See Tripura and Three.
THREE CLASSES OF BEINGS. [S]. Value = 3.
See Trailokya and Three.
THREE EYES. [S]. Value = 3. See Tryambaka
and Three.
THREE FORMS. [SJ. Value = 3. See Trimurti
and Three.
THREE HEADS. [S]. Value = 3. See Trishiras
and Three.
THREE JEWELS. [S]. Value = 3. See Triratna
and Three.
THREE LETTERS. [S]. Value = 3. See Trivarna
and Three.
THREE PRIMORDIAL PROPERTIES. [S).
Value = 3. See Triguna and Three.
THREE SACRED SYLLABLES. [SJ. Value = 3.
See Trivarna and AUM.
THREE STATES. [SJ. Value = 3. See Tripura
and Three.
THREE TIMES. [SJ. Value = 3. See Trikala and
Three.
THREE UNIVERSES [SI. Value = 3. See Jagat,
Loka, Trijagat and Three.
THREE WORLDS. [SJ. Value = 3. See Triloka
and Three.
TIBETAN NUMERALS. Symbols derived from
*Brahmi numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Gupta, Nagari and
Siddham numerals. Used in areas of Tibet since
the eleventh century CE. The symbols
correspond to a mathematical system that has
place values and a zero (shaped like a small
circle). However, it was not always thus: many
Tibetan manuscripts show that a structure
identical to the archaic Brahmi system was
used in former times. See Indian written
numeral systems (Classification of). See Fig.
24.16, 52 and 24.61 to 69.
TIL AKA. “Sesame”. Name given to the dot that
Hindus stick to their foreheads whcih represents
the third eye of *Shiva, the eye of knowledge. See
Poetry, zero and positional numeration.
TIME. [SJ. Value = 3. See Kala, Trikala and Three.
TITHI. Unit of time used in Babylonian tablets
which corresponds to a thirtieth of a synodic
revolution of the Moon. This length of time is
approximately the same as a day or nychthemer.
See Indian astronomy (History of).
TITHI. [SJ. Value = “Day”. 15. Allusion to the
15 days of each *paksha of the month. See Tithi
and Fifteen.
This symbol is notably found in
*Varahamihira: PnSi, VIII, line 4; Dvivedi and
Thibaut (1889); Neugebauer and Pingree
(1970-71).
TITILAMBHA. Name given to the number ten
to the power twenty-seven. See Names of
numbers and High numbers. Source:
*Lalitavistara Sutra (before 308 CE).
TONE. [S]. Value = 49. See Tana.
TOTAL. [Arithmetic], See Sarvadhana.
TRAI. (TRAYA, TRI). Ordinary Sanskrit terms
for the number three which form part of
several words which are directly related to the
number in question.
Examples: *Trailokya, *Trairashika, *Trayi,
*Triambaka, *Tribhuvana, *Tribhuvaneshvara,
*Trichivara, * Triguna, *Trijagat, *Trikala,
*Trikalajndna, *Trikandi, *Trikaya, *Triloka,
*Trimurti, *Trinetra, *Tripitaka, *Tripura,
*Tripura, *Tripurasundari, *Triratna, * Trishiras ,
*Trishula, *Trivamsha, *Trivarga, *Trivarna,
*Triveni, *Trividyd, * Tryambaka .
For words which are symbolically
associated with this number, see Three and
Symbolism of numbers.
TRAILOKYA. [S]. Value = 3. “Three classes”.
This name denotes the three classes of beings
envisaged by Hindu and Buddhist
philosophies: the kamadhatu, beings evolving
in desire; the rupadhatu, those of the world of
forms; and the arupadhatu, those of the world
of the formless. See Trai and Three.
TRAIRASHIKA. [Arithmetic]. Sanskrit name
for the Rule of Three. See Trai.
TRAYA. Ordinary Sanskrit name for the
number three. See Trai.
TRAYASTRIMSHA. Ordinary Sanskrit name for
the number thirty-three. For words which are
symbolically associated with this number, see
Thirty-three, Deva and Symbolism of numbers.
TRAYI. [SJ. Value = 3. “Triple science”.
Allusion to the Samhitd (Rigveda, Yajurveda,
Samaveda), who are the three first * Vedas. See
Trai, Veda and Three.
TRAYODASHA. Ordinary Sanskrit name for
the number thirteen. For words which are
symbolically associated with this number, see
Thirteen and Symbolism of numbers.
TRETAYUGA. Name of the second of the four
cosmic eras which make up a *mahdyuga. This
503
TRI
cycle, which is said to be worth 1,296,000 human
years, is regarded as the age during which human
beings would live no more than three quarters of
their life. See Mahdyuga and Yuga.
TRI. Ordinary Sanskrit word for the number
three. See Trai.
TRI AM B AKA. “With three eyes”. See
Tryambaka.
TRIBHUVANA. Name of the "three worlds” of
Hindu cosmogony: the skies ( svarga ), the earth
(*bhumi) and the hells ( *pdtala ). See Trai.
TRI BHUVANESH VARA. “Lord of the three
worlds”. One of the titles attributed to *Shiva,
♦Vishnu and ‘Krishna. See Trai.
TRICHiVARA. “Three garments”. Term
denoting the ritual costume comprising the
loincloth, sash and robe worn by Buddhist
monks of the schools of the South (Hinayana,
Theravada). See Trai.
TRIGUNA. [S]. Value = 3. “Three primordial
properties”, “three primary forces". Symbolism
which corresponds to the representation of the
group Vishnu-Sattva, Brahma-Rajas and
Rudra-Tamas, this group being thus composed
of the energies which personify the three main
divinities of the Brahmanic pantheon. See
Guna, Brahma, Vishnu, Shiva and Three.
TRIJAGAT. [S]. Value = 3. "Three universes”.
See Jagat, Triloka and Three.
TRIKALA. IS]. Value = 3. "Three times".
Allusion to the three divisions of time: the past,
the present and the future. See Kala and Three.
TRIKALAJNANA. From *trikala, “three times",
“three eras”, and from jhdna, ‘knowledge".
Name denoting the magic and occult power
which is given to the *Siddhi to enable them to
know the past, the present and the future all at
once. See Kdla, Trikala and Trai.
TRIKANDI. “Three chapters”. This name is
sometimes given to the Vakyapadiya of
Bhartrihari, famous text of “grammatical
philosophy" divided into three kdnda or
"chapters”. See Trai.
TRIKAYA. IS]. Value = 3. “Three bodies".
Allusion to the three bodies that a Buddha may
assume simultaneously: the “body of the Law'”
(dharmakaya), the “body of enjoyment”
{sambhogakaya) and the “body of magical
creation or transformation” ( nirmdnakdya ).
See Three.
TRILLION. See Akshiti, Antya, Madhya,
Mahapadma, Viskhamba, Vivara and Names
of numbers.
TRILOKA. [S]. Value = 3. “Three worlds”. In
allusion to the worlds of Hindu cosmogony: the
Skies {svarga), the earth ( *bhumi ) and the hells
( *pdtala ). See Three.
TRIMSHAT. Ordinary Sanskrit name for the
number thirty.
TRIMURTI. (SI. Value = 3. “Three forms”. See
Murti and Three.
TRINETRA. IS]. Value = 3. “Three eyes". See
Haranetra and Three.
TRIPITAKA. “Three baskets”. Term denoting
the Buddhist Law written in Sanskrit which
constitutes the sacred Scriptures of this
religion. The allusion is to the three different
baskets into which the three principal
compilers placed the three fundamental
Buddhist texts: the vinayapitaka, which deals
with monastic discipline; the sutrapitaka and
the abhidharmapitaka which deals with
Buddha’s teaching [see K. Friedrichs, etc,
(1989)]. See Trai.
TRIPLE SCIENCE. [SJ. Value = 3. See Trayi
and Three.
TRIPURA. [S]. Value = 3. Literally: “Three City-
fortress”. Name of a triple fortress-town (or
triple rampart) which was built by the *Asura
and destroyed by Shiva. See Pura and Three.
TRIPURA. [S]. Value = 3. Literally: “three
states”. Name which collectively denotes the
three states of consciousness of Hinduism. See
Pura and Three.
TRIPURASUNDARI. “Beauty of the three
cities”. One of the names given to ‘Parvati, the
“mountain dweller”, daughter of Himalaya,
sister of * Vishnu and wife of *Shiva. See Trai.
TRIRATNA. [S]. Value = 3. “Three jewels”. See
Ratna and Three.
TRISHATIKA. See Shridhardcharya.
TRISHIRAS. IS]. Value = 3. “He w-ho has three
heads". This is the name of the demon with
three heads, younger brother of ‘Ravana, who.
according to the legend of *Rdmdyana, was
killed by ‘Rama. See Ravana and Three.
TRISHULA. “Three points”. Name of ‘Shiva’s
Trident. See Shula and Trai.
TRIVAMSHA. Name which collectively
denotes the three principal castes of
Brahmanism (namely: the Brahmans, the
kshalriya and the vaishya). See Trai.
TRIVARGA. (S]. Value = 3. “Three aims". This
is an allusion to the three objectives of human
existence according to Hindu philosophy,
namely: material wealth {artha), love w'ith
desire ( *kama ) and duty {*dharma). See Three.
TRIVARNA. IS]. Value = 3. “Three letters”.
This refers to the letters A, U and M of the
Indian alphabet, which spell AUM, the sacred
Syllable of the Hindus, which means something
approximating “I bow”. This represents all of
the following at once: the divine Word in an
audible form; the fullblown ‘Brahman; the Fire
of the Sun; the Unity; the Cosmos; the
Immensity of the Universe; the past; the
present; the future; as well as all Knowledge.
According to Hindu religion, AUM contains
the very essence of all the sounds that have
been, that are, and that will be made, and
within it is reunited the three great powers of
the three great divinities of the Brahmanic
pantheon (see Frederic (1987)]. See AUM,
Akshara, Mysticism of letters, Trai and Three.
TRIVENI. “Three rivers”. Name sometimes
given to the town of Prayaga (now Allahabad)
where the following three rivers are said to
meet: the Ganges, the Yamuna and the
mythical Sarasvati. See Trai.
TRIVIDYA. Name given to the “three axioms”
of Buddhist philosophy: anitya, the
impermanence of all things; dukha, pain,
suffering; and andtma, the non-reality of
existence. See Trai.
TRIVIMSHATI. Ordinary Sanskrit name for
the number twenty-three. For words which are
symbolically connected with this number, see
Twenty-three and Symbolism of numbers.
TRUE NATURE. [SJ. Value = 5. See Tattva
and Five.
TRUE NATURE. [S]. Value = 7. See Tattva
and Seven.
TRUE NATURE. (Si. Value = 25. See Tattva
and Twenty-five.
TRUTH. (SJ. Value = 5. See Tattva and Five.
TRUTH. [S]. Value = 7. See Tattva and Seven.
TRUTH. [S]. Value = 25. See Tattva and
Twenty-five.
TRYAKSHAMUKHA. (SJ. Value = 5.
Synonymous with *Rudrasya, “faces of
*Rudra”. See Five.
TRYAMBAKA. IS]. Value = 3. “With three
eyes”, “with three sisters”. Epithet given to
many Hindu divinities, especially Shiva . See
Haranetra, Traya and Three.
TURAGA. IS]. Value = 7. “Horse”. See Ashva
and Seven.
TURANGAMA. [SJ. Value = 7. “Horse". See
Ashva and Seven.
TURIYA. [S]. Value = 4. “Fourth”. Epithet
occasionally given to the Brahman who
transcends the three states of consciousness.
See Tripura and Four.
TWELVE. Ordinary Sanskrit name: *dvddasha.
TWENTY. Ordinary Sanskrit name: *vimshati.
Here is a list of corresponding numerical
symbols: *Angu!i, *Kriti, *Nakha, *Ravanabhuja.
These words express: 1. The arms of Ravana
{Ravanabhuja). 2. The fingers {Anguli). 3. The
nails ( Nakha ). 4. An element of Indian
metrication (Kriti). See Numerical symbols.
TWENTY-ONE. Ordinary Sanskrit name:
*ekavimshati. Corresponding numerical
symbols: *Prakriti, Svaga (“heaven”), Vtkriti.
TWENTY-TWO. Ordinary Sanskrit name:
*dva vimshati. Corresponding numerical
symbols: *Akriti,]ati (“Caste"), Kritin.
TWENTY-THREE. Ordinary Sanskrit name:
*trayavimshati (or trivimshati). Corresponding
numerical symbol: *Vikriti.
TWENTY-FOUR. Ordinary Sanskrit name:
*chaturvimshati. Corresponding numerical
symbols: Arhat, *Gdyatri, Jina, Siddha.
TWENTY-FIVE. Ordinary Sanskrit name:
*pahchavimshati. Corresponding numerical
symbol: Tattva. This word expresses: 1.
The fundamental principles. 2. The “true
natures”. 3. The realities. 4. The truths.
TWENTY-SIX. Ordinary Sanskrit name:
*shadvimshati. Corresponding numerical
symbol: *Utkriti.
TWENTY-SEVEN. Ordinary Sanskrit name:
*saptavimshati. Corresponding numerical
symbols: *Bha, *Uda, * Nakshatra. These words
express or symbolise: 1. The “stars" ( Bha , Vda).
2. The “lunar mansions" ( Nakshatra ). 3. The
constellations ( Nakshatra ).
TWO. Ordinary Sanskrit names: *dva, dvc, dvi.
Corresponding numerical symbols: Akshi,
Ambaka, * Ash via, *Ashvina, *Ashivinau, Ay ana,
*Bdhu, *Chakshus, *Dasra, *Drishti, *Dvandva,
*Dvaya, *Dvija, Grahana, *Gulpha, Ishana,
Janghd, Jdnu, *Kara, Kama, Kucha, Kutumba,
*Lochana, Nadikuld, *\ 'dsatya, Nay a, * Nay ana,
*Netra, Oththa, *Paksha, Rdmananddana,
Ravichandra, Vishuvat, *Yama, *Yamala, *Yamau,
*Yuga, *Yugala, * Yugrna.
These terms symbolically refer to or
designate: 1. Twins, pairs or couples (Ashvin,
Ashvina, Ashvinau, Dasra, Dvandva, Dvaya,
Dvija, N dsatya, Ravichandra, Yam a. Yam ala,
Yugala, Yugrna). 2. Symmetrical organs {Bahu,
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
504
Gulpha, Kara, Nayana, Netra, Paksha). 3.
Wings (Paksha). 4. Arms (Bahu). 5. The
Horsemen (Ashvin, Ashvina, Ashvinau). 6.
Ankles (Gulpha). 7. The conception of the
world, contemplation, revelation, theory
( Drishti ). 8. The primordial couple (Yama). 9.
The epithet "twice born" (Dvija). 10. The twin
gods (Ashvin, Basra, Nasatya). 11. The hand
(Kara). 12. The pair (Dvaya). 13. The Sun-
Moon couple (Ravichandra). 14. The eye
(Netra, Chakshus). 15. Eyes (Lochana). 16.
Vision (Drishti). 17. The third age of a
mahayuga (Yuga). See Numerical symbols.
u
UCHCHAISHRAVAS. [S], Value = 1. This is the
name of a wonderful white horse which,
according to Brahmani and Hindu mythology,
came from the “churning of the sea of milk”
and which Indra appropriated. He is
considered to be the ancestor of alt horses, thus
the symbolism in question. See One.
UDA. [S]. Value = 27. “Star”. This is an allusion
to the twenty-seven -nakshatra. See Nakshatra
and Twenty-seven.
UDADHI. [SI. Value = 4. “Ocean”. See Sagara,
Four and Ocean.
UDARCHIS. [SJ. Value = 3. “Fire”. See Agni and
Three.
UIJAYIN1. Town situated in the extreme west
of what is now the state of Madhya Pradesh. It
defines the first meridian of Indian astronomy.
See Indian astronomy (History of) and Yuga
(Astronomical speculation on).
ULTIMATE REALITY. [S], Value = 1. See
Atman and One.
UNIQUE REALITY. [S], Value = 1. See Atman
and One.
UNIVERSE. [SJ. Value = 13. See Vishva,
Vishvada and Thirteen.
UPPALA. Pali word which literally means:
“(blue) lotus flower (half open)”. Name given to
the number ten to the power ninety-eight. See
Names of numbers. For an explanation of this
symbolism, see Lotus and High numbers
(Symbolic meaning of).
Source: *Vyakarana (Pali grammar) by
Kachchayana (eleventh century CE).
URVARA. [S]. Value = 1. “Earth”. See Prithivi.
UTKRITI. (SJ. Value = 26. In Sanskrit poetry,
this is a metre of four lines of twenty-six
syllables per stanza. See Indian metric.
UTPALA. Literally: “(blue) lotus flower (half
open)’’. In Hindu and Buddhist philosophies,
this lotus (which is never represented in full
bloom) notably symbolises the victory of the
mind over the body. Name given to the number
ten to the power twenty-five. See Names of
numbers. For an explanation of this
symbolism, see Lotus and High numbers
(Symbolic meaning of).
Source: *LaIitavistara Sutra (before 308 CE).
UTSANGA. Name given to the number ten to
the power twenty-one (= quintillion). See
Names of numbers and High numbers.
Source: *Lalitavistara Sutra (before 308 CE).
V
VACHANA. [S]. Value = 3. “Aphorism”. From
vach, “voice", “speech”, ’’spoken word”, and
form anna, “nourishment”. This is an allusion to
the creative and evocative power of sound and
acoustic resonance (especially through speech)
and to its “indestructible and imperishable”
nature, which correspond to the revelation of
the *Brahman, which is said to be resumed in
the three letters of the sacred Syllable *AUM.
See Akshara, Trivama and Three.
VACUITY. See Shunya, Shunyatd, Zero, Zero
(Graeco-Latin concepts of), Zero (Indian
concepts of) and Zero and Sanskrit poetry.
VADAVA. Name given to the number ten to the
power nine. See Names of numbers and High
numbers.
Source: *Kathaka Samhita (from the start of the
Common Era).
VADAVA. Name given to the number ten to the
power fourteen. See Names of numbers and
High numbers.
Source: *Pahchavimsha Brdhmana (date uncertain).
VAHNI. [S]. Value = 3. "Fire". See Agni and
Three.
VAIKUNTHA. Celestial home of *Vishnu and
* Krishna. See Bhuvana.
VAISHESHIKA. See Darshana.
VAISHVANARA. [SI. Value = 3. “that which
belongs to all humans". This is one of the Vedic
names for *Agni (= 3), the god of sacrificial fire,
who is said to possess the powers of fire,
lightning and light. See Agni and Three.
VAJASANEYI SAMHITA. This is a text which
forms part of the Yajurveda “white”, which is
one of the oldest Vedic texts. Passed down
through oral transmission since ancient times,
it only found its definitive form at the
beginning of Christianity. See Veda. Here is a
list of the main names of numbers mentioned
in this text:
*Eka (= 1), *Dasha (= 10), *Sata (= 10 2 ),
*Sahasra (= 10 3 ), *Ayuta (= 10 4 ), *Niyuta (= 10 5 ),
*Prayuta (= 10 6 ), *Arbuda (= 10 7 ), *Nyarbuda (=
10 8 ), *Samudra (= 10 9 ), *Madhya (= 10 10 ), *Anta
(= 10 u ), *Parardha (=10 12 ). See Names of
numbers and High numbers.
(See: Vajasaneyi Samhita, XVII, 2; Datta
and Singh (1938), p. 9; Weber, in: JSO, XV, pp.
132-40; Woepcke (1863).]
VAJIN. IS]. Value = 7. "Horse”. See Ashva and
Seven.
VAJRA. In Hindu and Buddhist philosophies, the
vajra is the “diamond" that symbolises all that
remains when appearances have disappeared.
Thus it is the vacuity ( *shunyata ) that is as
indestructible as a diamond. It is also the missile
“with a thousand points", which is said to never
miss its target, and made out of bronze by
Tvashtri for *Indra. This weapon is a symbol of
*linga and divine power. It also indicates a
strong, stable and indestructible mind. As a
word-symbol, vajra has several meanings: the
weapon is usually a short bronze baton, which
has three, five, seven or nine points at each end.
With three points, for example, vajra symbolises:
the *triratna (or “three jewels" of Buddhism);
time in its three tenses ( *trikdla)\ the three
aspects of the world (*tri bhuvana); etc. [see
Frederic, Dictionnaire (1987)]. See Shunyatd and
Symbols.
VALAJBHI NUMERALS. Symbols derived from
*Brahrm numerals and influenced by Shunga,
Shaka, Kushana, Andhra, Pallava, Chalukya, and
Ganga numerals. The system arose at the time of
the inscriptions of Valabhi, the capital city of a
Hindu-Buddhist kingdom that ruled over
present-day Gujurat and Maharastra. The
symbols correspond to a mathematical system
that is not based on place values and therefore
does not possess a zero. See Indian written
numeral systems (Classification of). See Fig.
24.44, 52 and 24.61 to 69.
VARA. [S]. Value = 7. “Day of the week”. This is
because of the seven days: ravivara or adivara
(Sunday), induvdra or somavdra (Monday),
mangalavdra (Tuesday), budhavara (Wednesday),
brihaspativara (Thursday), shukravdra (Friday),
and shanivara (Saturday). See Seven.
VARAHAMIHIRA. Indian astronomer and
astrologer c. 575 CE. His works notably include
Pahchasiddhantika (the “Five Siddhantas’j, where
there are many examples of the place-value
system [see Neugebauer and Pingree (1970-71)].
See Indian astrology, Indian astronomy
(History of) and Indian mathematics
(History of).
VARGA. Word used in arithmetic to denote the
squaring operation. Synonym: kriti. In algebra,
the same word is used for the square, in
allusion to cubic equations. See Ghana, Varga-
Varga and Ydvattdvat.
VARGAMULA. Word used in arithmetic to
describe the extraction of the square root. See
Patiganita, Indian methods of calculation
and Square roots (How Aryabhata
calculated his).
VARGA-VARGA. Algebraic word for quadratic
equations.
VARIDHI. [Si. Value = 4. “Sea”. See Sagara,
Four and Ocean.
VAR1NIDHI. [S]. Value = 4. “Sea”. See Sagara,
Four and Ocean.
VARNA. Literally “letter”, in mathematics
“symbol”. See AUM, Bija and Bijaganita.
VARNASAMJNA. “Syllable system”. Name
that Haridatta gave to the *katapaya system.
VARNASANKHYA. Literally: “letter-numbers”.
This word denotes any system of
representing numbers which uses the vocalised
consonants of the Indian alphabet, each
one being assigned a numerical value.
See Numeral alphabet.
VARUNA. Vedic and Hindu divinity of the
water, the sea and the oceans. See High
numbers (Symbolic meaning of).
VASU. [SJ. Value = 8. Name in the
* Mahabharata which is given to a group of
eight divinities, who are meant to correspond,
philosophically speaking, to the eight “spheres
of existence” of the Adibhautika, which in turn
represent the visible forms of the laws of the
universe. See Eight.
VASUDHA. [SJ. Value = 1. “Earth”. See Prithivi
and One.
VASUKI. In Brahmanic mythology, this is
the name given to the king of the *naga.
He is said to have been used by the *deva
(the gods) and the ‘asura (the anti-gods)
as a “rope” with which to spin *Mount
Meru on its axis in order to churn the sea
of milk and thus extract the “nectar
of immortality" (*amrita). See Serpent
(Symbolism of the).
505
VASUNDHARA
VASUNDHARA. [S]. Value = 1. “Earth". See
Prithivi and One.
VATTELUTTU NUMERALS. Symbols derived
from ‘Brahmi numerals and influenced by
Shunga, Shaka, Kushana, Andhra, Pallava,
Chalukya, and Ganga, Valabhi, Bhattiprolu and
Grantha numerals as well as by Ancient Tamil.
Used from the eighth to the sixteenth centuries
CE in the Dravidian areas of South India,
particularly the Malabar coast. The symbols
correspond to a mathematical system that is
not based on place values and therefore does
not possess a zero. See: Indian written
numeral systems (Classification of). See Fig.
24.52 and 24.61 to 69.
VAYU. “Wind”. This is a name for the god of the
wind. Other names include: Marut
(“Immortal”), Anila (“Breath of life”), Vdta
(“Wandering”, “He who is in perpetual
movement") or *Pavana (“Purifier”). According
to Brahmanic and Hindu cosmogonies, he is
one of the eight *Dikpala (divinities who guard
the horizons and points of the compass), whose
task is to guard the northwest “horizon”.
VAYU. [S]. Value = 49. “Wind”. This symbolism
can be explained by reference to tales of
Brahman mythology. One day Vayu revolted
against the *dcva, the “gods” who live on the
peaks of ‘Mount Meru. He decided to destroy
the mountain, and unleashed a powerful
hurricane. However, the mountain was
protected by the wings of Garuda, the carrier-
bird of ‘Vishnu, which rendered all the assaults
of the wind ineffectual. One day, in Garuda’s
absence, Vayu chopped off the peak of ‘Mount
Meru, and threw it into the ocean. That is how
the island of Sri Lanka was created. Mount
Meru was meant to be the place where the gods
lived and met. It was said to be situated at the
centre of the universe, above the seven *pdtdh
(or “inferior worlds”); it has seven faces, each
one facing one of the seven *dvipa (or “island-
continents) and the seven *sdgara (“‘oceans”).
When Vayu attacked the mountain, he created
seven strong winds, one for each face. Once the
summit of the sacred mountain had been rased,
the seven winds, thus placed at the centre of the
universe and no longer encountering any
barrier, each went to one of the seven
continents and the seven oceans. Thus: Vayu = 7
x 7 = 49. See other entry entitled Vayu.
VEDA. Name of the oldest sacred texts of India,
they are made up of four principal books
(namely: the Rigveda, the Yajurveda, the
Samaveda, and the Atharvaveda). These texts
and those of derived literature probably date
back to ancient times in the history of India. But
it is impossible to date them exactly, because
they were primarily transmitted orally before
being transcribed at a later date. In fact, it is only
possible to give them a chronological position in
relation to each other. The three Samhita (the
texts of the Rigveda, the Yajurveda and the
Samaveda) seem to have been composed first. As
for the fourth Veda, (the Atharvaveda), it was
followed by the Brdhmana, the Kalpasutra, and
lastly by the Aranyaka and the Upanishad {see
Frederic, Dictionnaire (1987)].
VEDA. [S], Value = 3. (Very rarely used as a
numerical symbol). The allusion here is probably
to the three Samhita (the Rigi'eda, the Yajurveda
and the Samaveda), which constitute the first
three texts of the Veda. See Trayi and Three.
VEDA. [SJ. Value = 4. (The most frequent value
of this word as a numerical symbol.) Here the
allusion is to the four principal books of which
the Veda is composed (the Rigveda, the Yajurveda,
the Samaveda, and the Atharvaveda). See Four.
VEDANGA. [SI. Value = 6. “Members of the
*Veda”. Group of six Vedic and Sanskrit texts
dealing principally with the Vedic ritual, its
conservation and its transmission. See
Darshana.
VEDIC RELIGION. See Indian religions and
philosophies.
VIBHUTANGAMA. Name given to the
number ten to the power fifty-one. See Names
of numbers and High numbers.
Source: *Lalitavistara Sutra (before 308 CE).
VIDHU. [S]. Value = 1. “Moon”. See Abja and
One.
VIKALPA. Word used in mathematics since
the eighth century to designate “permutations”
and “combinations”.
VIKRAMA. (Calendar). Formerly used in the
centre, west and northwest of India. Also called
vikramddityakdla, vikramasamvat, or quite
simply samvat. It began in the year 57 BCE. To
find an approximate corresponding date in the
Common Era, subtract 57 from a date in the
Vikrama calendar.
VI KR AM ADIT YAK ALA (Calendar). See
Vikrama.
VIKRAMASAMVAT (Calendar). See Vikrama.
VIKRITI. [S]. Value = 23. In Sanskrit poetry,
this is the metre of four times twenty-three
syllables per stanza. See Indian metric.
VILAYATI (Calendar). Solar calendar
commencing in the year 592 CE. Used in
Bengal and Orissa. To find a date in the
Common Era, add 592 to a date expressed in
the Vilayati calendar. See Indian calendars.
VIMSHATI. Ordinary Sankrit name for the
number twenty. For words which have a
symbolic relationship with this number, see
Twenty and Symbolism of numbers.
VINDU. [S]. Value = 0. *Prakrit word which
has the literal meaning and symbolism of
*bindu. See Zero.
VIOLENT. [S]. Value = 11. See Rudra-Shiva
and Eleven.
VIRASAMVAT (Calendar). Commencing in
the year 527 BCE, it is only used in ‘Jaina texts.
To find a corresponding date in the Common
Era, subtract 527 from a date expressed in this
calendar. See Indian calendars.
VISAMJNAGATI. Name given to the number
ten to the power forty-seven. See Names of
numbers and High numbers.
Source: *l.alilavistara Sutra (before 308 CE).
VISHAYA. [SJ. Value = 5. “Sense, sense organ”.
See Shara and Five.
VISHIKHA. [S]. Value = 5. “Arrow”. See Shara
and Five.
VISHNU. Name of one of the three major
divinities of the Brahmanic and Hindu
pantheon (‘Brahma, ‘Vishnu, ‘Shiva). See
Vishnupada, Piirna and High numbers
(Symbolic meaning of).
VISHNUPADA. [S]. Value = 0. Literally: “foot
of Vishnu”, and by extension (and
characteristically of Indian thought): “zenith”,
“sky”. This is an allusion to the “Supreme step
of Vishnu”, the zenith, which denotes ‘Mount
Meru, home of the blessed. The symbolism in
question also refers to the “Three Steps of
Vishnu”, symbols of the rising, apogee and
setting of the sun, which allowed him to
measure the universe. It is also from the “feet
of Vishnu” that, according to Hindu
mythology, the sacred Ganga (the Ganges)
springs and, before it divides into terrestrial
rivers, has its source at the summit of Mount
Meru (which is situated at the centre of the
universe and over which are the heavens or
“worlds of Vishnu”). Vishnu rests upon
‘Ananta, the serpent with a thousand heads
who floats on the primordial waters and the
“ocean of unconsciousness”, during the time
that separates two creations of the universe.
Thus this symbolism corresponds to the
connection in Indian philosophy between
infinity and zero, because Ananta is the
serpent of infinity, eternity and of the
immensity of space. Space also means sky,
which is considered to be the “void” which has
no contact with material things. Thus Vishnu
is identified with ether (*akasha), an
immobile, eternal and indescribable space. In
other words, Vishnu is synonymous with
vacuity ( *shunyata). See Abhra, Akasha, Kha,
Ananta, Zero and Zero (Indian concepts of).
VISHVA. [S]. Value = 13. Contraction of the
word Vishvadeva and a symbol for the number
13. See Vishvadeva and Thirteen.
VISHVADEVA. [SJ. Value = 13. This is an
allusion to the universe formed by thirteen
paradises or chosen lands (*bh uvana), and does
not include the *vaikuntha. See Bhuvana,
Vaikuntha and Thirteen.
VISION. IS]. Value = 2. See Drishti and Two.
VISION. [S]. Value = 6. See Darshana and Six.
VISKHAMBA. Name given to the number ten
to the power fifteen. See Names of numbers
and High numbers.
Source: *I.aIitavistara Sutra (before 308 CE).
VIVAHA. Name given to the number ten to the
pow r er nineteen. See Names of numbers and
High numbers.
Source: *I.a!itavistara Sutra (before 308 CE).
VIVARA. Name given to the number ten to the
power fifteen. See Names of numbers and
High numbers.
Source: *Lalitavislara Sutra (before 308 CE).
VOICE. IS]. Value = 3. See Vachana and Three.
VOID. [SJ. Value = 0. See Shunya, Shunyatd
and Zero.
VRINDA. Name given to a plant which
is similar to basil, the leaves of which are said
to have the power to purify the body and
mind. It is believed to be an incarnation of
Vishnu: according to the legend, Vrinda was
the wife of a Titan then was seduced by
Vishnu. She cursed her husband and
transformed him into a sh diagram a stone
before killing herself by throwing herself onto
a fire of live coals; the plant (still called tulasi)
was born out of the ashes. See Ananta,
Vishnupada, Samudra, Names of numbers
and High numbers.
VRINDA. Name given to the number ten to the
power nine. See Names of numbers and High
numbers. For an explanation of this
symbolism, see Vrinda (first entry) and High
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
506
numbers (Symbolic meaning of).
Source: Aryabhatiya (510 CE).
VRINDA. Name given to the number ten to the
power seventeen. See Names of numbers and
High numbers. For an explanation of this
symbolism, see Vrinda (first entry) and High
numbers (symbolic meaning of).
Source: *Kdmayana by Valmiki (early centuries CE).
VULVA. [SI. Value = 4. See Yoni and Four.
VYAKARANA. See Kachchayana.
VYAKTAGANITA. Name for arithmetic
(literally: “science of calculating the known”),
as opposed to algebra, which is called
*Avyaktaganita.
VYANT. [Sf Value = 0. “Sky”. The symbolism
can be explained by the fact that the sky (or
heaven) is the “void" in Indian beliefs. See
Shunya and Canopy of heaven.
VYARRUDA. Name given to the number ten to
the power eight (= hundred million). See
Names of numbers and High numbers.
Source: Kitab fi tahqiq i ma li’I hind by al-Biruni (c.
1030 CE).
VYASTATRAIRASHIKA. [Arithmetic]. Name
of the inverse of the Rule of Three. See
Trairdshika.
VYAVAHARA. Literally: “procedure". Term
used in algebra (since the seventh century CE)
to denote the solving of equations.
VYAVAKALITA. [Arithmetic]. Sanskrit term
for subtraction. Literally: “taken away”.
VYAVASTH AN APRAJN APATI . Name given to
the number ten to the power twenty-nine. See
Names of numbers and High numbers.
Source: *l.alitavistara Sutra (before 308 CE).
VYOMAN. [S]. Value = 0. Word meaning
“sky”, “space”. See Zero and Shunya.
VYUTTKALITA. [Arithmetic]. Sanskrit term
for subtraction. See Vyavakalita.
W
WAYS OF REBIRTH (The four). See
Chaturyoni and Yoni.
WEEK. [S]. Value = 7. See Vara and Seven.
WHEEL. IS]. Value = 12. See Chakra , Rdshi
and Twelve.
WHITE LOTUS. As a representation of the
numbers ten to power twenty-seven and ten to
power 112. See Pundarika, High Numbers
(Symbolic meaning of).
WIND. [S J. Value = 5. See Pavana.
WIND. [Sj. Value = 7. See Pavana.
WIND. (S]. Value = 49. See Vdyu.
WING [S]. Value = 15. See Paksha. Fifteen.
WING. (SJ. Value = 2. See Paksha. Two.
WORLD. [S]. Value = 3. See Bhuvana.
WORLD. [S]. Value = 7. See Loka.
WORLD. [Sf Value = 14. See Bhuvana.
Y
YAMA. [S]. Value = 2. “Primordial couple".
Allusion to the couple in Hindu mythology,
formed by Yama (the first mortal who became
god of death) and Yami, his twin sister, wife and
his feminine energy ( *shakti ). See Two.
YAMALA. [S]. Value = 2. Synonym of *Yama.
See Two.
YAMAU. [SJ. Value = 2. Synonym of *Yama.
See Two.
YAVATTAVAT. Literally : “as many as”. Term
used in algebra to denote the “equation” in
general.
YONI. [S]. Value = 4. “Vulva”. Allusion to the four
lips that form the entrance of the vulva. By
extension, the word also means “birth”. Here, the
reference is to the * Chaturyoni which, according to
Hindus and Buddhists, correspond to the “four
ways of rebirth”. According to this philosophy,
there are four different ways to enter the cycle of
rebirth ( *samsdra ) : either through a viviparous
birth (Jarayuva ), in the form of a human being or
mammal; or an oviparous birth ( andaja ), in the
form of a bird, insect or reptile; or by being born
in water and humidity ( samsvedaja ), in the form of
a fish or a worm; or even through metamorphosis
(i aupapdduka ), which means there is no “mother”
involved, just the force of Karma. See Four.
YUGA (Definition). "Period”. Generic names for
the cosmic cycles of Indian speculations which are
either based upon Brahmanic cosmogny or the
learned astronomy founded by * Aryabhata. The
principal cycle is the *mahdyuga (or “great
period") made up of 4,320,000 human years. This
is divided into four successive yugas. Thus the
words *mahdytiga and *chaluryuga (literally : four
periods) are treated as synonymous. These four
successive ages are named respectively : *kritayuga
(or *satyayuga), * t relay uga, *d\>aipayanayuga (or
*dvaparayuga), and *kaliyuga. The corresponding
lengths can considered to be equal or unequal
depending on which system of calculation is used.
See other entries entitled Yuga.
YUGA (Astronomical speculation on). Since
its emergence at the start of the sixth century
CE, learned Indian astronomy has been marked
by its amazing speculation about the
cosmic cycles (known as * yugas), which is
very different from the cosmogonical
speculations. See Yuga (Definition) and Yuga
(Systems of calculating).
According to this speculation, directly
linked to astronomical elements, the
*chaturyuga or cycle of 4,320,000 years is
the period at the beginning and end of which
the nine elements (namely the sun, the moon,
their apsis and node and the planets) are in
mean perfect conjunction at the starting point
of the longitudes. Thus the durations of the
revolutions, previously considered to be
the same lengths, are (in this astronomy)
subjected to common multiples and general
conjunctions. See Indian astronomy (History
of) and Indian mathematics (History of). This
speculation seems so audacious because it is
obviously devoid of any physical meaning.
As for the cycle called *kalpa, which
constitutes an even longer period of time of
4,320,000,000 years, it is delimited, according
to * Brahmagupta (628 CE), by two perfect
conjunctions in true longitude of the totality of
elements, themselves each matched by a total
eclipse of the Sun on the stroke of six in the
secular time in *Ujjayini. In practice, however,
these fictional eras can be reduced to the age of
the *kaliyuga, the present age, which
traditionally starts at a theoretical point of
departure of the celestial revolutions
corresponding to the 18 February 3101 BCE at
zero hours. (This moment is fixed itself at the
general conjunction in mean longitude at the
starting point of the sidereal longitudes of the
sun, the moon and the planets, the apogees and
node ascending from the moon being
respectively at 90° and 180° from these
longitudes.) Literally, the word *yuga signifies
“yoke", “link". In ancient Indian astronomy,
this term was employed in the very limited
sense of the simple “cycle”. Thus in the
*Jyotish a vedanga (“Astronomic Element of
Knowledge"), a yuga of five years is used, this
being a period at the end of which the sun and
the moon are considered to have each
completed a whole number of revolutions. On
the other hand, in the Romakasiddhdnta (start
of the fourth century CE), the yuga is a lunar-
solar cycle, the length of which is 2,850 years.
These cycles, however, do not constitute an
“astronomical speculation" like the one that
began to be developed in Aryabhata’s time. No
speculative system relating to yugas is found in
the texts of the * Vedas. This means that the
yuga speculations were probably unknown in
India during Vedic times and until the early
centuries of the Common Era.
Nevertheless, purely arithmetical speculative
calculations on these cycles appear in the
*Manusmriti (a significant religious work
considered to form the basis of Hindu society),
as well as in the much later texts of the
Ydjhavalkyasmriti and the epic of the
*Mahabharata. It is difficult, however, if not
impossible, to glean from this a chronology
for the history of speculative yugas, since a great
deal of uncertainty presides over the dates of
these documents.
On the other hand, the work of Aryabhata,
in which astronomical speculation of yugas
appears for the first time, is dated in a rather
precise manner, to within a few years of 510 CE.
In fact, as far as it is possible to tell, it was
Aryabhata who, after the beginning of the sixth
century CE, introduced speculative yugas into
mathematical astronomy and made them
generally known in India.
None of the Indian speculative canons (on
yugas) that are known today is dated before
Aryabhata's time. Aryabhata’s astronomical
speculations on yugas use basic numbers, some
of which can be seen in the following calculations
(*nakshatra here denoting the twenty-seven
lunar mansions divided into equal lengths) :
1 *mahayuga = 4,320,000 years = 12,000
(moments) x 360 = 27 ( nakshatra ) x 4 x 4
(phases) x 10,000 = “great period". 1 *yugapada
= 1,080,000 years = 3,000 (moments) X 360 = 27
( nakshatra ) x 4 (phases) x 10 000 = one quarter
of a “great period”. 1 *kaliyuga = 432,000 years =
1,200 (moments) x 360 = 27 ( nakshatra ) x 4 x 4
(phases) x 1,000 = 1,200 (“divine years”) x 360 =
one tenth of a “great period”.
According to Censorious, Heraclitus's
“great year" was 10,800 years long. On the
other hand, the surviving fragments of the
work of Babylonian astronomer Berossus
(fourth-third century BCE) contain mention of
a cosmic period 432,000 years long, which is
also called “Great Year” :
1 * Great Year of Heraclitus = 10,800 years =
30 (moments) x 360. 1 * Great Year of Berossus=
432,000 years = 1,200 (moments) X 360. In
other words, in all the cycles there is the
following arithmetical relationship: 1
*yugapada = 100 times the Great Year of
507
Y UC A
Heraclitus = 2.5 times the Great Year of
Berossus. 1 *kaliyuga = one Great Year
of Berossus = 40 times the Great Year of
Heraclitus. 1 * mahayuga = 400 times the Great
Year of Heraclitus = 10 times the Great Year
of Berossus.
From what is known today, it is impossible
to establish whether there is any link between
Aryabhata’s yugas and the cosmic periods of
the Mediterranean world. What is known is
that Heraclitus belonged to the time when
Persia dominated certain countries of the
Greek world as well as part of India, whilst
Berossus belonged to the end of the Persian
rule and the beginning of the conquests of
Alexander the Great ... So why did Aryabhata
develop his remarkable speculation? “As far as
Aryabhata was concerned, speculation about
yugas was just a theory. Convinced of the
existence of common multiples of the different
revolutions, he had set himself the task of
researching the cycles of this astronomy, which
was the most advanced of his time, and of
which he was fully aware of the value. Whether
it was a spontaneous idea, or drawn from a
revival of the the Great Year 432,000 years long
of the Babylonian astronomer Berossus, or
even inspired by a wholly verbal, strictly
arithmetical speculation, in any case Aryabhata
drew out the constants of the mean movements
in order to construct these common multiples
and general conjunctions, from a single reality
in time, that is to say the astronomical reality of
510 CE almost to the year. Of course, the theory
was regrettable, but we must not forget the
serious and extreme rigour he showed in
undertaking such a work.” (Billard)
YUGA (Cosmogonical speculations on).
According to speculations developed by
cosmogonies on what is referred to as the
decline of Proper moral and cosmic Order
over the course of time”, the *mahayuga
corresponds to the appearance, evolution and
disappearance of a world, and the whole cycle
is followed by a new mahayuga, and so on until
the destruction of the universe. The four ages
of this “great cycle" are considered to be
unequal in terms of both length and worth.
Qualitatively, this is how things are meant to
unfold [see Friedrichs, etc (1989), Dictionnaire
de la sagesse orientate ] :
1- The * kritayuga, the first of the fouryw^s, is
the golden age, during which humans enjoy
extremely long lives and everything is
perfect. This is the ideal age, where virtue,
wisdom and spirituality reign supreme,
and there is a total absence of ignorance
and vice. Hatred, jealousy, pain, fear and
menace are unknown. There is only one
god, one sole *Veda, one law and one
religion, each caste fulfilling its duties with
the utmost selflessness. This age is said to
have lasted 4,800 divine years
( * divyavarsha ), which is equal to 1,728,000
human years.
2. The * tretayuga is the age during which
humans are only believed to live three
quarters of their lives. They are now
marked by vice, there are the beginnings of
laxity in their behaviour and the first ritual
sacrifices are carried out. During this age,
humans begin to act with intention and in
self-interest. Rectitude diminishes by a
quarter. The age is said to last 3,600 divine
years, or 1,296,000 human years.
3. The *dvaparayuga is said to be the age
during which the forces of Evil equal those
of Good, and where honest behaviour,
virtue and spirituality are reduced by half.
Illnesses have made their appearance and
humans now only live half their lives. This
age is said to last 2,400 divine years, or
864.000 human years.
4. Finally, the *kaliyuga or "iron age” is the age
we are living in now. “True virtue” is
something which has all but disappeared and
conflicts, ignorance, irreligion and vice have
increased manifold. Illnesses, exhaustion,
anger, hunger, fear and despair reign supreme.
Living things only live for a quarter of their
existence and the forces of evil triumph over
good. Only a quarter of the original rectitude
displayed by humans remains. This age is
meant to have begun in the year 3101 BCE,
and is meant to last 1,200 divine years, or
432.000 human years. It is meant to end with
a pralaya (destruction by fire and water)
before a new * chaturyuga begins.
See Yuga (Definition), Yugas (Systems of
calculating) and any other entry entitled Yuga.
YUGA (Systems of calculating). In the
traditional system, the four ages of a *chaturyuga
are of unequal length, with the ratios of 4, 3, 2
and 1, from the *kritayuga to the *kaliyuga
whose length is equal to 1/10 of the * mahayuga.
In other words, the four successive ages of a
chaturyuga are divided unequally as follows :
1 mahayuga = 4/10 + 3/10 + 2/10 + 1/10.
Thus the corresponding values in “divine”
years: 1 *kritayuga = 4,800 divine years (= 4/10
of mahayuga)', 1 'tretayuga = 3,600 divine years
(= 3/10 of mahayuga ); 1 'dvdparayuga = 2,400
divine years (= 2/10 of mahayuga)-, 1 'kaliyuga
= 1,200 divine years (= 1/10 of mahayuga)-, 1
mahayuga - 12,000 divine years.
As one divine year is said to be equal to 360
human years, these cycles have the following
durations in human time:
1 kritayuga = 1,728,000 human years; 1
tretayuga = 1,296,000 human years; 1
dvdparayuga - 864,000 human years; 1 kaliyuga
= 432,000 human years; 1 mahayuga =
4.320.000 human years.
See Divyavarsha. The system for
calculating unequal yugas was used by a
considerable number of Indian astronomers
(including *Brahmagupta), as well as by a great
many cosmographers, philosophers and
authors of religious texts (traditional system).
However, the system used by * Aryabhata
consists in dividing the mahayuga in the
following manner :
1 kritayuga = 1,080,000 human years; 1
tretayuga = 1,080,000 human years; 1
dvdparayuga = 1,080,000 human years; 1
kaliyuga = 1,080,000 human years; 1 mahayuga
- 4,320,000 human years.
In other words, the four cycles of the
chaturyuga are all considered to be equal here.
This is the system of the 'yugapadas or
"quarters of a yuga". However, the mahayuga is
not the longest unit of cosmic time in these
systems of calculation. There is also the cycle
called a *kalpa, which corresponds to
12,000,000 divine years:
1 kalpa = 12,000,000 divine years =
12.000. 000 X 360 = 4,320,000,000 human
years. Bearing in mind the length of the
mahayuga (= 12,000 divine years), this cycle is
thus also defined by :
1 kalpa = 1,000 mahayuga = 4,320,000 x
1.000 = 4,320,000,000 human years.
An even longer period exists, the
'mahakalpa, or “great kalpa", which is the length
of twenty “ordinary” kalpas (20,000 mahayugas) :
1 mahdkalpa - 12,000,000 x 20 =
240.000. 000 divine years = 240,000,000 x 360
= 86,400,000,000 years.
YUGA. [S]. Value = 2. (This symbol is very
rarely used to represent this value.) The
allusion here is to the cycle called
'Dvaipayanayuga, where, according to
Brahmanic cosmogony, men have only lived
half of their lives and where the forces of Evil
are counteracted by the equal strength of the
forces of Good. The duality ( *dvaita ) between
Good and Evil is at the root of this symbolism.
See Yuga (Definition) and Two.
YUGA. [SJ. Value = 4. The allusion here is to
the most important of the cosmic cycles of this
name : the * mahayuga (or “Great Age”), also
called * chaturyuga (or "four ages”). Composed
of four successive “ages”, in Hindu cosmogony
the mahayuga is said to correspond to the
appearance, evolution and disappearance of a
world. See Yuga (Definition) and Four.
YUGALA. [SJ. Value = 2. Synonym of *Yama.
See Two.
YUGAPADA. “Quarter of a *yuga'\ Name given
to each of the four cycles of a * chaturyuga,
subdivided into four equal parts, according to
the system of calculation used by * Aryabhata.
A yugapada thus corresponds to 1,080,000
human years. See Yuga (Definition) and Yuga
(Systems of calculating).
YUGMA. [S]. Value = 2. Synonym of *Yama.
See Two.
z
ZENITH. [S]. Value = 0. See Vishnupada and
Zero.
ZERO. Ordinary Sanskrit name for zero :
'shunya. Here is a list of corresponding
numerical symbols: *Abhra, 'Akdsha,
*Ambara, 'Ananta, *Antariksha, *Bindu,
*Gagana, 'Jaladharapatha, *Kha, *Nabha,
*Nabhas, 'Purna, *Randhra, *Vindu,
'Vishnupada, 'Vyant, *Vyoman. These words
translate or symbolically express :
1. The void ( Shunya ). 2. Absence (Shunya). 3.
Nothingness (Shunya). 4. Nothing (Shunya).
5. The insignificant (Shunya). 6. The
negligible quantity (Shunya). 7. Nullity
(Shunya). 8. The “dot” (Bindu, Vindu). 9. The
“hole” (Randhra). 10. Ether, or “element
which penetrates everything” (Akdsha). 11.
The atmosphere (Abhra, Ambara, Antariksha,
Nabha, Nabhas). 12. Sky (Nabha, Nabhas,
Vyant, Vyoman, Vishnupada). 13. Space
(Akdsha, Antariksha, Kha, Vyant, Vyoman).
14. The firmament (Gagana). 15. The canopy
of heaven (Gagana). 16. The immensity of
space (Ananta). 17. The “voyage on water”
( Jaladharapatha ). 18. The “foot of Vishnu”
(Vishnupada). 19. The zenith (Vishnupada).
20. The full, the fullness (Puma). 21. The
state of that which is entire, complete or
finished (Purna). 22. Totality (Purna). 23.
Integrity (Purna). 24. Completion (Purna).
25. The serpent of eternity (Ananta). 26. The
infinite (Ananta, Vishnupada). See
Numerical symbols.
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
508
ZERO (Graeco-Latin concepts of)- Western
cultures have obviously had a concept of the
void since Antiquity. To express it, the Greeks
had the word ouden (“void”). As for the
Romans, they used the term vacuus (“empty"),
vacare (“to be empty”), and vacuitas
(“emptiness”); they also had the words absens,
absentia, and even nihil (nothing), nullus and
nullitas. But these words actually corresponded
to notions that were understood very distinctly
from each other. With the help of some
appropriate examples, an etymological
approach will enable us hereafter to form quite
a clear idea of the evolution of the concepts
down the ages and to perceive better the
essential difference which exists between these
diverse notions and the Indian concept of the
zero. “Presence” (from the Latin praesens,
present participle of praesse, “to be before
[prae]", “to be facing") is properly speaking the
fact of being where one is. But the adjective
present also means "what is there in the place
of which one is speaking”; this meaning is
applicable then both to an object and to a
living being. In the figurative sense, applied to
people, present means “that which is present in
thought to the thing being spoken of” (to be
present in thought at a ceremony, despite the
physical absence); applied to things, however,
it means “that which is there for the speaker, or
for what he is aware of”. It is thus a moral or
deliberate presence. Another meaning of
presence, in opposition this time to the past
and the future, is “that which exists or is really
happening, either at the moment of speaking
or at the moment of which one is speaking”.
Consequently, this meaning corresponds to the
present situation. Figuratively, it is rather a
matter of “that which exists for the
consciousness at the moment one is speaking",
somewhat like a scene one witnessed and
which remains present in one’s mind.
This preamble allows a better
understanding of “absence”, since it is a term
that is opposed to presence. The word comes
from the Latin absentia, which derives from
abest, “is far”. Thus it expresses the character of
“that which is far from”. It is thus by definition
the fact of not being present at a place where
one is normally or one is expected. And the
absent is the person or the thing which is
lacking or missing. As for non-presence, it is
simply the void left by an absence, since it is the
space that is not occupied by any being or any
thing. If it is an unoccupied place, it is this that
is empty, whether it be a seat, an administrative
post or even one of the “places” of the place-
value system. By dint of thinking solely of the
void, some thinkers have arrived at vacuism, a
type of physics, according to which there exist
spaces where all material reality is void of all
existence. It was developed notably by the
Epicureans, who conceded the existence of
places where all matter, visible or invisible, was
absent. Others opted rather for anti-vacuism,
like Descartes, who considered an absolute void
to be a contradictory notion. Nowadays, it is
still sometimes said that nature abhors a
vacuum. This aphorism was created by those
who held to the physics of the ancient world in
order to make sense of the existence of certain
phenomena for which they were incapable of
providing a satisfactory explanation. It was not
until the experiments of the Italian physicist
Torricelli on the laws of atmospheric weight,
that the lie was given to this idea.
ZERO (Indian concepts of). In Sanskrit, the
principal term for zero is Shunya, which
literally means “void’or “empty”. But this
word, which was certainly not invented for this
particular circumstance, existed long before
the discovery of the place-value system. For, in
its meaning as “void”, it constituted, from
ancient times, the central element of a veritable
mystical and religious philosophy, elevated
into a way of thinking and existing. The
fundamental concept in *shunyatdvada, or
philosophy of “vacuity”, *shunyala, this
doctrine is in fact that of the “Middle Way”
( Madhymakha ), which teaches in particular
that every made thing ( samskrita ), is void
(*shunya), impermanent (anitya), impersonal
(i anatman ), painful ( dukh ) and without original
nature. Thus this vision, which does not
distinguish between the reality and non-reality
of things, reduces the same things to total
insubstantiality. See Shunya and Shunyatd.
This is how the philosophical notions of
“vacuity”, nihilism, nullity, non-being,
insignificance and absence, were conceived
early in India (probably from the beginning
of the Common Era), following a perfect
homogeneity, contrary therefore to the
Graeco-Latin peoples (and more generally to
all people of Antiquity) who understood
them in a disconnected and totally
heterogeneous manner.
The concepts of this philosophy have been
pushed to such an extreme that it has been
possible to distinguish twenty-five types of
shunya, expressing thus different nuances,
among which figure the void of non-existence,
of non-being, of the unformed, of the unborn,
of the non-product, of the uncreated or the
F i g . 2 4 d . 1 o . The Western concept of nought
Abest, “is far”
Absentia, “absence”,
“non-presence”,
the quality of that
which is not there
(but somewhere
else)
Nullus, “not any”,
“not a. . .”
Nullus, in the
sense of “none”
Nullitas, “nullity”,
the quality of that
which is null or
void. (The Latin
word arose in the
Middle Ages, and
its meaning was
influenced by the
zero imported
from Arabic
culture)
509
ZERO
F1G.24D11. The perfect homogeneity of the Indian zero
non-present; the void of the non-substance, of
the unthought, of immateriality or
insubstantiality; the void of non-value, of the
absent, of the insignificant, of little value, of
no value, of nothing, etc. In brief, the zero
could have hardly germinated in a more fertile
ground than the Indian mind. Once the place-
value system was born, the shunya, as a
symbol for the void and its various synonyms
(absence, nothing, etc.), naturally came to
mark the absence of units in a given order [see
Fig. D. 11]. It is important to remember that
the Indian place-value system was born out of
a simplification of the *Sanskrit place-value
system as a consequence of the suppression of
the word-symbols for the various powers of
ten. This was a decimal positional numeration
which used the nine ordinary names of
numbers and the term shunya ("void”) as the
word that performs the role of zero. Thus the
Indian zero has meant from an early time not
only the void or absence, but also heaven,
space, the firmament, the canopy of heaven,
the atmosphere and ether, as well as nothing,
a negligible quantity, insignificant elements,
the number nil, nullity and nothingness, etc.
This means that the Indian concept of zero by
far surpassed the heterogenous notions of
vacuity, nihilism, nullity, insignificance,
absence and non-being of all the
contemporary philosophies. See all other
entries entitled Zero.
The Sanskrit language, which is an
incomparably rich literary instrument,
possessed more than just one word to
express “void". It possessed a whole panoply
of words which have more or less the
same meaning; these words are related, in a
direct or indirect manner, to the universe
of symbolism of Indian civilisation.
See Sanskrit, Numerical symbols,
Numeration of numerical symbols.
Thus words which literally meant the
sky, space, the firmament or the canopy
of heaven came to mean not only the void
but also zero. See Abhra, Akdsha, Ambara,
Antariksha, Gagana, Kha, Nabha, Vyoman
and Zero. In Indian thought, space is
considered as the void which excludes
all mixing with material things, and, as
an immutable and eternal element, is
impossible to describe. Because of the
elusive character and the very different
nature of this concept as regards ordinary
numbers and numerals, the association
of ideas with zero was immediate. Other
Indian numerical symbols used to mean
zero were: *purna “fullness”, “totality”,
“wholeness”, "completion”; *ja!adharapatha,
“voyage on the water”; *vishnupada, “foot of
Vishnu; etc. To find out more about this
symbolism, see the appropriate entries. Such a
numerical symbolism has played a role that
has been all the more important in the history
of the place-value system because it is in fact at
the very origin of a representation that we are
very familiar with. The ideas of heaven, space,
atmosphere, firmament, etc., used to express
symbolically, as has just been seen, the concept
of zero itself. And as the canopy of heaven is
represented by human beings either by a
semi-circle or a circular diagram or again
by a complete circle, the little circle that
we know has thus come, through simple
transposition or association of ideas, to
symbolise graphically, for the Indians, the idea
of zero itself. It has always been true that
"The circle is universally regarded as the very
face of heaven and the Milky Way, whose
activity and cyclical movements it indicates
symbolically” [Chevalier and Gheerbrant
(1982)]. And so it is that the little circle was
put beside the nine basic numerals in the
place-value system, to indicate the absence of
units in a given order, thereafter acquiring its
present function as arithmetical operator (that
is to say that if it is added to the end of a
numerical representation, the value is thus
multiplied by ten). See Numeral 0 (in the form
of a little circle), Shunya-chakra.
The other Sanskrit term for zero is the
word *bindu, which literally means “dot". The
dot, it is true, is the most elementary
geometrical figure there is, constituting a circle
reduced to its centre. For the Hindus, however,
the *bindu (in its supreme form of a
* paramabindu ) symbolises the universe in its
non-manifest form and consequently
constitutes a representation of the universe
before its transformation into the world of
appearances ( rupadhatu ). According to Indian
philosophies, this uncreated universe is
endowed with a creative energy capable of
engendering everything; it is thus in other
terms the causal point whose nature is
consequently identical to that of “’’vacuity”
(*shunyatd). See Bindu, Paramanu,
Paramabindu, Akdsha, Shunyata and Zero.
Thus this natural association of ideas with
this geometrical figure, which is the most basic
of them all, yet capable of engendering all
possible lines and shapes ( rupa ). It is the
perfect symbol for zero, the most negligible
quantity there is, yet also and above all the
DICTIONARY OF INDIAN NUMERICAL SYMBOLS
510
most basic concept of all abstract mathematics.
Thus the dot came to be a representation of
zero (particularly in the Sharada system of
Kashmir and in the notations of Southeast
Asia; see Fig. 24. 82) which possesses the same
properties as the First symbol, the little circle.
See Numeral 0 (in the form of a dot) and
Shunya-bindu.
This is the origin of the eastern Arabic zero
in the form of a dot : when the Arabs acquired
the Indian place-value system, they evidently
acquired zero at the same time. This is why, in
Arabic writings, sometimes the sign is given in
the form of a dot, sometimes in the form of a
small circle. It is the little circle that prevailed
in the West, after the Arabs of the Maghreb
transmitted it themselves to the Europeans
after the beginning of the twelfth century. To
return to India, this notion was gradually
enriched to engender a highly abstract
mathematical concept, which was perfected in
‘Brahmagupta’s time (c. 628 CE); that of the
“number zero” or “zero quantity”, it is thus
that the shunya was classified henceforth in the
category of the *samkhya, that is to say the
“numbers", so marking the birth of modern
algebra. See Shunya-samkhya. So, from
abstract zero to infinity was a single step which
Indian scholars took early and nimbly. The
most surprising thing is that amongst the
Sanskrit words used to express zero, there is
the term *ananta, which literally means
“Infinity". Ananta, according to Indian
mythologies and cosmologies, is in fact the
immense serpent upon which the god ‘Vishnu
is said to rest between two creations; it
represents infinity, eternity and the immensity
of space all at once. Sky, space, the atmosphere,
the canopy of heaven were, it is true, symbols
for zero, and it is impossible not to draw a
comparison in these conditions, between the
void of the spaces of the cosmos with the
multitude represented by the stars of the
firmament, the immensity of space and the
eternity of the celestial elements. As for the
ether ( *akasha ), this is said to be made up of an
infinite number of atoms ( *anu , *paramanu).
This is why, from a mythological, cosmological
and metaphysical point of view, the zero and
infinity have come to be united, for the Indians,
in both time and space. See Ananta, Shesha,
Sheshashirsha, Infinity (Indian mythological
representation of) and Serpent (Symbolism
of the).
But from a mathematical point of view,
however, these two concepts have been very
clearly distinguished, Indian scholars having
known that one equalled the inverse of the
other. See Infinity, Infinity (Indian concepts
of) and Indian mathematics (History of).
To sum up, the Indians, well before and
much better than all other peoples, were able
to unify the philosophical notions of void,
vacuity, nothing, absence, nothingness, nullity,
etc. They started by regrouping them (from the
beginning of the Common Era) under the
single heading *shunyata (vacuity), then (from
at least the fifth century CE) under that of the
*shunyakha (the sign zero as empty space left
by the absence of units in a given order in the
place-value system) before recategorising them
(well before the start of the seventh century CE)
under the heading of shunya-samkhya (the
“zero number”) [see Fig. D. 111. Once again,
this indicates the great conceptual advance and
the extraordinary powers of abstraction of the
scholars and thinkers of Indian civilisation.
The contribution of the Indian scholars is not
limited to the domain of arithmetic; by
opening the way to the generalising idea of
number, they enabled the rapid development
of algebra and consequently played an essential
role in the development of mathematics and all
the exact sciences. It is impossible to
exaggerate the significance of the Indian
discovery of zero. It constituted a natural
extension of the notion of ‘vacuity, and gave
the means of filling in the space left by the
absence of an order of units. It provided not
only a word or a sign, it also and above all
became a numeral and a numerical element, a
mathematical operator and a whole number in
its own right, all at the point of convergence of
all numbers, whole or not, fractional or
irrational, positive or negative, algebraic or
transcendental.
ZERO AND INFINITY. See Zero, Infinity,
Infinity (Indian concepts of), Infinity
(Mythological representations of). Serpent
(Symbolism of the), Zero (Graeco-Latin
concepts of), Zero (Indian concepts of) and
Indian mathematics (History of).
ZERO AND SANSKRIT POETRY. In India,
the use of zero and the place-value system has
been a part of the way of thinking for so long
that people have gone as far as to use their
principal characteristics in a subtle and very
poetic form in a variety of Sanskrit verse. As
proof, here is a quotation from the poet
Biharilal who, in his Satsai, a famous collection
of poems, pays homage to a very beautiful
woman in these terms : “The dot [she has] on
her forehead Increases her beauty tenfold, Just
as a zero dot (* shunya-bindu) Increases a
number tenfold. “ [See Datta and Singh, in:
AMM, XXXIII, (1926), pp. 220ff.].
First of all, it should be remembered that the
dot that the woman has on her forehead is none
other than the *tilaka (literally: sesame), a mark
representing for the Hindus the third eye of
‘Shiva, that is the eye of knowledge. While
young girls put a black spot between their
eyebrows by means of a non-indelible colouring
matter, married women put a permanent red dot
on their foreheads; it would seem then that the
homage was being paid to a married woman. It
is known that the dot (*bindu) figures among the
numerous numerical symbols with a value equal
to zero, and is even used as one of the graphical
representations of this concept. See Zero,
Shunya-bindu, Numeral 0 (in the form of a dot).
This is a very clear allusion to the arithmetical
operative property of zero in the place-value
system, because if zero is added to the right of
the representation of a given number, the value
of the number is multiplied by ten (see Fig. 23.26
and 27). Another quotation, taken this time
from the Vasavadatta by the poet ‘Subandhu
(a long love story, written in an extremely
elaborate language, swarming with word plays,
implications and periphrases):
“And at the moment of the rising of the Moon
With the darkness of the falling night,
It was as if, with folded hands
Like closed blue lotus blossoms,
The stars had begun straightway
To shine in the firmament (*gagana)...
Like zeros in the form of dots {* shunya-bindu).
Because of the emptiness ( *shunyata ) of the
*samsara,
Disseminated in space (kha),
As if they had been [dispersed]
In the dark blue covering the skin of the
Creator [= ‘Brahma],
Who had calculated their sum total
By means of a piece of Moon in the guise of
chalk.”
[See Vasavadatta of*Subandhu, Hall, Calcutta
(1859), p. 182; Datta and Singh (1938), p. 81.]
Here too the metaphor used leaves the reader
in no doubt; the void ( *shunya ) - which is
placed in relation to the emptiness (*shunyata)
of the cycle of rebirths (*samsara) - is also
implied in its representation in the form of a
dot (* shunya-bindu), as an operator in the art of
written calculation. These concepts really had
to have been part of the way of thinking for a
long time for the subtleties used in this way to
have been understood and appreciated by the
wider public of the time.
ZODIAC. [S]. Value = 12. See Rashi and Twelve.
511
CHAPTER 25
INDIAN NUMERALS AND
CALCULATION IN THE
ISLAMIC WORLD
As we saw in the previous chapter, it was indeed the Indians who invented
zero and the place-value system, as well as the very foundations of written
calculation as we know it today.
These highly significant inventions date back at least as far as the fifth
century CE.
However, it was not until five centuries later that the nine basic numer-
als appeared in Christian Europe.
Another two or three centuries elapsed before zero was first used in
Europe, along with the afore-mentioned methods of calculation, and it was
later still that these revolutionary new ideas were propagated and fully
accepted in the Western world.
Thus the Indian inventions were not transmitted directly to Europe:
Arab-Muslim scholars (amongst their numerous fundamental roles) played
an essential part as vehicles of Indian science, acting as “intermediaries”
between the two worlds.*
Therefore, before we proceed with our history, it is worth knowing a
little about the Arabs, in terms of their culture, their way of thinking, their
own science and their fundamental contributions to the evolution of
science the world over. This will give the reader a clearer idea of the condi-
tions under which this transmission of ideas took place, which led to the
internationalisation of Indian science and methods of calculation.
THE SCIENTIFIC CONTRIBUTIONS OF
ARAB-ISLAMIC CIVILISATION
In the century following the death of the prophet Mohammed the
Islamicised Arabs built up an enormous empire through their conquests.
At the beginning of the eighth century CE, the Empire stretched from the
Pyrenees to the borders of China, and included Spain, southern Italy, Sicily,
North Africa, Tripolitania, Egypt, Palestine, Syria, part of Asia Minor and
Caucasia, Mesopotamia, Persia, Afghanistan and the Indus Valley.
Words preceded by an asterisk have entries in the Dictionary (pp. 445-510).
THE SCIENTIFIC CONTRIBUTION OF ARABIC-ISI.AMIC CIVILISATION
Fig. 25 . 1 . Detail of a page from Al bahir fi ‘ilm a! hisab (The Lucid Book of Arithmetic) by As
Samaw'al ibn Yahya al-Maghribi (died c. 1180 in Maragha), a Jewish mathematician, doctor and
philosopher from the Maghreb, who converted to Islam (Istanbul, Aya Sofia Library, Ms. ar. 2,718.
See Rashed and Ahmed 1972]. This document, which uses “Hindi" numerals to reproduce what is
known as “Pascal's triangle'’, shows that Muslim mathematicians knew about the binomial expan-
sion (a + b)m, where "m "is a positive integer, as early as the tenth century. The author admits that
this triangle is not his, and attributes it to al-Karaji, who lived near the end of the tenth century
[Anbouba; Rashed in DSBJ.
INDIAN NUMERALS AND CALCULATION
THE ISLAMIC WORLD
Nevertheless, the advance came to a halt when it met with successful resis-
tance: in 718 by the Byzantine army near Constantinople; in 732 by Charles
Martel at Poitiers; and in 751 by the Chinese on the border ofSogdiana.
Once the political influence of the “Son of the Arabian Desert" fell into
decline, the Empire was controlled for nearly a century by the caliphs of the
Omayyad Dynasty (661-750), with Damascus as their capital. Power then
went to the Abbasid caliphs (750-1258) who made Baghdad their capital
in 772 and reigned over the empire for the next 500 years.
There followed a period of exaltation characteristic of expansion, and
this was a highly fertile era of cultural assimilation and scientific develop-
ment. Arabic culture dominated the world for several centuries, before
Mongol invasions, the Crusades, the division of the Empire and the anar-
chy of internal wars brought it to an end in the thirteenth century.
THE ASSIMILATION OF OTHER CULTURES
When the Arab nomads who had been converted to Islam left the desert to
conquer this immense territory, they lived from trading spices, medicines,
cosmetics and perfume. Their level of literacy and numeracy was very
basic. The little that they knew of science was based on practical applica-
tions involving simple formulae, and was often tinged with arithmology,
mysticism and all kinds of magical and divinatory practices.
Thus the first Islamicised Arabs initially possessed none of the intellec-
tual means they would need to realise their desire to conquer other lands
and to deal with the enormous revenue that taxation and capitation would
soon bring to this vast new Empire.
However, through their conquests and trade relations, they found them-
selves increasingly in contact with people from different cultures: Syrians,
Persians, Greek emigres, Mesopotamians, Jews, Sabaeans, Turks,
Andalusians, Berbers, peoples from Central Asia, inhabitants of the shores
of the Caspian, Afghans, Indians, Chinese, etc. Thus they discovered cul-
tures, sciences and technologies far superior to their own. They were quick
to adapt and to get to grips with the new concepts and knowledge, which
scientists, intellectuals and engineers from the conquered lands had accu-
mulated over the ages, and in some cases had developed to quite an
advanced level.
THE METROPOLIS OF NEAR EASTERN SCIENCE
BEFORE ISLAM
Long before the Arabic conquest, the philosophy of Aristotle and the sci-
ences of nature, mathematics, astronomy and medicine, according to the
doctrines of Hippocrates and Galen, were all taught in Syria and
Mesopotamia, notably at the schools of Edesse, Nisibe and Keneshre.
At the same time, Persia constituted an important crossroads and centre
of influence for the meeting of Greek, Syrian, Indian, Zoroastrian,
Manichaean and Christian cultures.
The Persian King Khosroes Anushirwan (531-579) sent a cultural
mission to India and brought many Indian scientists to Jundishapur.
Nestorian Christians, who had been expelled from the school at Edesse by
Byzantine orthodoxy, found refuge in the same town. This is also where
the Neo-Platonist philosophers of Athens (such as Simplicius who wrote
commentaries on the works of Aristotle and Euclid) were welcomed by
King Anushirwan when their academy was closed in 529 under the orders
of Emperor Justinian (527-565). It was at Edesse, Nisibe, Keneshre and
Jundishapur that Greek works were first translated into Syrian or Persian,
and that the first works in Sanskrit were discovered. The first translations
into Arabic were undertaken at Jundishapur shortly after the constitution
of the Islamic Empire [see L. Massignon and R. Arnaldez in HGS;
A. P. Youschkevitch (1976)].
BAGHDAD, FIRST ISLAMIC CENTRE OF
SCIENTIFIC LEARNING
The importance of these cultural and scientific centres gradually declined
during the Abbassid Dynasty, and so the town of Baghdad became the
centre of intellectual activity in the Near East, thus playing a vital role in
this history.
Founded in 762, then elevated to capital of the Arabic Empire in 772,
Baghdad was initially the obvious centre for international trade. Then,
owing to both its privileged location, and to the generous action of sovereign
patrons, such as caliphs al-Mansur (754-775), Harun al-Rashid (786-809)
and al-Ma’mun (813-833), whose subsidies contributed to the development
of science and culture in Islam, Baghdad became the most vivacious intellec-
tual centre of the East. This is where Arabic science truly began.
If we put together the religious and social conditions, we shall under-
stand the position of Islamic intellectuals and the fillip they gave to
intellectuals of all creeds and races, by their mobilisation for a common
labour in the Arabic tongue. For science is one of the Islamic city’s insti-
tutions. Not only do patrons encourage it, but caliphs work to create and
develop it. It is sufficent to cite Khalid, the “philosopher prince”, whose
actions were “perhaps legendary” or al-Mansur, the founder of
Baghdad, and al-Ma’mun “who eagerly sent out emissaries to look for
manuscripts and have them translated” [L. Massignon and R. Arnaldez].
“ARABIC” OR “ISLAMIC SCIENCE?
THE GOLDEN AGE OF ARABIC SCIENCE
One of the most outstanding periods in the history of science took place in
Islam between the eighth and thirteenth centuries of the Common Era.
This was at a time when Western civilisation was devastated by epi-
demics, famine and war, and was in no position to relay the cultural
heritage of Antiquity. The Arab-Muslim scholars were able to develop not
only mathematics, astronomy and philosophy, but also medicine, phar-
macy, zoology, botany, chemistry, mineralogy and mechanics.
Through a collective effort, the Muslims and the peoples conquered by
Islam collected together all the Greek works that they could find on philo-
sophy, literature, science and technology.
It is sufficient to cite the names of Euclid, Archimedes, Ptolemy,
Aristotle, Plato, Galen, Hero of Alexandria, Apollonius, Menelaus, Philo of
Byzantium, Plotinus and Diophantus to give an idea of the variety and
richness of the works that were translated into Arabic.
These translations and collected works grew in number and circulation,
as universities and libraries sprang up all over the Islamic world. Towns
such as Damascus, Cairo, Kairouan, Fez, Granada, Cordoba, Bokhara,
Chorem, Ghazni, Rey, Merv and Isfahan soon became intellectual and
artistic centres which were centuries ahead of the Christian capitals.
“ARABIC” OR “ISLAMIC” SCIENCE?
Arabic science is not necessarily the same thing as Muslim science. The
Arabic language was a vehicle for science, which, during that long period of
time, became the international language of the scholars of the Muslim
world, and consequently the intellectual link between the different races.
Amongst the diverse cultures which were conquered or influenced by
Islam was Persia, the birthplace of many brilliant minds, including al-
Fazzari, al-Khuwarizmi, al-Razi, Avicenna, al-Biruni, Kushiyar ibn Labban
al-Gili, Umar al-Khayyam, Nasir ad din at Tusi and Ghiyat ad din
Ghamshid ibn Mas’ud al-Kashi.
During the assimilation of Indian science, the Arabs were helped by
many Hindu Brahmans, who were often received at the court of Baghdad
by enlightened caliphs.
They were assisted by Persians and Christians from Syria and
Mesopotamia, who, being fervent admirers of Indian cultures, had gone so
far as to learn Sanskrit.
The Buddhists also greatly contributed, especially those converted to
Islam, such as Barmak who was sent to India to study astrology, medicine
and pharmacy and who, on his return to Muslim territory, translated many
Sanskrit texts into Arabic [see L. Frederic (1989)].
There were also non-Muslim Arabic scholars, such as the Christian and
Jewish intellectuals, who were often referred to as ahl al kitab, the “people
of books”, and whom the caliphs of Baghdad and Cordoba integrated to a
certain extent amongst the members of their empires, sometimes allowing
them the privileged right to hospitality which they called dhimma.
Often mistranslated as "tolerance”, dhimma really means “right to hos-
pitality” a “protection” that the caliphs sometimes gave to non-lslamic
residents. They did also show a certain “tolerance” towards their conquered
peoples, sometimes even “allowing” them to profess a different religion.
But this tolerance had its limits. The expression of ideas contrary to official
dogma, and even more, living by non-orthodox ideas, was vigorously
repressed. Non-believers were often considered to be “internal emigrants”
and not permitted to rise to the same rank as Muslims. The “Pact of ‘Umar”
even forced Jews and Christians to wear a “circlet”: a round piece of cloth,
yellow for the former and blue for the latter. Conversion to Islam offered a
number of social, pecuniary and fiscal advantages.
Even the brilliant culture of the Kharezm Province was discriminated
out of existence, as al-Biruni (a native of Kharezm) explained: “Thus
Qutayba did away with those who knew the script of Kharezm, who under-
stood the country’s traditions and taught the knowledge of its inhabitants;
he submitted them to tortures so that they were wrapped up in shadows
and no one could know (even in Kharezm) what had (preceded) or followed
the birth of Islam” [Youschkevitch],
The case of the Maghreb and especially that of Islamic Spain (before the
virulence of the Almohads) do still prove that “tolerance” was practised for
almost six centuries, in the sense of a greater liberty for Jews and for
Mozarabes (“Arabic” Christians) who lived peacefully according to their
own philosophies, organisations and traditions, with their synagogues,
churches and convents [V. Monteil (1977)].
Thus the Christian scholars of the Arabic world often worked as “cata-
lysts” by collecting, translating and commenting on, in Syriac or Arabic,
many scientific and philosophical works of Greek or Indian origin.
Amongst these men were: the astrologer Theophilus of Edesse, who trans-
lated many Greek medical texts into Syriac; the doctor Ibn Bakhtyashu,
head of the Jundishapur hospital; the doctor Salmawayh ibn Bunan; the
astronomer Yahya ibn Abi Mansur; the doctor Massawayh al-Mardini; the
philosopher, doctor, physician, mathematician and translator Qusta ibn
Luqa, from Baalbek; and the translators Yahya ibn Batriq, Hunayn ibn
Ishaq, Matta ibn Yunus and Yahya ibn ‘Adi.
As for Jewish intellectuals, or those issued from Judaism, it is worth
mentioning the astronomer Ya’qub ibn Tariq, one of the first scholars of
the Empire to study Indian astronomy, arithmetic and mathematics;
INDIAN NUMERALS AND CALCULATION IN T H F. ISLAMIC WORLD
astronomers Marshallah and Sahl at Tabari; the astrologer Sahl ibn Bishr;
the mathematician converted to Islam As Samaw’al ibn Yahya ibn ‘Abbun
al-Maghribi, who continued al-Karaji’s work on algebra; and the converted
doctor and historian Rashid ad din, who compiled a history of China.
There was also the philosopher-rabbi Abu ‘Amran Musa ibn Maymun
ibn ‘Abdallah, better known as Rabbi Moshe Ben Maimon or Maimonides,
whose encyclopaedic interests embraced not only philosophy, but also
mathematics, astronomy and medicine. Born in Cordoba in 1135, he was
initially one of the victims of religious persecution at the hands of the
Almohad sovereigns, who forced him to proclaim himself a Muslim for six-
teen years. The rabbi remained a Jew, and at the end of this period of time,
he went first to Fez, then to Palestine, before settling in Egypt where he
became a doctor at the court of the Fatimids in Cairo, until his death in
1204. He wrote many works on medicine ( Aphorisms of Medicine, Tract of
Conservation and of the Regime of Health and Rules of Morals being the only
ones to have survived). These works were mainly concerned with;
the treatment of haemorrhoids (a surgical operation which should
only be carried out as a last resort), of asthma by a dry climate, ner-
vous depression or “melancholy” through psychotherapy; recovery
being seen as a return to equilibrium; and diets, all embraced by a
global vision of the human being, always presented in a spirit of com-
passion and charity [V. Monteil (1977)].
He also wrote the famous Moreh Nebukhim ( Guide for the Lost), in which his
Aristotelian philosophy seeks to reconcile faith and reason, and he asserts
himself as one of the first intermediaries between Aristotle and the doctors
of scholasticism. Another of his fundamental contributions, this time to
Judaism, was his Commentary on Mishna (1158-1165) and his Second Law or
the Strong Hand (1170-1180). Before they were even translated into Hebrew
or Latin, the medical and philosophical works of Maimonides were generally
written in Arabic first. In other words, despite their profound attachment to
Judaism, scholars such as Maimonides were authentic Arabic thinkers.
After the Abbasid school of Baghdad (ninth to eleventh century CE),
there came the schools of Toledo and Seville, and Jewish scholars such as
Yehuda Halevi, Salomon ibn Gabirol (Avicebron) and Abraham ibn Ezra or
Abraham bar Hiyya (who would have spoken Hebrew, Arabic, Castilian
and even Latin or Greek) acted as intermediaries between the Arabic and
Christian worlds.
Of course, Arabic science was also and above all the creation of
Muslim scholars. Amongst these men were: al-Fazzari, al-Kindi, al-Razzi,
al-Khuwarizmi, Thabit ibn Qurra, al-Battani, Abu Kamil, al-Farabi,
al-Mas’udi, Abu’l Wafa, al-Karayi, al-Biruni, Ibn Sina (Avicenna), Ibn al-
514
Haytham, ‘Umar al-Khayyam, Ibn Rushd (Averroes) and Ibn Khaldun (see
the Chronology, pp. 519fF. for further information).
Islamic religion played an important role in the scientific discoveries of
this civilisation. The Koran preaches humanism in the search for knowl-
edge; one of the necessities imposed by the study of this holy book and of
Islamic thought is “the development of scientific research where
Revelation, Truth and History are considered in their dialectic relationship
as structural terms of human existence” [M. Arkoun (1970)].
The Koran frequently encourages the faithful to look for signs of proof
of their faith in the heavens and on Earth: “Search for science from the
cradle to the grave, even if you have to travel as far as China . Those that
follow the path of scientific research will be led by God on the path to
Paradise” [L. Massignon and R. Amaldez in HGS]. (We have not been able
to find the source for this advice, which many attribute to Mohammed. But
it would seem that it forms an integral part of Islamic culture, at least since
the time of Ibn Rushd.)
It is true that the science in question here is knowledge of religious Law
(‘ilm), but in Islam this is not set apart from secular science. Thus we find a
whole series of hadith about medicine, remedies and the legitimacy of their
use. Moreover, scholars and philosophers have not hesitated to quote the
texts in order to defend their activities.
Averroes wrote in his Authoritative Text: “It is clear in the Koran that the
Law invites rational observation of living beings in the search for an under-
standing of these beings through reason.”
This is the opinion of all Muslims who have accepted and cultivated sci-
ence. It is because the Koran invites the faithful to contemplate the power
of Allah in the organisation of the universe that astrology has always been
considered the “highest, noblest and most beautiful of sciences” [al-
Battani] in the Islamic world.
The patient assimilation of observations and calculations relative to the
positions of the planets, the moon’s phases, equinoxes, etc., v/ere often
directly related to the demands of Islamic religion: the calculation of the
exact times of the ritual prayer of the ‘asr, the dates of religious ceremonies,
the month of Ramadan, orientation towards Mecca, etc.
This is why, despite the considerations above, the science and culture of
the Islamic world should be more accurately termed “Arab-Islamic”
THE SPREAD OF SCIENTIFIC KNOWLEDGE:
ANOTHER ACHIEVEMENT OF ISLAM
Other sciences existed before Islam (in Ancient Greece, Persia, India,
China, etc.), but although these were all mainly concerned with the same
515
THE DEVELOPMENT OF ARABIC ISLAMIC SCIENCES
problems, they all had their own unique way of dealing with them. In other
words, before Islam, there was no universal science as we know it today.
In fact, different cultures sought to preserve their knowledge and keep it
a secret from the outside world. An example of this is the Neo-
Pythagoreans in Greece.
Part of the reason why the Muslims were responsible for the unification
of science is their success in conquering other peoples.
International trade played an important role, as did the Arabic geogra-
phers, travellers and cosmographers, translators, philologists, lexicographers
and writers of commentaries:
By describing different areas of the globe, those unusual men described
the marvels of nature, products of the earth, fauna, agriculture and crafts.
This was a considerable source of information. Some geographers were also
great scholars, experts in all fields, such as the famous al-Biruni
[Massignon and Arnaldez],
Another factor was the cultural assimilation by the Muslims of the most
diverse of cultures: this began at the time of the caliphs of Damascus, but it
was not until the time of the enlightened caliphs of Baghdad and Cordoba
that the results were felt.
The “tolerance” of these caliphs towards other cultures, beliefs, customs
and traditions for nearly six centuries was also an important factor.
The promotion of study and research in the Koran has already been
mentioned in this chapter. This was not only a fundamental condition for
the development of Arabic Islamic science, but also one of the main causes
for Islam’s ready acceptance of the most diverse of cultures. (But it should
also be noted that Arab-Islamic science, despite its universal nature, was
always oriented towards knowledge of divine Law. It is necessary to wait
until the European Renaissance before science gains the non-religious
character we now recognise.)
THE DEVELOPMENT OF ARABIC
ISLAMIC SCIENCES
The Islamic conquerors were not always in favour of science and culture.
Caliph ‘Umar (634-644 CE) ordered the destruction of countless works
seized in Persia. His argument was as follows: “If these books contain the
key to the truth, Allah has given us a more reliable way to find it. And if
these books contain certain falsehoods, they are useless” [see A. P.
Youschkevitch (1976)].
There were certainly other similar cases of religious or xenophobic
opposition, leading to great cultural losses. In the Islamic world, scholars
suffered from the whims of totalitarian leaders. They had to avoid direct
confrontation with official dogma if they did not want to lose their state
subsidies and risk even greater repression. At the end of the eleventh cen-
tury, the famous poet, astronomer and mathematician Omar Khayyam
reported, in his Mathematical Treatise :
We have witnessed a decline in scholarship, few scholars are
left, and those who remain experience vexations. Their troubled
times stop them from concentrating on deepening and bettering
their knowledge. Most so-called scholars today mask the truth
with lies.
In science, they go no further than plagiarism and hypocrisy and
use the little knowledge they have for vile material ends. And if they
come across others who stand apart for their love of the truth and
rejection of falsehood and hypocrisy, they attack them with insults
and sarcasm.
But according to Youschkevitch, “this situation could not in the long term
stop the triumph of scientific progress. Schools, libraries and observato-
ries were built in the cities. To make a name for themselves, enlightened
sovereigns set up academies similar to those founded by European mon-
archs in the seventeenth and eighteenth centuries. The transmission of
knowledge was thus assured; but it was only later that the discovery of
printing facilitated it.”
However, such opinions were exceptional and not held by caliphs ruling
later in Islam. In fact, the role of Islam and of Arabic scholars in the fields
of science and culture can never be overstated.
The famous library of Alexandria, the most important one of Ancient
Greece, was pillaged and destroyed twice: the first time in the fourth cen-
tury CE by the Christian Vandals, and the second time (through a perverse
paradox of history) by the Muslims in the seventh century. Many original
manuscripts of inestimable value disappeared. Many Greek literary and
scientific masterpieces would have been lost forever if they had not been
collected and translated into Arabic.
It was thanks to the work of the North African philosopher Ibn Rushd
(Averroes) that Saint Thomas Aquinas could study and understand the
importance of Aristotle’s work. Similarly, it is thanks to Avicenna that
Albertus Magnus could develop the philosophy of universality. It is largely
thanks to the work of Arabic translators that the works by Ancient Greeks
are known to us today.
Moreover, the Arabs have never hesitated to underline the importance
of Greek science and to express their admiration for it: “The language of the
Hellenistic people is Greek; it is the most vast and the most robust of lan-
guages” [Sa’id al-Andalusi, Tabaqat al umam, in R. Taton (1957), I, p. 432].
Greek culture played a huge part in the development of Arabic science.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
But it would be a mistake to believe that the latter was nothing more than
the continuation of Greek science. This would be as far-fetched as the opin-
ion that “India, and not Greece, formed the religious ideals of Arabia and
inspired its art, literature and architecture”
Of course the framework of Arabic scientific thinking constituted an
obvious extension of, and was largely based upon Hellenic science.
However, the Arabs used the discoveries of Ancient Greece as a source of
inspiration and actually expanded upon them. Moreover, Greece was not
the only civilisation to inspire the Arabic scholars. They were also inter-
ested in oriental culture, from which they borrowed different elements
which they adopted to suit their needs.
Thus their numeral alphabet was forged from a combination of Jewish,
Greek and Syriac systems by adopting the corresponding principle to the
twenty-eight letters of their own written alphabet.
Through the Christians of Syria and Mesopotamia they discovered the
place-value system of the Babylonians, which they used in their tables and
their astronomical texts to record sexagesimal fractions. Through trading
with Persia and parts of the Indian sub-continent, they also came into con-
tact with Indian civilisation. Thus they discovered Indian arithmetic, algebra
and astronomy. Sa’id al-Andalusi (see above) expressed his admiration for
Indian culture. He recognised its precedence over Islam and went as far as to
call it “a mine of wisdom and the source of law and politics” He also wrote
that “the Indian scientists devote themselves to the science of numbers (‘ ilm
al ‘adad), to the rules of geometry, to astronomy and generally to mathemat-
ics .. . they are unrivalled in medicine and the knowledge of treatments”. His
conclusion, however, is a little subjective. He claims that the intellectual tal-
ents and qualities of the Indians are nothing more than the product of “good
fortune ( hazz :)”, due to “astral influences” [R. Taton (1957), I, p. 432].
The Chinese were another foreign influence. After the battle of Talas in
751, the Arabs learned the secrets of making paper from linen or hemp
from their Chinese prisoners. The first factory was built in Baghdad c. 800.
It would be another four centuries before paper was used in Europe,
through the intermediary of Spain.
At the beginning of the fourteenth century, Rashid ad din, grand vizier
of the Mongolian sultan Ghazan Khan a Tabriz, and himself a converted
Jew, compiled a library of 60,000 manuscripts, many of which came from
Chinese and Indian sources.
In his Universal History (Jami'at tawarikh), he carefully described how
Chinese characters were engraved on wood, and gave their transcription
in Arabic. He translated extracts from the best known medical works of
China and Mongolia into Arabic and Persian, including Mejing, a text on
sphygmology (or science of the arterial pulse) by Wang Shuhe (265 -
516
317), which identifies four standard methods of medical examination,
namely observation, auscultation, interrogation and palpation. These
would not be studied in Western Europe until the eighteenth century [see
V. Monteil (1977)].
However, the Arabs were not content merely to preserve the discoveries
of Greece, Babylon, China and India. They wanted to make their own con-
tribution to the world of science.
As they carefully collected, translated and studied works from the past,
they added various commentaries, after mixing explanations with original
developments, and always maintaining a critical perspective which rejected
fixed dogmatism [see A. P. Youschkevitch (1976)].
Thus in mathematics, Greek methods were often mixed with Indian
methods, sometimes with Babylonian ones or even, at a later date, with
Chinese approaches.
The Arabs combined the strict systematisation of Greek mathematics
and philosophy with the practicality of Indian science. This enabled them
to make significant progress in the fields of arithmetic, algebra, geometry,
trigonometry and astronomy. Through collecting, propagating and
teaching the use of Indian numerals and calculation, and by pushing the
study of certain remarkable properties of numbers towards the first seeds
of a theory of numbers, the Arabs made considerable progress in the field
of arithmetic.
Scholars such as al-Khuwarizmi, Abu Kamil, al-Karaji, As Samaw’al al-
Maghribi, ‘Umar al-Khayyam, al-Kashi and al-Qalasadi led arithmetic
towards algebrised operations.
The Arabs (and more generally the Semites) “personalised” the number.
Instead of an object which had various properties, it became an active
being. They did not see numbers as being static and limited, as the Greeks
did. The Arabs were interested in the ordinal, rather than the cardinal
numbers: they were not deterred as the Greeks were by the irrational [see
L. Massignon and R. Arnaldez].
The assimilation of the classical heritage allowed the mathematicians
of Islamic countries to develop algorithms and corresponding problems,
and thus achieve a higher level than that reached by Indian or Chinese
mathematicians. It also enabled them to find more efficient ways to
resolve and generalise these problems than the Chinese and the Indian
methods. Where the latter were content to establish a specific rule of cal-
culation, the mathematicians of Islam often managed to develop an entire
theory [A. P. Youschkevitch (1976)].
In short, the work of the Arabic scholars involved objectivity, the ques-
tioning of the doctrines of the ancient scientists and systematic recourse to
analysis, synthesis and experimentation.
517
THE ARABIC LANGUAGE
The progress of sciences, in terms of knowledge, is dependent on the sci-
entific mind . Perhaps some thought that all of science had already been
discovered, and that all that remained to do was to assimilate all the infor-
mation. But this gathering of knowledge was in fact an excellent prelude to
methodical research and progress. The need for an inventory led to classifi-
cations of the sciences (such as those of al-Farabi or Avicenna) which was
enough to cause an evolution of the concept of science. Under the influence
of Plato and Aristotle, the Ancient Greeks classified the sciences according
to their method, and the degree of intelligibility of their purpose. The
Arabs took a more straightforward stance: the sciences exist and they must
be ordered so that none is forgotten. The lack of conceptual analysis which
characterises Arabic classification of the sciences was in fact an advantage
from a purely scientific point of view. Knowledge itself promotes learning
and marks out the direction to follow towards the acquisition of further
knowledge [L. Massignon and R. Arnaldez],
For the Arabs, then, to know was not to contemplate, but to do; in
other words to verify, challenge, experiment, observe, rethink, describe,
identify, measure, correct, even complete and generalise. This is the Arab
influence on science: it became an operating science following the develop-
ment of “scientific reason” The Arabs had a great deal of curiosity, love
and estimation for knowledge [L. Massignon and R. Arnaldez], which
meant that they not only preserved and transmitted the science of
Antiquity, but they transformed it and established it along new lines,
giving it a new lease of life and originality.
THE ARABIC LANGUAGE: THE AGENT AND
VEHICLE OF ISLAMIC SCIENCE
Right from the start of the history of Arab-Islamic science, anything con-
cerning the science had to be written in Arabic if it was to be of any
consequence, this language having become the permanent intellectual link
between the scholars of various origins during this long period of time.
For many philosophers, mathematicians, physicians, chemists, doctors
and astronomers, this language was more than a mere obligation: it was a real
passion. It was the preferred language for expressing science and knowledge.
For example, the Persian scientists Avicenna and al-Biruni, rather than
writing in Turkish or Persian wrote in Arabic, despite having been born in
Kharezm, to the north of Iran in what is now Uzbekistan (formerly in the
USSR). Al-Biruni explains his preference for Arabic in his Kitab as saydana
(“ Treatise on Drugs" [see V. Monteil (1977), p. 7]:
It is in Arabic that, through translation, the sciences of the world were
transmitted [to us] and were embellished and found a place in our
hearts. The beauty of the Arabic language has circulated with them in
our arteries and veins. Of course, every nation has its own language,
the one used for trading and talking to our friends and companions.
But personally I feel that if a science found itself eternalised in my own
mother-tongue, it would be as surprised as a camel finding itself in a
gutter of Mecca or a giraffe in the body of a thoroughbred. When
I compare Arabic with Persian (and I am equally competent in both
languages) I admit that I prefer invective in Arabic to praise in Persian.
You would agree with me if you saw what happens to a scientific work
when it is translated into Persian: it loses all its brilliance and has
less impact, it becomes muddled and quite useless. Persian is only
good for transmitting historical stories about kings or telling tales
at evening gatherings.
Of course, al-Biruni’s description of Farsi is totally unjustified. Many
Muslim scholars from Persia, Afghanistan and the Indus Valley also wrote
in Arabic, although Persian is perfectly capable of expressing any concept,
as well as the rigour, nuances and foundations of scientific thought.
However, al-Biruni’s preference for Arabic was not brought about by
chance, and was certainly not due to a passing fad.
In terms of structure, Arabic became a much richer language and gradu-
ally acquired its scientific character in order to meet the demands of the
translation of foreign works and the transposition of scientific texts.
When a scientific text is translated from its original language into an
equally well-equipped language, there might be grammatical problems but
there are no technical or conceptual difficulties. This was not the case when
Greek was first translated into Arabic: vocabulary had to be created, or
existing words adjusted to meet the needs of science. There was often an
intermediate Syriac word which prepared the way for Arabic. The creation
of the scientific Arabic language was not only philological, it also involved
two scientific methods: the identification and verification of concepts
[L. Massignon and R. Arnaldez].
It was in this scientific spirit that the lexicographers made an inven-
tory of the Arabic language, as scholars had made an inventory of
knowledge by attempting to classify different fields of learning through
rethinking and evaluating concepts, then organising them in relation
to one another. As for those who translated or commented on texts, they
looked for Arabic equivalents for foreign terms in lexicons and in nature,
and also in the different elements of knowledge, either to introduce
new words and concepts, or even to express new ideas using the most
ancient vocabulary.
This is how Arabic acquired its unique aptitude for expressing scientific
thought, and for developing it in the service of the exact sciences.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
This language, which was originally considered to be the language of the
Revelation and the fundamental criterion for anyone wishing to belong to the
Muslim religion and the Islamic community (Umma), became not only the
vehicle of international science, but also and above all the essential agent of
the Renaissance and the dominant factor in the Arabic scientific revolution.
OTHER ARABIC CONTRIBUTIONS TO THE
WORLD OF SCIENCE
The Arabs also contributed significantly in the field of technology, develop-
ing upon the knowledge passed down by the Ancient Greeks.
The Greek school (which had turned out such prestigious scholars as
Archimedes, Ctesibios, Philon of Byzantium and Hero of Alexandria) had
seen the discovery of quite advanced mechanical technology: the endless
screw, the hollow screw, pulley blocks, mobile pulleys, levers and weapons;
clepsydras (types of clocks activated by water); astrolabes (for observing the
positions of the stars and determining their height above the horizon);
the construction of automata (devices capable of moving by themselves); the
use of the odometer (an instrument designed to measure distances,
comprising a series of chains and endless screws, moved by chariot wheels
and pulling a needle along a graduated scale which indicated the distance
travelled); etc. [see A. Feldman and P. Fold (1979); B. Gille (1980, 1978);
C. Singer (1955-7); D. de Sofia Price (1975)].
The Greeks of Byzantium carried on the work of the Greeks of
Alexandria, and, to a certain extent, were one of the transmission links
with mediaeval Europe.
However, the handing-on of the Greek tradition was also and above all
the work of the engineers of the Muslim world. Here again, the Arabs gath-
ered all the information, then made improvements and even innovations.
Under orders from the caliph Ahmad ibn Mu’tasim, Qusta ibn Luqa al-
Ba’albakki translated Hero of Alexandria’s work on the traction of heavy
bodies into Arabic; others translated or were inspired by the work of Philo
of Byzantium [see B. Gille (1978)]. The Arabs also distinguished themselves
in the art of clock-making. They even created their own inventions, above
all in the field of automata and astronomical clocks, this being not only the
legacy of the Greeks but probably also the Chinese, who were likewise
experts in this field.
The following were amongst the most famous of the Arabic-Muslim
engineers: the Banu Musa ibn Shakir brothers, whose works notably
include Al’alat illati tuzammi bi nafsiha ( The Instrument Which Plays Itself,
written c. 850), largely inspired by the ideas of Hero of Alexandria; Ibn
Mu’adh Abu ‘Abdallah al-Jayyani, who wrote Kitab al asrarfinata’ij al afkar,
518
which describes several water clocks (second half of the tenth century);
Badi’al-Zaman al-Asturlabi, famous for the automata he built for the
Seleucid monarchs (first half of the twelfth century); ‘Abu Zakariyya Yahya
al-Bayasi, known for his mechanical pipe organs (second half of the twelfth
century); and Ridwan of Damascus, made famous by his automata acti-
vated by ball-cocks (1203).
The most famous and most productive of the Arabic engineers was
lsma’il ibn al-Razzaz al-Jazzari, whose Kitab fi ma’rifat al hiyat al handasiyya
( Book of the Knowledge of Ingenious Mechanical Instruments, 1206) shows a
perfect knowledge of Greek traditions and records apparently hitherto
unknown innovations. This work not only contains the plans for construct-
ing perpetual flutes, water clocks, mechanical pump systems for fountains,
automata activated by ball-cocks and movements transmitted by chains
and cords, it also contains descriptions of sequential automata, mainly
using camshafts, thus transforming the circular movement of a type of
crankshaft into the alternating movement of distribution instruments.
As well as the diverse instruments, there is also the astrolabe which
became known in the West (at the same time as the “Arabic” numerals)
thanks to Pope Sylvester II (Gerbert of Aurillac), who acquired
the astrolabe from the Arabs when he lived as a monk in Spain from
967 to 970 CE.
There was also the compass, that ingenious device which has a magnetic
needle and made navigation possible. It was invented by the Chinese at the
beginning of the Common Era and was retrieved by the Arabs (in all likeli-
hood in 752 during the battle of Talas), who in turn passed it on to the
Europeans during the Renaissance.
The scholars of Islam also made their mark on the science of optics,
which led to the invention of the mirror:
Optics was particularly studied by Ibn al-Haytham. His work
included physiological optics and a philosophical discussion of the
nature of light, but he is known above all for his research in the field
of geometry. He knew about reflection and refraction; he experi-
mented with different mirrors, planes, spheres, parabolas, cylinders,
both concave and convex. He wrote a text about the measurement of
the paraboloid of revolution. He embarked on actual physical
research through his work on the light of the stars, the rainbow, the
colours, shadows and darkness. For a scientist of this calibre there
was no fixed distinction between mathematical sciences and natural
sciences, and Ibn al-Haytham was always shifting between the two
[L. Massignon and R. Arnaldez],
Alchemy, too, was a fanciful art, the aim of which was to find the so-called
“philosopher’s stone” from which could be extracted a miraculous property
519
SIGNIFICANT DATES IN THE HISTORY OF A R A B I C - 1 S I. A M I C CIVILISATION
which would at once cure all illnesses, give eternal life and transform
metals: it was a vain science whose basis was denounced by great minds
such as al-Kindi, Avicenna and Ibn Khaldun. However, as Diderot pointed
out, alchemy, in spite of its frivolous nature, “often led to the discovery of
important truths on the great path of the imagination” By stripping it of
some of its arithmology and magic, the early Arabian scholars began to
prepare the way for the creation and expansion of modern chemistry.
THE FORERUNNERS OF
CONTEMPORARY SCIENCE?
Certain Arabic scholars, such as al-Biruni and Averroes, and doctors such
as ‘Ali Rabban at Tabari and Ibn Massawayh were well ahead of their time.
Perhaps the most significant contribution of the Arabic world, however,
came from the historian ‘Abd ar Rahman ibn Khaldun (who was born in
Tunis in 1332 and died in Cairo in 1406), a visionary of modern science,
who was gifted with truly extraordinary insight. One only need read this
extract from his Prolegomena to appreciate his foresight: “The human world
is the next step after the world of apes ( qirada ) where sagacity and wisdom
may be found, but not reflection and thought. From this point of view, the
first human level comes after the ape world: our observation ends here”
[s eeMuqaddimah, p. 190; V. Monteil (1977), p. 101].
This is a surprising opinion for a time when such ideas were practically
inconceivable. It would not be until 1859, in Darwin’s Origin of Species, that
these ideas would be presented and even then some time elapsed before
they were accepted and developed in the Western world.
Thus, we can see how much Europe owes to this civilisation which is
largely unknown or at least unrecognised by the Western public.
SIGNIFICANT DATES IN THE HISTORY OF
ARABIC-ISLAMIC CIVILISATION
The following chronology is divided into sections, each representing
half a century in the golden age of Arab-Islamic civilisation. Its aim is to
give an idea of cultural, literary, scientific and technical activity which
ran parallel to military and religious events. The list (which, of course,
is not exhaustive) is of scholars and intellectuals, including the most
illustrious poets, writers, mathematicians, physicians, astronomers,
geographers, engineers, chemists, naturalists and doctors of the Arab
world. In some cases, a summary of their fundamental contributions is
supplied [see A. A. al-Daffa (1977); M. Arkoun (1970); O. Becker and J.
E. Hoffman (1951); E. Dermenghem (1955); EIS; O. Fayzoullaiev (1983);
A. Feldman and P. Fold (1979); L. Frederic (1987 and 1989); L. Gille
(1978); C. Gillespie (1970-80); L. Massignon and R. Arnaldez in HGS;
A. Mazaheri (1975); A. Mieli (1938); V. Monteil (1977); R. Rashed
(1972); G. Sarton (1927); C. Singer (1955-7); H. Suter (1900-02);
G. J. Toomer; K. Vogel (1963); H. J. J. Winter (1953); A. P.
Youschkevitch (1976)].
Second half of the sixth century
571 CE. “Year of the Elephant” Supposed birth-date of the prophet
Mohammed.
First half of the seventh century
612. Year of the “Revelation” when Mohammed began his prophecy in
Mecca.
622. Mohammed and the first followers of the new faith, the “Muslims” ( al
muslimin, from the Arabic word “believers”) were expelled from
Mecca. They found refuge in Yahtrib, which then became the “Town”
of the Prophet or “Medina” ( madinah ). This year marked the begin-
ning of the Muslim calendar, which is called the Hegira (from hijra,
"expatriation”).
624. Battle of Badr. The qibla is established, the symbol of the “new people
of God”. Beginning of the “Muslim institutions”
628. Mecca is seized by Mohammed and his followers.
632. The death of Mohammed.
632-661. Time of the “orthodox” caliphs (Abu Bakr, ‘Umar, ‘Uthman and
‘Ali); capital: Medina.
632-634. Abu Bakr is caliph, the successor of Mohammed.
634. The conquest of Syria by the Arabs, who defeat Heraclius’s Byzantine
army near Jerusalem.
634-644. ‘Umar (Omar) is caliph.
635. The Arabs take Damascus and overturn the Persian Empire.
637. Battle ofKadisiya, defeat of the Persians.
637-638. Founding of the towns Basra and Kufa. The writing of the Koran
begins.
637-640. Conquest of Mesopotamia, Khuzistan, Azerbaijan and Media.
638. Jerusalem is surrendered to Omar.
639. Arabs attack Armenia.
640. The conquest of Palestine.
641. Egypt is conquered by the Arabs.
642. Victory over the Persians.
642-646. The Arabs attack Armenia.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
643. The Arabs complete their conquest of Persia and Tripolitania, and
arrive in Sind (now Pakistan).
644-656. Rule of ‘Uthman (Ottman).
647. Barka in Tripolitania is taken (now Libya).
649. Cyprus is conquered by the Arabs.
Second half of the seventh century
655. Battle of Lycia, where the Muslim fleet destroys the Byzantine fleet.
656-661. Rule of ‘Ali, the son-in-law of the prophet.
657. Battles of Jamal and Siffin, where the followers of ‘Ali (then considered
to be the first man converted to Islam) fought the followers of
Mu’awiyah (rival and hostile descendants of Mohammed’s family).
661-750. The Omayyad Dynasty. Capital: Damascus. Rule henceforth
becomes hereditary. Effort to centralise Omayyad administration.
665. First attacks in the Maghreb.
670. Successful campaigns in North Africa. Founding of the town of
Qairawa (Kairouan, in present-day Tunisia). Appearance and begin-
ning of Shiite and Kharajite movements.
673-678. Siege of Constantinople by the Arabs.
680. Death ofHusayn in Kerbala. Martyrdom of the Shiites.
695. First use of coins by the Arabs.
Culture, Science and Technology
Period of:
• the poet Imru’ al-Qays.
• the poet Yahya ibn Nawfal al-Yamani.
• Khalil ibn Ahmad (one of the founders of Arabic poetry).
First half of the eighth- century
707. Development of political, “courtly” and urban poetry. First theological-
political discussions.
707-718. The Muslims seize the mouths of the River Indus (Sind) and part
of the Punjab (India).
709. The Maghreb surrenders to Arabic domination.
711. Musa Ben Nusayir dispatches Tariq ibn Ziyad, who crosses the Straits
of Gibraltar (called Jabal Tariq), then successively occupies Seville,
Cordoba and Toledo, before continuing north.
712. Arabic conquest of Samarkand (now Uzbekistan).
715. The Arabic Empire extends its confines to China and the Pyrenees.
718. The Arabs meet resistance from the Byzantine army at Constantinople.
Thus the Arabic advance comes to a halt at the Taurus mountains.
720. The Arabs cross the Pyrenees and penetrate the kingdom of the
Franks. First Arabic colony in Sardinia.
520
732. The Arabs are defeated at Poitiers by Charles Martel; the end of the
Arabic advance in Europe.
Culture, Science and Technology
Period of:
• the Christian doctor Yuhanna ibn Massawayh.
• the poets al-Farazdaq, al-Akhtal and Jarir.
• the mystic thinker Hasan al-Basri.
• the Arabic version of the Kalila wa Dimna fables by Ibn al-Muqafa’
(ancient Persian tale inspired by the Indian *Pahchatantra).
• the first paintings of Islamic art.
Second half of the eighth century
750. Abu’l ‘Abbas founds the Dynasty of the same name.
750-1258. Abbasid Dynasty. Capital: Baghdad (from 772).
751. Battle of Talas in present-day ex-Soviet Kyrghyzstan, where the
Chinese armies are defeated by Arab troops. But Chinese reprisals
later stop the Arabic advance at the limits of Sogdania.
754-775. Reign of the Abassid caliph al-Mansur.
756-1031. Omayyad Dynasty in Spain. Capital: Cordoba.
760. Arabic expedition against Kabul (in Afghanistan).
761-911. Rustamid Dynasty in Tiaret.
762. Caliph al-Mansur founds the town of Baghdad.
768. Sind is governed by the Arabs.
786. The Arabs seize Kabul.
786-809. Reign of the Abassid caliph Harun al-Rashid.
786-922. Idrissid Dynasty in the Maghreb. Capital: Fez.
795. Disorder in Egypt.
Culture, Science and Technology
Period of:
• the introduction of Indian science, numerals and calculation to the
Islam world.
• the Persian astronomers Abu Ishaq al-Fazzari, and Muhammad al-
Fazzari (his son), and of Jewish astronomer Ya’qub ibn Tariq. These
are the men who would translate the Brahmasphutasiddhanta by
Brahmagupta and study, for the first time in Islam, Indian astronomy
and mathematics.
• the Persian astrologer al-Nawabakht and his son al-Fadl, chief
librarian of caliph Harun al-Rashid.
• the Jewish astronomer Mashallah.
• the Christian Abu Yahya, translator of Tetrabiblos by Ptolemy.
521
SIGNIFICANT DATES IN THE HISTORY OF A R A B I C - I S I. A M I C CIVILISATION
• the Persian Christian Ibn Bakhtyashu’, first of a large family of doc-
tors, head of the hospital at Jundishapur.
• the Sabaean alchemist Jabir ibn Hayyan (Gebir in mediaeval Latin)
who studied chemical reactions and bonds between chemical bodies.
• the alchemist Abu Musa Ja’far al-Sufi who wrote that there are two
types of distillation, depending on whether or not fire is used.
• the Christian astrologer Theophilus of Edesse, translator of Greek
works.
• the philologist and naturalist al-Asma’i.
• Abu Nuwas, one of the greatest Arabic poets.
• the poet Abu al-’Atahiya.
• the mystic thinker Abu Shu’ayb al-Muqafa.
• and the first production of paper in Islam.
End of the eighth century
At this time, the provinces of Africa, the Maghreb and Spain freed them-
selves from the links with the caliph of Baghdad.
Ninth - tenth century
This was the time of the development of the Sunni (Hanbali, Maliki,
Hanafi, Shaft, Mutazili, Zahiri, etc.) and Shiite (Immami, Zayidi, Ismaeli,
etc.) religious movements and of the mystical philosophy of the Sufi; popu-
lar Islam prevailed over classical Islam, which was reduced to a few
common cultural and religious signs. This time was also marked by the
rapid development of Arab-Islamic civilisation in all fields. It was also the
era when the Alflayla wa layla, the Thousand and One Nights was written
(anonymous masterpiece of Arabic literature, a collection of tales and leg-
ends, such as those of Scheherazade, Ali Baba, Sinbad the sailor, the magic
lamp, etc., which have become an integral part of universal mythology).
First half of the ninth century
800. Charlemagne is named Emperor of the West.
800-809. Aghlabite Dynasty in “Iffiqiya” (territory composed of present-
day Tunisia and part of Algeria).
813-833. Reign of the Abassid caliph al-Ma’mun who, as a grand patron,
would favour cultural and scientific translations.
820-999. Independent indigenous dynasties in eastern Persia: Tahirid
(820-873), Saffarid (863-902), Samanid (874-999).
826. Crete is taken by the Arabs.
827-832. Sicily is taken.
846. Sacking of Rome by the Saracens.
Culture, Science and Technology
Period of:
• the founding of the “House of Wisdom” ( Bayt al-Hikma) in
Baghdad, a kind of academy of sciences, where the cultural heritage of
Antiquity was welcomed with enthusiasm and where the development
of Arab-Islamic science began.
• the Persian astronomer and mathematician al-Khuwarizmi. His work
on the Indian place-value system and on algebra with quadratic equa-
tions contributed greatly to the knowledge and propagation of Indian
numerals, methods of calculation and algebraic procedures, not only in
the Muslim world but also in the Christian West. He also wrote an
interesting series of problems with examples taken from the methods of
merchants and executors which required a great deal of mathematical
skill due to the complex structure of the legacies of the Koran.
• the mathematician ‘Abd al-Hamid ibn Wasi ibn Turk.
• the Christian translator Yahya ibn Batriq.
• al-Hajjaj ibn Yusuf, translator of Euclid’s Elements.
• the astronomer and mathematician al-Jauhari, who carried out
some of the first work on the parallel postulate.
• the converted Jewish astronomer Sanad ibn Ali, who had the obser-
vatory built in Baghdad.
• the philosopher al-Nazzam.
• the great philosopher and physician al-Kindi, who was interested in
logic and mathematics, and sought to analyse the essence of definition and
demonstration; he also wrote about geometrical optics and physiology.
• the philosopher al-Jahiz, author of the famous Book of Animals.
• the Persian Christian astronomer Yahya ibn Abi Mansur, who drew
up Al zij al muntahan ( Established Astronomical Tables).
• the astronomer Abu Sa’id al-Darir, from the Caspian region, who
wrote about the course of the meridian.
• the astronomer al-Abbas, who introduced the tangent function.
• the astronomer Ahmad al-Nahawandi of Jundishapur.
• the astronomer Hasbah al-Hasib, from Merv, who established a
table of tangents.
• the astronomer al-Farghani, who wrote an Arabic version of
Ptolemy’s Almagestus.
• the astronomer al-Marwarradhi, from Khurasan.
• the astronomer ‘Umar ibn al-Farrukhan, from Tabaristan.
• the Jewish astronomer Sahl at Tabari, from Khurasan.
• the Jewish astronomer Sahl ibn Bishr, from Khurasan.
• the astrologer Abu Ma’shar, from Balkh (Khurasan).
• Ali ibn ‘Isa al-Asturlabi, famous maker of astronomical instruments.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
• al-Himsi, who made the work of Apollonius known.
• the Banu Musa ibn Shakir brothers, translators, mathematicians
and engineers, who wrote a work on automata.
• Ibn Sahda, who translated medical works.
• the Christian doctor Jibril ibn Bakhtyashu’.
• the Christian doctor Salmawayh ibn Bunan.
• the surgeon Abu’l Qasim az Zahraw'i (Abulcassis in mediaeval
Latin), from Cordoba.
• the Christian pharmacologist Ibn Massawayh, author of Aphorisms.
• the writer As Suli.
• the doctor and philosopher ‘Ali Rabban at Tabari, author of
Paradise of Wisdom, inspired by the Aphorisms of the Indian Brahman
heretic Chanakya of the third century BCE.
• the mystical thinkers Dhu ‘an Nun Misri, al-Muhasibi, Ibn Karram,
Bistami.
• and the poets Abu Tammam and Buhturi.
Second half of the ninth century
868-905. Tulunid Dynasty in Egypt and Syria.
869. Malta is taken by the Arabs.
875-999. Samanid Dynasty in the north and east of present-day Iran,
Tadjikistan and Afghanistan. Capital: Bokhara.
880. Italy is recaptured from the Arabs by Basil I.
Culture, Science and Technology
Period of:
• al-Mahani the geometer and astronomer from the region of Kirman,
who studied the problems of the division of the sphere using the cubic
equation which bears his name.
• al-Nayrizi (Anaritius in mediaeval Latin), astronomer and mathe-
matician from the Shiraz region, who wrote commentaries on Euclid
and Ptolemy.
• the Egyptian mathematician Ahmad ibn Yusuf, who wrote a work
dealing with proportions.
• the mathematician Thabit ibn Qurra, who translated Archimedes’s
treatise on the sphere and the cylinder and who did important work
on conic sections; he also produced a brilliantly clear proof of
Pythagoras’s theorem, the first general rule for obtaining pairs of ami-
cable numbers' and a method for constructing magic squares.
* Two numbers are “amicable” if the sum of the distinct divisors of each one (including 1 but excluding the
number itself) is equal to the other number. For instance, 220 has divisors 1, 2, 4, 5, 10, 11, 20, 22, 44 , 55,
110, which add up to 284; while 284 has divisors 1, 2, 4, 71, 142, which add up to 220. The numbers 220
and 284 form an "amicable pair”, and they are the smallest to do so.
• the mathematicians Abu Hanifa Ahmad and al-Kilwadhi.
• al-Battani (Albategnus in mediaeval Latin) the astronomer who
accompanied his theory of planets with insights into trigonometry,
which were later to be thoroughly investigated by Western
astronomers; he determined the inclination of the ecliptic and the pre-
cession of the equinoxes with great accuracy using cotangents.
• the astronomer Hamid ibn ‘Ali.
• the Persian astrologist Abu Bakr.
• Qusta ibn Luqa al-Ba’albakki, the Christian mathematician and
engineer of Greek origin, who in particular translated Hero of
Alexandria’s Mechanics which deals with the traction of heavy objects,
as well as works by Autolycus, Theodosius, Hypsicles and Diophantus.
• the Christian doctor Hunayn ibn Ishaq, who translated Greek med-
ical works into Arabic, as well as works by Archimedes, Theodosius
and Menelaus.
• the Christian Yahya ibn Sarafyun, who wrote a medical encyclopae-
dia in Syriac.
• the pharmacologist Sabur ibn Sahl, from Jundishapur, author of a
book of antidotes.
• Muhammad Abu Bakr Ben Zakariyya al-Razi (Razhes in mediaeval
Latin) the great Persian clinician, alchemist and physician who was
thought to be the greatest doctor of his age; he first differentiated
between German measles and measles; he described how to equip a
chemical “laboratory” and his Sirr al Asrar ( The Secret of Secrets), con-
tains important work on distillation.
• the philosopher Abul Hasan ‘Ali ibn Ismail al-Ash’ari, founder of
Muslim scholasticism and of the Mutaqallimin school. He expounded
a theological system based on an atomism similar to that of Epicurus.
• the geographer al-Ya’qubi.
• the Persian geographer Ibn Khurdadbeh, alias Ibn Hauqal, author
of the Book of Routes and Provinces.
• the mystical thinker Tirmidhi, known as “the philosopher”
• the poets Mutanabbi and Ibn Sa’ad.
First half of the tenth century
905. End of Tulunid Dynasty in Egypt, power taken by the governors of
the caliphs.
909. Beginning of the rule of the Fatimid caliphs in Ifriqiya.
932-1055. The Buyid Dynasty, unifying eastern Persia and Media.
935. Muhammad ibn Tughaj reconquers Alexandria and southern Syria.
943. The Caliphate of Baghdad confers the rule of Egypt to Ibn Tughaj for
thirty years.
523
SIGNIFICANT DATF.S IN THE HISTORY OF A R A B I C - 1 S L A M I C CIVILISATION
945. The Buyids enter Baghdad. The Caliphate is now no more than a
“legal fiction”
Culture, Science and Technology
Period of:
• Abu Kamil, the great algebraist from Egypt, who continued the
work of al-Khuwarizmi, and whose algebraic discoveries were to be
used, c. 1206, by the Italian mathematician Leonard of Pisa (or
“Fibonacci”); also devised interesting formulas related to the pentagon
and decagon.
• the geometer Abu ‘Uthman, translator of the tenth book of Euclid’s
Elements and of Pappus’s Commentary.
• the Christian translators Matta ibn Yunus and Yahya ibn ‘Adi.
• Sinan ibn Thabit, mathematician, physician, astronomer and doctor.
• the mathematician Ibrahim ibn Sinan ibn Thabit, who dealt with
the problem of constructing conic sections, and who studied the sur-
face of the parabola and conoids.
• the mathematician Abu Nasr Muhammad, who made an interest-
ing discovery with his theorem of sines in plane and spherical
trigonometry.
• the mathematician Abu Ja’far al-Khazini, from Khurasan, who
worked on algebra and geometry, and who solved al-Mahani’s cubic
equation by using conic sections.
• the astronomer al-Husayn Ben Muhammad Ben Hamid, known as
Ibn al-Adami.
• the astrologist and mathematician al-Imrani, who wrote a commen-
tary on Abu Kamil’s algebra.
• the arithmeticians Ali ibn Ahmad and Nazif ibn Yumn al-Qass.
• Bastulus, the famous maker of astronomical instruments.
• the great geographer and mathematician al-Mas’udi.
• the geographer Qudama.
• the geographer Abu Dulaf.
• the geographer Ibn Rusta of Isfahan.
• the Persian geographer Ibn al-Faqih, from Hamadan.
• the geographer Abu Zayd, from Siraf (Arabic-Persian gulf).
• the geographer al-Hamdani, from the Yemen.
• the philosopher al-Farabi (Alpharabius in mediaeval Latin), from
Turkestan, who devised a metaphysics based on Aristotle, Plato and
Plotinus and who, in his Ihsa’ al ‘ulum, came up with a “Classification
of the Sciences” in five branches: linguistics and philology; logic; math-
ematical sciences, subdivided into arithmetic, geometry, perspective,
astronomy, mechanics and gravitation; physics and metaphysics; and
finally the political, legal and theological sciences.
• the alchemist and agronomist known as Ibn Wahshiya.
• the mystic thinkers Junayd and Abu Mansur ibn Husayn al-Hallaj.
• the poet Ibn Dawud.
• and the Persian poet Rudaki.
Second half of the tenth century
957. The Byzantines in northern Syria.
961-969. The Byzantines reconquer Crete and Cyprus, as well as Antioch
and Aleppo (Syria).
962. A Turkish tribe conquers the Afghan kingdom of Ghazna.
969. The Fatimids of Tunisia occupy Egypt, which puts up no resistance,
then settle there.
972-1152. Zirid and Hammadid Dynasties in Iffiqiya.
973. Foundation of Al Kahira (Cairo).
998-1030. Reign of Mahmud, or “the Ghaznavid” (because he settled in
Ghazna), over what is now Afghanistan, Khurasan and various
annexed regions in the north of India.
Culture, Science and Technology
Period of:
• the founding in Cairo of the Dar al Hikma (House of Wisdom), a
sort of scientific academy similar to that of Baghdad.
• the founding of the al-Azhar university in Cairo.
• the blossoming of the sciences in the Caliphate of Cordoba, thanks
to Caliph al-Hakam II, who put together an immense library.
• the mathematician Abu’l Wafa’ al-Bujzani, from Quistan, who
wrote commentaries on Euclid, Diophantus and al-Khuwarizmi. He
introduced the tangent function, and his work on trigonometry led to
great improvements in methods of solving spherical triangles where,
instead of Ptolemy’s formula (derived from Menelaus’s theorem)
which involved the six great-circle arcs of a quadrilateral, a formula
involving the four arcs generated on a transversal of the figure com-
posed of a spherical triangle and the perpendiculars dropped from
two of its vertices to opposite sides is used. Thanks to him, the Arabs
acquired Diophantus’s Arithmetica with its studies of algebra and
number theory.
• the mathematician al-Uqlidisi (whose name means “the Euclidean”)
who published an important study of decimal fractions.
• the mathematician Ibn Rustam al-Kuhi, from Tabaristan, who stud-
ied geometrical problems posed by Archimedes and Apollonius.
• the Persian mathematician/astronomer Abu’l Fath from Isfahan,
who revised the Arabic translation of much of Apollonius’s work.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
• the mathematician al-Sijzi, from Sigistan, who studied problems of
conic intersections and the trisecting of angles.
• the mathematician al-Khujandi, from the Sir Daria region, who
established a proof concerning the sine in spherical triangles, and
proved that the sum of two cubes cannot equal a cube.
• the mathematicians Sinan ibn al-Fath and Abu Nasr.
• the secret society of the Brothers of Purity ( Ikhwan al-Safa), whose
Epistles were a sort of encyclopaedia based on Pythagorean and Neo-
Platonic mysticism, and which divided the sciences into four sorts:
mathematics; science of physical bodies; science of rational souls; and
science of divine laws.
• the Andalusian astronomer and mathematician Maslama ibn
Ahmad, based in Cordoba.
• the Persian astronomer and mathematician ‘Abd ar-Rahman al-
Sufin, who drew up a catalogue of stars containing the first
observation of the Andromeda nebula.
• the great doctor and surgeon Abu’l Qasim from Zakna, near
Cordoba, author of the Kitab al-Tasrif, which deals with practical
surgery, cauterising wounds, tying up arteries, operating on bones, the
eyes, etc.
• ‘Ali ibn ‘Abbas, a doctor from southern Persia.
• Abu Mansur Muwaffak, a doctor who wrote an important medical
treatise in Persian.
• the Andalusian doctor Ibn Juljul.
• the Persian geographer al-Istakhri, from near Persepolis.
• the geographer Buzurg ibn Shahriyyar, from Khuzistan.
• the Palestinian traveller and geographer al-Muqaddasi, from
Jerusalem.
• the philosopher Abu Bakr Ahmad ibn ‘Ali al-Baqilani.
• the historian Ya ‘qub ibn al-Nadim, author of Kitab al-Fihrist al
‘ulum ( The Book and Index of Sciences) containing biographies of con-
temporary thinkers.
• the mechanical and hydraulic engineer Ibn Mu’adh Abu ‘Abdallah
al-Jayyani, author of an important treatise on water clocks.
• and the mystic thinker Tawhidi.
First half of the eleventh century
1000. First clashes in Khurasan between the Ghaznavids and the Seljuks
(Turkomans pushed out of Central Asia by the Chinese).
1001-1018. Mahmud the Ghaznavid takes possession of Peshawar, crushes
a Hindu coalition and sacks Muttra, one of India’s holy cities.
1030. The Ghaznavids crushed by the Seljuks.
524
1030-1050. The Seljuks occupy various towns in eastern then western
Persia (where they clash with the Buyids), before turning away towards
Syria and Asia Minor.
Culture, Science and Technology
Period of:
• the mathematician, astronomer, physician and geographer al-
Biruni, from Khiva in Kharezm, who travelled widely in India, where
he learnt Sanskrit and became acquainted with Indian science; he later
took back what he had learnt and wrote numerous works on astron-
omy, arithmetic and mathematics; he also made a new calculation of
trigonometric tables based on Archimedes’s premises, an equivalent
of Ptolemy’s theory.
• the mathematician al-Karaji, who did important work on the arith-
metic of fractions; basing himself on the work of Diophantus and Abu
Kamil, he devised an algebra in which, alongside the standard forms of
second degree equations, he dealt with certain 2n degree equations;
his work showed how a rigorous approach can, by using irrational
numbers, arrive at forms that are more supple than those of Greek
geometric algebra; this was, in fact, the start of a development which
would lead to the elimination of geometrical representations in Arabic
arithmetic and algebra thanks to the use of symbols.
• the mathematician Kushiyar ibn Labban al-Gili, from the south of
the Caspian Sea, who worked on Indian arithmetic and sexagesimal
calculations.
• the Persian mathematician An Nisawi, from Khurasan, who contin-
ued the work of al-Khuwarizmi in arithmetic and algebra.
• the mathematician Abu’l Ghud Muhammad ibn Layth.
• the mathematician Abu Ja’far Muhammad ibn al-Husayn.
• the astronomer Ibn Yunus, appointed to the Dar al-Hikma observa-
tory in Cairo.
• the mathematician, physician and doctor Ibn al-Haytham (Alhazen
in mediaeval Latin) whose Book of Optics contains important discover-
ies about eyesight, the theory of the reflection and refraction of light,
the fundamental laws of which he established.
• the Andalusian mathematician al-Kirmani, from Cordoba.
• the Andalusian mathematician and astronomer Ibn al-Samh, from
Granada.
• the mathematician and astronomer Ibn Abi’l Rijal (Abenragel in
mediaeval Latin), from Cordoba but working in Tunis.
• the Andalusian mathematician and astronomer Ibn al Saffar, from
Cordoba.
525
SIGNIFICANT DATES IN THE HISTORY OF A R A B I C - 1 S L A M I C CIVILISATION
• the great philosopher A1 Husayn ibn Sina (Avicenna), a universal
mind as interested in medicine as in mathematics; based on Aristotle’s
ideas, his philosophy rejected mysticism and theology and dwelt
instead on science and nature; his Canon Medecinae remained a text-
book in Europe until the seventeenth century; in his Aqsam al ‘ulum al
‘aqliyya (or Division of the Rational Sciences) he drew up a consistent
classification by means of an analytical division to allow a hierarchy of
the sciences to be established.
• the Christian philosopher Miskawayh whose rational thought
makes him one of Ibn Khaldun’s precursors.
• the chemist al-Kathi.
• the Christian doctor Massawayh al-Mardini, settled in Cairo.
• the doctor ‘Ali ibn Ridwan, from Cairo.
• the doctor Abu Sa’id ‘Ubayd Allah.
• the Andalusian doctor Ibn al-Wafid, from Toledo.
• the Jewish doctor Ibn Janah, author of a treatise on medicinal herbs.
• the doctor Ibn Butlan.
• the oculist Ammar.
• the oculist ‘Ali ibn ‘Isa, author of an important treatise on ophthal-
mology.
• the jurist and poet Ibn Hazem.
• the atheist Syrian poet Abu’l ‘ala al-Ma’ari.
• and the Persian poet Abu’l Qasim Firduzi, author of the famous
Book of Kings.
Second half of the eleventh century
1050. Troubled times in Egypt. Order re-established by Badr al-Jamali who
then ruled over Egypt until 1121 for the Fatimids.
1055. Tughril Beg, the Seljuk, enters Baghdad as the protector of the
Empire of the Caliphs.
1055-1147. Almoravid Dynasty in the Maghreb.
1062. Yusuf Ben Tashfin (the true founder of modern Morocco) founds
Marrakech, which becomes the Almoravids’ capital.
1069. Yusuf Ben Tashfin takes Fez, an Arab-Islamic centre, then develops its
intellectual, artistic and economic activities.
1076. The Seljuks take Damascus and Jerusalem.
1085. The Christians occupy Toledo.
1086. Faced with a dangerous situation in Andalusia, Yusuf Ben Tashfin
declares a "Holy War” against Christian Spain, stops the activities of
Alphonsus VI of Castile, then annexes the whole of the south of Spain,
uniting it with the Maghreb and thus forming the Almoravid Empire.
1090. The Turks arrive between the Danube and the Balkans.
1096. Start of the First (People’s) Crusade. The badly organised crusaders
are massacred in Asia Minor.
1097-1099. The crusaders take Nicea, defeat the Turks at Dorylaeum then
take Jerusalem.
Culture, Science and Technology
Period of:
• the great Persian poet and mathematician ‘Umar al-Khayyam (Omar
Khayyam), from Nishapur, author of the Rubaiyat, the famous collec-
tion of poems; he also wrote commentaries on Euclid’s Elements,
worked on the theory of proportions and studied third-degree equa-
tions, suggesting geometric solutions for some of them.
• the mathematician Yusuf al-Mu’tamin (the enlightened king of
Saragossa).
• the mathematician Muhammad ibn ‘Abd al-Baqi.
• the Andalusian astronomer al-Zarqali, from Cordoba, who
reworked Ptolemy’s Planisphaerium.
• the poet, philosopher, mathematician and astronomer al-Hajjami,
who played a vital part in reforming the calendar; he also provided an
overview of third-degree equations and made an important study of
Euclid’s postulates.
• the Persian oculist Zarrin Dast.
• the Andalusian geographer and chronicler al-Bakri, from Cordoba.
• the doctors Ibn Jazla and Sa’id ibn Hibat Allah.
• the Andalusian agronomist Abu ‘Umar ibn Hajjaj, from Seville.
• the mystic philosopher Abu Hamid al-Ghazali (Algazel in mediaeval
Latin), whose teachings stood against Islam’s scientific progress.
• the “sociologist” al-Mawardi.
• and the Persian poet Anwari.
First half of the twelfth century
1100. Foundation of the Christian Kingdom of Jerusalem.
1125. Revolt of the Masmudas of the Atlas under Ibn Tumert, the inventor
of the Almohad doctrine.
1136. Cordoba, Western Islam’s cultural capital, is taken by Ferdinand III,
king of Castile and Leon.
1147. ‘Abd al-Mu’min, the successor of Ibn Tumert, destroys the power of
the Almoravids and proclaims himself Caliph after taking Fez (in 1146)
and Marrakech (in 1147). He then extends his conquests to Iffiqiya
and reaches Spain.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
526
1147-1269. The Almohad Dynasty in the Maghreb and Andalusia.
1148. The crusaders defeated at Damascus.
Culture, Science and Technology
Period of:
• the Andalusian mathematician Jabir ibn Aflah, from Seville, partic-
ularly famous for his work on trigonometry.
• the great Andalusian geographer al-Idrisi, who made important
contributions to the development of mathematical cartography.
• the Jewish mathematician from Spain Abraham Ben Mei'r ibn ‘Ezra
(better known as Rabbi Ben Ezra).
• the engineer Badi al-Zaman al-Asturlabi, famous for the automata
he made for the Seljuk kings.
• the philosopher Ibn Badja (Avempace in mediaeval Latin and
during the Renaissance).
• the philosopher and doctor Abu al-Barakat, author of the Kitab al
mu'tabar ( Book of Personal Reflection).
• and the Andalusian philosopher Ibn Zuhr (alias Avenzoar).
Second half of the twelfth century
1150. Allah ud din Husayn, Sultan of Ghur, destroys the Ghazni Empire.
1169-1171. Salah ad din (Saladin), a Muslim of Kurdish origins, succeeds
his uncle as Vizier of Egypt then ends the reign of the Fatimids by
recognising only the suzerainty ofNur ad din, the unifier of Syria, and
the Abbasid Caliph of Baghdad.
1174. Salah ad din succeeds Nur ad din and founds the Ayyubid Dynasty
which dominated Egypt and Syria thenceforth. Leaning on the Arab
traditionalists, he declares “Holy War” against the Christians of the
West, hence reinforcing the links between the eastern peoples and
their Arab-Islamic traditions.
1187. Salah ad din takes back Jerusalem. Victory of the Almohads at Gafsa
under the Maghribi Sultan Abu Yusuf Ya’qub al-Mansur.
1188. Genghis Khan unifies the Mongols.
1191. Under Muhammad of Ghur, the Islamic Afghan and Turkish tribes of
Central Asia try to conquer the north of India, but are pushed back at
the very gates of Delhi.
1192. Battle of Tarain: Muhammad of Ghur defeats Prithiviraj and takes
Delhi.
1192-1526. Sultanate of Delhi.
1193. The Muslims take Bihar and Bengal.
1195. Victory of the Almohads at Alarcos.
Culture, Science and Technology
Period of:
• the mathematician al-Amuni Saraf ad din al-Meqi.
• the converted Jewish mathematician, philosopher and doctor As
Samaw’al ibn Yahya al-Maghribi, from the Maghreb, who continued
the work of al-Karaji.
• the Persian encylopaedist and mathematician Fakhar ad din al-Razi.
• the Persian engineer Abu Zakariyya Yahya al-Bayasi, famous for his
mechanical pipe organs.
• the great Jewish philosopher Maimonides (Rabbi Moshe Ben
Maimon), from Cordoba, whose encyclopaedic interests included
astronomy, mathematics and medicine.
• the great Andalusian philosopher Ibn Rushd (Averroes), born in
Cordoba and died in Marrakech, the finest flowering of Arab philoso-
phy and a profound influence on the West.
• the Maghrebi philosopher Ibn Tufayl (Abubacer in mediaeval Latin
and during the Renaissance).
• the mystical thinker Ruzbehan Baqli.
• the Persian mystical poet Nizami.
• and the Persian poet Khaqani.
First half of the thirteenth century
1202. The Muslims arrive on the banks of the Ganges, at Varanasi
(Benares).
1203. Continuation of Muhammad of Ghur’s conquest of northern India.
1206-1211. Reign of Qutb ud din (Sultanate of Delhi).
1208. The Albigensian Crusade.
1212. Defeat of the Almohads at Las Navas de Tolosa.
1211-1222. Under Genghis Khan, the Mongols invade China, Transoxiana
and Persia, before continuing their migration under Hulagu Khan
towards Mesopotamia and Syria.
1211-1227. Reign of Iltumish (Sultanate of Delhi) who obtains recognition
of his authority over India from the Caliph of Baghdad. Under this
domination, India will remain relatively stable until 1290.
1214-1244. The Banu Marin (Merinids) conquer the north of the
Maghreb.
1221. The Mongols press against the borders of the Sultanate of Delhi, but
are held back by Iltumish.
1227. Death of Genghis Khan, whose empire stretched from the Pacific to
the Caspian Sea.
527
SIGNIFICANT DATES IN THE HISTORY OF ARABIC-ISLAMIC CIVILISATION
1248. The Christians take back Seville from the Muslims.
Culture, Science and Technology
Period of:
• the mathematician Muwaffaq al din Abu Muhammad al-Baghdadi.
• the leading court official and patron Abu’l Hasan al-Qifti, author of
Tarikh al huqama (Chronology of the Thinkers).
• Muhammad ibn Abi Bakr, famous maker of astronomic instruments.
• the engineer Ridwan of Damascus, best known for his ball-cock
automata.
• the great Persian engineer al-Jazzari, author of the Book of the
Knowledge of Ingenious Mechanical Instruments, in which he provided
plans for perpetual flutes, water clocks, and different sorts of sequen-
tial automata using ball-cocks and camshafts.
• Ya’qub ibn 'Abdallah ar Rumi, who produced an important ency-
clopaedia of Arab geography.
• the esoteric Muslim Ibn ‘Arabi.
• and the poets Ibn al-Farid and Shushtari.
Second half of the thirteenth century
1250. The Mamelukes take power in Egypt. The Kingdom of Fez created by
the Banu Marin (Merinids).
1254-1517. Reign of the Mamelukes in Egypt.
1258. Hulagu Khan’s Mongols retake and sack Baghdad.
1259. Mongol invasion of Syria.
1260. Mongols crushed at the border of Egypt by the Mameluke monarch.
1261. Egypt becomes the centre of the Arab world and also, to a certain
degree, of the Islamic world.
1269. In the Maghreb, the Banu Marin take Marrakech and found their
own dynasty (the “Merinids”).
1291. The Mamelukes take Acre and eliminate the Christians on the
Syrian-Palestinian coast.
1297. ‘Ala ud din Khalji (Sultanate of Delhi) defeats the Mongols then starts
sacking Gujarat and Rajasthan.
Culture, Science and Technology
Period of:
• the mathematician and astronomer Nasir ad din at Tusi, from Tus
in Khurasan, who did important work on arithmetic, algebra and
geometry; his work undoubtedly marks the high point of Arabic
trigonometry, dealing thoroughly with spherical right-angled triangles
and successfully broaching the study of spherical triangles in general,
even bringing in the polar triangle; in astronomy, he published his
famous “Ikhanian” Tables; and, in geometry, he corrected the transla-
tions of Greek geometrical works and his discussion of Euclid’s
propositions was later to inspire the Italian mathematician Saccheri in
his initial research in 1773 into non-Euclidean geometry.
• the doctor Ibn al-Nafis of Damascus, who wrote a commentary on
Avicenna’s Canon, with important developments concerning pul-
monary circulation.
• the pharmacologist and botanist Ibn al-Baytar.
• the Persian mystical poet and hagiographer Farid ad din ‘Attar.
• the Persian poet Sa’adi of Chiraz, author of the Gulistan.
First half of the fourteenth century
1306. The Sultans of Delhi repulse the Mongols once more.
1307-1325. The Sultans of Delhi attack the kingdoms of Deccan and reach
the south of India, conquering the lands of the Maratha, Kakatiya and
Hoysala.
1333. The Moors recapture Gibraltar from the Kingdom of Castile.
Culture, Science and Technology
Period of:
• the great Maghrebi arithmetician Ibn al-Banna al-Marrakushi.
• the converted Jewish doctor and historian Rashid ad din, author of
a Universal History, in which he reproduced large extracts of the best-
known medical works of China and Mongolia.
• the historian al-Umari.
• the moralist Ibn Taymiyya.
• the Andalusian mystic thinker Ibn Abbad of Ronda.
• the great Maghrebi traveller Ibn Battutua who, in thirty years, cov-
ered more than 120,000 kilometres in the Islamic world, from
Northern Africa to China, via India.
• and the Persian poets Hamdallah al-Mustawfi and Tebrizi.
Second half of the fourteenth century
1356. India is “given” by the Caliph of Baghdad to Firuz Shah Tughluq.
1371. The Ottomans defeat the Serbs at Chirmen.
1389. The Ottomans crush the Serbs at Kosovo Polje.
1390. The Ottomans occupy the remaining territories of the Byzantine
Empire in Asia Minor.
INDIAN NUMERALS AND CALCULATION IN THF. ISLAMIC WORLD
1392. The Ottomans arrive in the Balkans.
1398-1399. Timur (Tamerlane) sacks Delhi.
Culture, Science and Technology
Period of:
• the great thinker Ibn Khaldun, from Tunis, remarkable for his ratio-
nalism, his feeling for general laws and his extraordinarily acute
scientific insights; in many ways a precursor of Auguste Comte.
• the writer Ibn al-Jazzari.
• the writer Taybugha.
• and the Persian poet Hafiz of Chiraz, author of Bustan.
First half of the fifteenth century
1400-1401. Incursion of Timur and sacking of Baghdad.
1405. Return to Baghdad of the Jalayrid leaders.
1422. The Ottomans besiege Constantinople.
1400-1468. Constant disputes between Turkomans and Mongols.
1444. The Viceroy of the Baghdad Timurid Dynasty founds his empire in
Mesopotamia and Kurdistan.
1447. End of the empire of Timur, independence of Persia and of the
Afghan and Indian regions.
Culture, Science and Technology
Period of:
• Ulugh Bek, the enlightened monarch of Samarkand, builder of an
observatory equipped with the finest instruments of the age; author of
trigonometric tables, among the most precise of the numeric tables
produced by Islam’s thinkers.
• the Persian mathematician Ghiyat ad din Ghamshid ibn Mas’ud al-
Kashi, who did important work on algebra, sexagesimal calculations
and arithmetic, especially on the binomial formula, decimal fractions,
exponential powers of whole numbers, n roots, the theory of propor-
tions and irrational numbers.
• the historian al-Maqrizi.
Second half of the fifteenth century
1453. Constantinople falls to Sultan Mehmet II. The beginning of the
Ottoman Empire, which will later cover Anatolia, Rumelia, Bulgaria,
Albania, Greece, the Crimea, Syria, Mesopotamia, Palestine, Egypt,
Hejaz, Armenia, Kurdistan and Bessarabia and which, after 1520, will
extend its frontiers as far as Hungary, southern Mesopotamia, the
Yemen, Georgia, Azerbaidjan, with Tripoli and the whole of Ifriqiya as
528
dependencies (excepting the Maghreb which managed to remain
autonomous during this period).
1468. The Turkoman al-Koyunlu establishes his authority in Mesopotamia.
1492. The Catholics Ferdinand and Isabella retake Granada.
1499-1722. Reign of the Safavids in Persia; Shi’ism becomes the official religion.
Culture, Science and Technology
Period of:
• the mathematician al-Qalasadi, who did important work on arith-
metic, especially algebra, greatly developing its symbols.
• and the Persian historian Mirkhond.
The sixteenth century
1508. The Safavids push the Turkomans out of Mesopotamia.
1516. Turkish corsairs establish themselves in Algiers.
1517. Ottoman conquest of Syria and Egypt, thus ending the Caliphate of
Baghdad and bringing about the fall of the Mamelukes in Egypt.
1524. Babur, a descendant of Timur, invades the Punjab and takes Lahore.
1526. Babur kills the last Sultan of Delhi and takes the throne. The begin-
ning of the Mogul Empire in India and Afghanistan (1526-1707).
1571. Turks defeated by Holy League in naval battle of Lepanto.
1574. The central and eastern regions of North Africa come under
Ottoman control.
1578-1603. Beginning of the Sa’adian Dynasty with the reign of al-Mansur
(Maghreb).
Culture, Science and Technology
Period of:
• the Turkish arithmetician Tashkopriizada.
• and the Turkish poets Baki and Fuzuli.
The seventeenth century
1672-1727. Beginnings of the 'Alawite Dynasty in the Maghreb with the
reign of Mulay Ismail, contemporary of Louis XIV.
Culture, Science and Technology
Period of:
• the mathematician Beha ad din al-Amuli.
• the arithmetician and commentator al-Ansari.
• the writers Hajji Khalifa, ‘Abd al-Qadir al-Baghdadi, ‘Abd ar Rashid
Ben ‘Abd al-Ghafur and Ad Damamini.
• the encylopaedist Jamal ad din Husayn Indju.
529
NDIAN NUMERALS IN THE ISLAMIC WORLD
• the Turkish poets Nefi, Nabi and Karaja Oghlan.
• and the Turkish traveller and writer Evliya Chelebi.
The eighteenth century
1799. Start of the Nahda (“Renaissance”).
Culture, Science and Technology
Period of:
• the Turkish poet Nedim.
• the Turkish writer and historian Naima.
The nineteenth century
1804. The Wahhabis take Mecca and restore Hanbali Islam.
1805-1849. Reign of Muhammad ‘Ali, Pasha of Egypt.
1811-1818. ‘Ali defeats the Wahhabis.
Culture, Science and Technology
Period of:
• the Turkish thinkers Namik Kemal, Ziya Pasha, Ahmet Mithat,
Chinassi and Avdiilhak Hamit.
Beginning of the twentieth century
1918-1922. Reign of Sultan Mehmed IV (whom the Treaty of Sevres
obliged to accept the dismemberment of the Turkish Empire. Turkey
was reduced to the landmass of Anatolia).
1922. Mehmed IV overturned by Mustafa Kemal, founder of modern,
republican Turkey.
1924. Official end of the Ottoman Empire.
THE ARRIVAL OF INDIAN NUMERALS IN THE
ISLAMIC WORLD
How were Indian numerals and calculating methods introduced into
Islam?
The Arabs possibly encountered them at the beginning of the eighth
century CE, when Hajjaj sent out an army under Muhammad Ben al-Qasim
to conquer the Indus Valley and the Punjab.
But it is far more likely that the army had nothing to do with it, and that
it was necessary to wait for a delegation of scholars before Indian science
was transmitted to the Islamic world.
This is, indeed, Ibn Khaldun’s explanation, who says in his Prolegomena
that the Arabs received science from the Indians, as well as their numerals
and calculation methods, when a group of erudite Indian scholars came to
the court of the caliph al-Mansur in year 156 of the Hegira (= 776 CE) [see
Muqaddimah, trans. Slane, III, p. 300].
This is a late source, dating from about 1390. But Ibn Khaldun’s version
corresponds closely with earlier texts, especially with a tale told by the
astronomer Ibn al-Adami in about 900, which is referred to by the court
patron Hasan al-Qifti (1172-1288) in his Chronology of the Scholars:
Al-Husayn Ben Muhammad Ben Hamid, known as Ibn al-Adami,
tells in his Great Table, entitled Necklace of Pearls, that a person from
India presented himself before the Caliph al-Mansur in the year 156
[of the Hegira = 776 CE] who was well versed in the sindhind method
of calculation related to the movement of heavenly bodies, and
having ways of calculating equations based on kardaja calculated in
half-degrees, and what is more various techniques to determine solar
and lunar eclipses, co-ascendants of ecliptic signs and other similar
things. This is all contained in a work, bearing the name of Fighar,
one of the kings of India, from which he claimed to have taken the
kardaja calculated for one minute. Al-Mansur ordered this book to
be translated into Arabic, and a work to be written, based on
the translation, to give the Arabs a solid base for calculating the
movements of the planets. This task was given to Muhammad
Ben Ibrahim al-Fazzari who thus conceived a work known by
astronomers as the Great Sindhind. In the Indian language sindhind
means “eternal duration” The scholars of this period worked accord-
ing to the theories explained in this book until the time of Caliph
al-Ma’mun, for whom a summary of it was made by Abu Ja’far
Muhammad Ben Musa al-Khuwarizmi, who also used it to compose
tables that are now famous throughout the Islamic world
[F. Woepcke (1863)].
Much can be learned from this. The repetition of the word sindhind is
significant; it is the Arabic translation of the Sanskrit *siddhanta, the
general term for Indian astronomic treatises, which contained a com-
plete set of instructions for calculating, for example, lunar or solar
eclipses, including the trigonometric formulae for true longitude [see R.
Billard in IJHS]. The “sindhind” method thus stands for the set of ele-
ments contained in such treatises. As for the word kardaja, which is also
frequently used, it means “sine” and derives from an Arabic deformation
of the Sanskrit ardhajya (literally “semi-chord”) which Indian
astronomers had used, from the time of *Aryabhata, for this trigono-
metric function which is the basis of all calculations in the Indian
siddhanta system.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
This method is presented in the mathematician and astronomer
Brahmagupta’s (628) Brahmasphutasiddhanta and the astrologer
‘Varahamihira’s (575) Pahchasiddhantika. But it was explained long before
these treatises in the astronomer ‘Aryabhata’s Aryabhatiya (c. 510).
Now, apart from the Aryabhatiya (which uses a special form of alpha-
betic numeration), all Indian astronomers noted their numbers by using
Sanskrit numerical symbols: this notation gave them a solid base for noting
numeric data and was based on a decimal place-value system using zero. As
for their calculations, they used a system quite similar to our own one with
their nine numerals plus a tenth sign written as a circle or point and acting
as a true zero (see ‘Zero, etc).
In other words, when the Arabs learnt Indian astronomy, they
inevitably came up against Indian numerals and calculation methods, so
that the arrival of the two branches of knowledge precisely coincided. This
is confirmed by al-Biruni’s Kitab fi tahqiq i ma li’l hind ( c . 1030), which tells
of his thirty-year stay in India.
We must now try to date this transmission.
Now, al-Qifti, Ibn al-Adami and other authors agree on the date
mentioned in the quotation above; i.e. 156 of the Hegira, or 776 CE.
Several facts about Arabic science make this date plausible. According to
A. P. Youschkevitch:
If the arrival of Indian scholars gave the astronomers of Baghdad
the possibility of acquainting themselves with the astronomy of the
siddhanta, there was already much interest in the subject. Three
astronomers who worked during the reign of Caliph al-Mansur
are known to us, thanks to al-Qifti: Abu Ishaq Ibrahim al-Fazzari
(died c. 777) who first made Arabic astrolabes, his son Muhammad
(died c. 800), and finally Ya’qub ibn Tariq (died c. 796), who
wrote works dealing with spherical geometry and who also compiled
various tables.
All we now have to discover is which of the Indian siddhanta was adapted
by al-Fazzari during the reign of al-Mansur. Now, the Fighar who is men-
tioned in the text is none other than Vyagramukha (abbreviated to Vyagra
then deformed into Fighar), an Indian sovereign of the Chapa Dynasty
who, according to an inscription, was defeated by Pulakeshin II, king of the
Deccan in about 634. His capital was Bhillamala (now Bhinmal), in the
southwest of what is now Rajasthan. And it was precisely under the reign of
Vyagramukha, in the year 550 of the *Shaka era (i.e. 628 CE), that
‘Brahmagupta composed his Brahmasphutasiddhanta ( Brahma's Revised
System) at the age of thirty.
530
Thus, one or other of the Indian scholars who arrived in Baghdad in 773
probably gave the caliph a copy of the Brahmasphutasiddhanta, along with
other Sanskrit works.
It thus seems quite likely that not only Indian astronomy, but mathe-
matics too, were introduced to the Muslims through the work of
Brahmagupta.*
What led these Indian scholars to give such a present to al-Mansur?
They had been kept for some time in his palace, which gave that enlight-
ened monarch, with his lifelong thirst for knowledge, the opportunity to
learn some Indian astronomy and arithmetic. Thus it was that these
Brahmans, as worthy representatives of Indian culture, were led to demon-
strate to him what seemed to them to be most important, original and
ingenious in their science. They then, quite probably, gave the caliph copies
of Brahmagupta’s Brahmasphutasiddhanta and Khandakhadyaka, which
contained not only the siddhanta method, but also the principle of the deci-
mal place-value system, the zero, calculation methods and the basics of
Indian algebra.
It is easy to imagine the enthusiasm of al-Khuwarizmi, Abu Kamil,
al-Karaji, al-Biruni, An Nisawi and others, too, who could appreciate
the superiority of the Indians’ place-value system and methods
of calculation.
In his Chronology of the Scholars, Abu’l Hasan al-Qifti speaks of their
admiration:
Among those parts of their sciences which came to us, [1 must men-
tion] the numerical calculation later developed by Abu Ja’far
Muhammad Ben Musa al-Khuwarizmi; it is the swiftest and most
complete method of calculation, the easiest to understand and the
simplest to learn; it bears witness to the Indians’ piercing intellect,
fine creativity and their superior understanding and inventive genius
[F. Woepcke(1863)].
We must, in passing, admire this author’s objectivity and lack of
chauvinism, his ability to recognise the superiority of a discovery made
by foreigners and his praise for a civilisation which had produced such a
superior system to his own culture’s.
* Even if Brahmagupta made some mistakes (he argued against the rotation of the earth demonstrated by Aryabhata
in 520, for example), he was incontestably the greatest mathematician of the seventh century - a reputation he
would keep for several centuries among Indian mathematicians and astronomers, and also among many Arabic-
Islamic scholars, such as al-Biruni. His work, first presented in his Brahmasphutasiddhanta (628) then expanded in
his Khandakhadyaka (664), made considerable progress compared to earlier work, including that of Aryabhata and
Bhaskara, particularly in algebra, one of his main innovations. Among his fundamental contributions can be cited
his own system of a negative or zero arithmetic (with a clear and accurate statement of the rules of algebraic sym-
bols), and his presentation of general solutions to quadratic equations with positive, negative or zero roots.
531
INDIAN NUMERALS IN THE ISLAMIC WORLD
This quotation also leads us to look at one of the Islamic world’s most
famous mathematicians: al-Khuwarizmi, who was born in 783 in Khiva
(Kharezm) and died in Baghdad in about 850 [see 0. Fayzoullaiev (1983);
G. J. Toomer in DSB; K. Vogel (1963)]. Little is known about his life, except
that he lived at the court of the Abbasid caliph al-Ma’mun, shortly after the
time when Charlemagne was made Emperor of the West, and that he was
one of the most important of the group of mathematicians and
astronomers who worked at the “House of Wisdom” (Bayt al-Hikma),
Baghdad’s scientific academy.
His fame is due to two works which made significant contributions to
the popularisation of Indian numerals, calculation methods and algebra in
both the Islamic world and the Christian West. One of them, Al jabr wa'l
muqabala ( Transposition and Reduction ), dealt with the basics of algebra. It
has come down to us both in its original Arabic and in Geraldus
Cremonensis’s mediaeval Latin translation, entitled Liber Maumeti filii
Moysi Alchoarismi de algebra et almuchabala. This book was extremely
famous, to such an extent that we owe to it the term for that fundamental
branch of mathematics, “algebra” The first word of its title stands for one
of the two basic operations which must be made before solving any alge-
braic equation. Al jabr is the operation of transposing terms in an equation
such that both sides become positive; later compressed into aljabr, it was
translated into Latin as “algebra”, giving us the term we know today. As for
Al-muqabala, it stands for the operation consisting in the reduction of all
similar terms in an equation.
According to Ibn al-Nadim’s Fihrist, al-Khuwarizmi’s other work was
called Kitab al jami’ wa’l tafriq bi hisab al hind ( Indian Technique of Addition
and Subtraction). The original has, unfortunately, been lost but several post-
twelfth century Latin translations of it survive. It is the first known Arabic
book in which the Indian decimal place-value system and calculation meth-
ods are explained in detail with numerous examples. Like his other book, it
became so famous in Western Europe that the author’s name became the
general term for the system. Latinised, al-Khuwarizmi first became
Alchoarismi, then Algorismi, Algorismus, Algorisme and finally Algorithm.
This term originally stood for the Indian system of a zero with nine digits
and their methods of calculation, before acquiring the more general and
abstract sense it now has.
Unbeknown to him, al-Khuwarizmi provided the name for a
fundamental branch of modern mathematics, and gave his own name
to the science of algorithms, the basis for one of the practical and
theoretical activities of computing. What more can be said about this
great scholar’s influence?
In,. 25 . Muhammad Hen Musa al-Khuwarizmi (c. 783-850). Portrait on wood made in 1983
from a Persian illuminated manuscript for the 1200th anniversary of his birth. Museum of the Ulugh
Begh Observatory. Urgentsch (Kharezm), Uzbekistan (ex USSR). By calling one oj its fundamental
practices and theoretical activities the “algorithm " computer science commemorates this great
Muslim scholar.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
532
1234567890
Mathematical treatise copied in Shiraz in
969 by the mathematician 'Abd Jalil al-
Sijzi. Paris, BN, MS. ar. 2547, P 85 v-86
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Bodleian, Ms. Or. 516, P 12 v
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Eleventh-century astronomical treatise.
Paris, BN, Ms. ar, 2511, P v 10, 14,19
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Eleventh-century astronomical tables.
Paris, BN, Ms. ar. 2495, P 10
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BN, Ms. ar. 2494, P 10
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manuscript. Paris, BN, Ms. ar. 4457 P 20 v
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Kushyar ibn Labban’s astronomical treatise,
copied in 1203 in Khurasan. University of
Leyden, Ms. al madkhal
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Thirteenth-century astronomical tables.
Paris, BN, Ms. ar. 2513, P 2 v
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Princeton University, ELS 373
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arithmetic. Paris, BN, Ms. ar. 2475, P 25,
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THE GRAPHIC EVOLUTION OF INDIAN
NUMERALS IN EASTERN ISLAMIC COUNTRIES
When the Arabs learnt this number-system, they quite simply copied it
(Fig. 25.3).
In the middle of the ninth century, the Eastern Arabs’ 1 (f ), 2 (^), 3 ( f),
4 (y*), 5 (£l), 6 (^) and 9 (.A) could easily be confused with their Indian
Nagari prototypes, thus:
Midi / o $ ?
123456789
But Arabic scribes gradually modified them, until they no longer resem-
bled their prototypes (Fig. 25.3).
Such a development was a normal adaptation of the Indian models to
the style typical of Arab writing. In other words, as they became integral
parts of the writing system and associated with its graphic style, the Indian
numerals gradually changed until they looked like a set of original symbols.
But these stylistic changes cannot explain everything. A close examina-
tion of Arab manuscripts, dating from the early centuries of Islam, shows
that the Indian numerals became inverted.
And thus, in Islamic countries of the Near East:
Indian 1
( \ )
became:
I
Indian 2
( 4 )
became:
<
then:
r
and finally:
r
Indian 3
(?)
became:
?
then:
r
and finally:
r
Indian 4
(^)
became:
¥•
then:
f
and finally:
l
Indian 5
( £ )
became:
ti
then:
A
and finally:
a
Indian 6
<J>
became:
S
then:
f
and finally:
Indian 7
( n >
became:
s
then:
and finally:
V
Indian 8
( S )
became:
then:
<
and finally:
A
Indian 9
( 5> )
became:
y
then:
1
and finally:
This inversion came about for practical, material reasons.
During the early centuries of the Hegira, eastern Arabic scribes used to
write the characters of their cursive script from top to bottom, rather than
from right to left, in successive lines from left to right. They wrote some-
what as follows:
Fig. . The " Hindi ” numerals, used by Eastern A rabs
533
Then to read, they turned their manuscript clockwise through 90°, so that
the lines could be read from right to left:
Top of scroll
Bottom of scroll
Fig. 25. 4B.
This was the old custom of Aramaic scribes of the ancient city of Palmyra,
perpetuated then transmitted to the Arabs by Syriac scribes [see M. Cohen
(1958)].
It came about for the following reasons, essentially to do with manu-
script writing on papyrus, which, until the ninth century, was widely used
in the Islamic world.
First of all, stalks were cut into sections, the length of which determined
the height of the sheet. The tissue was then cut open with a knife, ham-
THK GRAPHIC EVOLUTION OF INDIAN NUMERALS
mered flat, then the strips thus obtained were laid side by side in two layers
at right-angles to each other. They were then struck repeatedly. The
finished sheets were glued along the longer sides so that the horizontal
fibres were on one side (the facing page) and the vertical ones on the
other. Once the horizontal fibres had been placed on the inside and the
vertical ones on the outside, the sheet could be rolled up into a scroll [see
L. Cottrell (1962)].
In order to write, Arabic scribes (like their Palmyrenean and Syriac pre-
decessors) sat cross-legged, with their robe pulled up as a writing table.
Bearing in mind this position and the fragility of the sheet, it is easy to
understand why scribes held their manuscripts lengthways, perpendicular
to their bodies, with the head of the scroll to their left, thus writing their
cursive script from top to bottom, in successive lines from left to right.
This explains the inversion of most Indian numerals in Arabic manu-
scripts dating from the early centuries of Islam.
As for zero, it was originally written as a “little circle resembling the
letter ‘O’,” to borrow al-Khuwarizmi’s explanation, who was referring to
the Arabic letter ha(&), shaped like a small circle [see A. Allard (1957);
B. Boncompagni (1857); K. Vogel (1963); A. P. Youschkevitch (1976)].
Several Arabic manuscripts prove that this usage continued in certain
places until the seventeenth century.
Here is a pun, typical of twelfth-century Arabic poetry. It occurs in
two lines taken from the poem Khaqani composed in praise of
Prince Ghiyat ad din Muhammad (c. 1155), to exhort him to free
the province of Khurasan from its Oghuzz Turkoman invaders [see
A. Mazaheri (1975)]:
Your enemy will be mutawwaq (“captured with a metal collar”)
Like zero ( al sifr ) on the earthen tablet ( takht al turab)\
At his side will be the units (“of soldiers”)
Like a sigh ( aah ) of regret.
It is true; among your subjects, your enemy is nothing.
If we did pay attention to him,
He would merely be a zero to the left of the figures ( arqam ).
The meaning of this fine passage is clearer if we consider that:
• the Arabic word for “sigh” is aah, composed of a double alif ( 1 ) and
a single ha($);
• the first of these two letters looks like the vertical line representing
the number 1, while the other resembles zero;
• the phrase “your enemy will be mutawwaq" means "your enemy will be
captured with a metal collar, as the zero which is shaped like an 0” the
(hence, by extension: “your enemy will be imprisoned, then hanged”).
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
The poet’s metaphor thus plays on the graphic resemblance between the
word aah (a sigh) and the numerical notation Oil to give the image of the
leader of the opposing army being dragged by the neck (0) by the victorious
troops (11):
Oil o M
< <
H A A Oil
These verses thus mean: “The Turkoman will have a chain round his
neck, and be dragged by the troops in front of Sultan Muhammad.”
This confirms that the small circle still stood for zero in the twelfth
century in certain eastern provinces of the Muslim empire.
This is not surprising, for it is the Shunya-chakra (the “zero-
circle”), one of the Indian ways of depicting zero (see *Shunya\ * Shunya-
chakra) *Zero).
But, in the long term, this circle became so small that it was reduced to a
point (Fig. 25.3).
The point is, in fact, the second way the Indians used to depict zero. It
appeared at an early period in India and Southeast Asia (see *Shiinya ;
*Bindu ; * Shunya-bindw, *Zero). Al-Biruni also speaks of this in his Kitab
fi tahqiq i ma li’l hind, where he discusses Indian numerals and the Sanskrit
numeric symbol system and lists the words symbolising zero: he cites
the Sanskrit words *shunya (“vacuum” “zero”) and *kha (“space” “zero”)
before adding “wa huma ‘n naqta” (“they mean ‘point’”) [see F. Woepcke
(1863)].
To conclude, it was in this stylised and slightly modified form that the
nine Indian numerals spread across the eastern provinces of Islam, in a
fixed series that was only to be changed in insignificant ways throughout
the succeeding centuries, particularly for the numbers 5 and 0 (Fig. 25.3).
And these were what Arab authors have always referred to as arqam al hindi
(“Indian numerals”):
\ X X ^ or 0 or £ or V A A
123 4 5 6 7890
These forms can be found in 'Abd Jalil al-Sijzi (951-1024),
al-Biruni (c. 1000), Kushiyar ibn Labban al-Gili (c. 1020) and As
Samaw’al al-Maghribi (c. 1160) (Fig. 25.1), and they are still used in
all the Gulf countries, from Jordan and Syria to Saudi Arabia, the
Yemen, Iraq, Egypt, Iran, Pakistan, Afghanistan, Muslim India, Malaya
and Madagascar.
534
THE WESTERN ARABS’ “GHUBAR” NUMERALS
But this was not exactly the origin of our “Arabic” numerals. We inherited
them from the Arabs, true enough, but from the Arabs of the West (the inhab-
itants of North Africa and Spain) and not from the Arabs of the Near East.
Before proceeding further, we should like to quote three revealing pas-
sages from manuscripts in the Bibliotheque nationale and translated by
Woepcke [F. Woepke (1863), pp. 58-69].
They are three commentaries on mathematical works. In each of them,
the commentator’s explanations are mixed in with the original text, which
is written in red ink to distinguish it from the commentary, which is writ-
ten in black ink. Thus, in the following extracts, the original text is printed
in italics and the commentary in Roman.
First passage
The nine Indian numerals [arqam al hindi] are as follows:
123456789
irrfafyA*
Or like this:
123456789
I zTf (foe i
which are the “Ghubar" numerals.
Second passage
The author says: The first order goes from one to nine and is called the
order of units.
These nine symbols, called “ghubar” [ “dust”] numerals, are
widely used in the provinces of Andalusia and in the lands of the
Maghreb and Iffiqiya. Their origin is said to have occurred when an
Indian picked up some fine dust, spread it over a board ( luha ) made of
wood, or of some other material, or else over any plane surface, on
which he marked the multiplications, divisions or other operations he
wanted to carry out. When he had finished his problem, he put it [the
board] away in its case until he needed it again.
[In order to memorise their shapes] the following verses have
been written about these numerals [in which the shapes of the
letters, words and figures mentioned evoke the numerals being
referred to]:
535
THE WESTERN ARABS' "gHUBAR” NUMERALS
12 3 4567890
Practical arithmetical treatise by Ibn
al-Banna al-Marrakushi. Fourteenth
century. University of Tunis, Ms.
10 301, f° 25 v. CF. M. Souissi
f
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Guide to the Katib (work which gives
details of the various number-systems
used by scribes, accountants, officials
etc.) Manuscript dated to 1571-72
(see Fig. 25.10). Paris, BN, Ms. ar.
4441, P 22
/
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Sharishi, Kashfal talkhis
(“Commentary on the Arithmetical
Treatise. . .). Manuscript dated 1611.
University of Tunis, M. 2043, P 16r
1
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Bashlawi, Risala fi’l hisab (“Letter
Concerning Arithmetic”).
Seventeenth-century manuscript.
University of Tunis, Ms. 2043,
P 32 r. Cf. M. Souissi
]
y
6
7
*
3
Anonymous. Arithmetical treatise
entitled Fath a! wahhab 'ala nuzhat at
husab ‘at ghubar (“Guide to the Art of
Ghubar Calculations”). Commentary
by al-Ansari, written in 1620 and
completed byl629. Paris, BN, Ms. ar.
2475, P 46 r, 152 v and 156 v
1
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Copy of a treatise of practical
arithmetic by Ibn al-Banna ( Talkhis a
1 mal al hisab, "Concise Summary of
Arithmetical Operations”)
Seventeenth century. Paris, BN, Ms.
ar. 2 464, P 3v
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7
1
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7
As Sakhawi, Mukhtasar Fi ‘ilm al
hisab ("Summary of Arithmetic”).
Eighteenth century. Paris, BN, Ms.
ar. 2463, f“ 79 v - 80
i
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6
if
s
fi
Fig. 25 . 5 . The Western Arabs ' numerals (“ Ghubar ’ script)
These are an alif( | ) [for number 1],
Andaya («£_)[for2],
Then the word hijun (^ ) [for 3].
After that the word ‘awun ( jt-) [for 4];
And after ‘awun, one traces an ‘ay in (t ) [for 5],
Then a ha [final] ( 4 ) [for 6].
And after the ha, appears a number [7], which,
When it is written, looks like an iron with a bent head (1 ).
The eighth (of these signs is made of) two zeros [sifran]
[Connected by] an alif($). And the waw ( 2 ) is the
Ninth, which completes the series.
The shape of the ha (C) [sometimes given to number 2] is not pure.
Here are the nine signs (which must be written so that) the one
appears in the highest place, with the two below it, as follows:
I L ^ ^ ^
12 34 56789
Third passage
The preface deals with the shape of the Indian signs, as they were drawn
up by the Indian nation, and these are, i.e. the Indian signs, nine figures
which must be formed as follows, that is: one, two, three, four, five, six,
seven, eight, nine, with the following forms:
1 r r f a * v a 4
1234 56789
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
which are most often used by us, i.e. the Easterners, but others too are used.
Or, they must be formed as follows :
1234 56789
which are not much used by us, while their use is widespread among the
Western [Arabs].
Note. The author’s meaning is clearly that both series come from
India, which is true. The learned al-Shanshuri says in his commentary
on the Murshidah: and they are called, i.e. the second way [of forming
these signs], Indian, because they were devised by the Indian nation.
End of quotation. But they are distinguished by different names, the
former are called Hindi and the latter Ghubar, and they are termed
Ghubari because people used to spread flour over their board and trace
figures in it.
The following verses have been written about these signs [the
same as those quoted in the second passage above, with one slight dif-
ference which is described in Fig. 25.6].
(But) they have been brought together better in one single verse, as
follows:
An alif{ I ) (for numberl),
a /w(C) [for 2],
hizun ( ^ ) (for 3),
‘awun ( •£■) [for 4],
an ‘ay in ( £_ ) (for 5),
a ha (final) ( ^ ) [for 6],
an inverted waw(t_ ) (for 7),
two zeros [linked by an alif ( 9 ) (for 8),
and a waw (J) [for 9].
Certain points are worthy of note in these passages.
Firstly, we learn that the Ghubar numerals were used in the Maghreb
(the western region of North Africa, between Constantine and the
Atlantic), in Muslim Andalusia and in Iffiqiya (the eastern region of North
Africa, between Tunis and Constantine). And it can be observed that they
536
are written in a completely different way from the eastern provinces’ Hindi
numerals.
We have also learnt about a means of calculation: a sort of wooden
board sprinkled with dust, the use of which was, as we shall see, linked with
Ghubar numerals.
We can also see that the tradition of an Indian origin for these numerals
had been transmitted by Arab and Maghrebi arithmeticians.
But the most important point concerns the verses written about Ghubar
numerals, and which ingeniously fix their shapes. The stability of these
verses from one manuscript to another is remarkable when one considers
that they are not copies of the same source, but two completely indepen-
dent manuscripts from different periods and locations.
They are an excellent way of memorising the nine numerals, by associ-
ating them with certain Arabic letters (or groups of letters), written in the
typical style of the old Maghribi and Andalusian script. They were pre-
sumably composed to teach pupils how to write the nine Indian numerals
in the style of their native province; it is rather as though we gave the
shapes of the Roman letters 0, 1, Z etc. to our children for them to learn
the numbers 0, 1, 2 etc.
Figure 25.6 contains further explanations of each line, as it appears in
manuscript. The exact forms have been recreated, with reference to
local scripts and drawing on parallels with the numerals contained in
these manuscripts.
The two oldest known documents which refer to Ghubar numerals
and calculation date back to 874 and 888 CE [see JASB 3/1907; SC
XXIV/1918]. The shapes of the numerals they contain are close to those
in Fig. 25.6 and, of course, to those described in the verses quoted above.
And, as the most recent manuscript containing these verses comes from
the beginning of the nineteenth century, it can be supposed that the
forms of the Ghubar numerals were fixed centuries ago and passed down
from generation to generation in this manner. In other words, an attempt
was made to prevent the Ghubar numerals from being altered by scribes.
These verses can also be found in numerous other arithmetical treatises.
The original forms of these numerals were conserved no doubt
because the Maghrebi are attached to traditions coming from the Muslim
conquest of Andalusia and North Africa. And that is when these
numerals arrived in these regions and were then adapted to the local
cursive scripts.
537
THE TRANSMISSION OF INDIAN NUMERALS TO WESTERN ARABS
Reconstruction of Ghubar script numerals,
from the style of the Maghribi letters and
the mnemonic poem
Ghubar numerals as they appear in
manuscripts
Letters, words
or images in
the poem:
from the
2nd passage
cited
from the
3rd passage
cited
from the
1st passage
cited
from the
2nd passage
cited
from the
3rd passage
cited
1
an alif
i
f
I
f
I
2
a ya
a ha 1
L.
C
% A. > x.
Z—
7
3
the word hijun
5
E
tu )
*
f
4
the word 'awun
y. y
r
5
an 'ay in
t
t
d i i
t
t
6
a final ha 2
<r
r
Q i <
S
(
7
an iron with a
bent hand an
upturned waw*
0
^ ? ? T
0
\3
8
two zeros
linked by al alif
a
a
6 t ft
t
*
9
a waw
3
3
y j *
r
1
1: The author of the second passage notes that the ha "is not pure" This remark, referring
here of course to the number 2, seems to mean that the variant similar to this letter
(which is also found in manuscripts) was not the original shape of 2 and that it had ini-
tially been more like the final form of ya, which is often written in this way in the
Maghribi script (Fig. 25.8).
2: Such is, in fact, the final form of ha, as it occurs in Maghrebi and Andalusian manu-
scripts (Fig. 25.8A).
3: The existence of this variant of the number 7 (as an upturned waw) is confirmed in
a marginal note which occurs in the manuscript of the first passage.
. 25.6.
THE TRANSMISSION OF INDIAN NUMERALS TO
WESTERN ARABS
The question that now needs answering is how and when Indian arithmetic
arrived in North Africa and Spain.
Woepcke provides us with part of the answer:
Even though the unity of the caliphs came to an early end,
pilgrimages to Mecca, flourishing trade, individual travels,
migrations of entire populations and even wars kept up a constant
communication between the various lands inhabited by Muslims.
Once Indian arithmetic was known in the East, it inevitably became
introduced into the West. A lack of precise information concerning
this event in the history of science makes dating it impossible,
but we are probably not far from the truth if we say that
Indian arithmetic arrived in North Africa and Spain during the
ninth century.
It is important to remember the special relationship the Caliphate of
Cordoba had with Byzantium, which allowed the circulation of certain
ancient texts. It can also be supposed that this facilitated contacts and
meetings with representatives of Indian culture in the cosmopolitan world
of Byzantium. But we should also bear in mind the contact that the
Andalusians and Maghrebi must have had with their eastern cousins, with-
out passing through Byzantium.
The arrival of Indian arithmetic in these regions could easily have come
about either through texts written by eastern Arabs, or via more direct con-
tacts with Indian scholars; thus in a similar way to what happened between
India and the eastern Arabs.
But we must not overlook the vital role Jewish tradesmen and mer-
chants probably played in this transmission. This is, in fact, suggested by
Abu’l Qasim ‘Ubadallah, a Persian geographer working in Baghdad. Better
known as Ibn Khurdadbeh, he wrote as follows in his work entitled Book of
Routes and Provinces (c. 850 CE):
Jewish merchants speak Arabic as well as Persian, Greek, Latin and
all other European languages. They travel constantly from the
Orient to the Occident and from the West to the East, by both land
and sea. They take ship from the land of the Latins [franki] by the
western sea [the Mediterranean] and sail towards Farama; there,
they unload their merchandise, place it in caravans and take the
overland route to Colzom, on the edge of the eastern sea [the Red
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
5 3 8
Sea]. From there, they take ship again and sail towards Hejaz
[Arabia] and Jidda, before moving on to Sind, India and China.
Then they return, bringing with them goods from the east
These travels are also made by road. The merchants leave the land
of the Latins, go towards Andalusia, cross the patch of sea [the
Straits of Gibraltar] and travel across the Maghreb before reaching
the African provinces and Egypt. They then travel towards Ramalla,
Damascus, Kufa, Baghdad and Basra, before coming to Ahwaz, the
Fars, Kerman, the Indus, India and China [quoted in Smith and
Karpinski (1911)].
Similar information about these merchants can be found in this extract
from the Gulistan ( Rose Garden), written by the Persian poet Sa’adi in the
first half of the thirteenth century [see E. Arnold (1899); D. E. Smith and
L. C. Karpinski (1911)]:
I met a merchant who had a hundred and forty camels
And fifty porters and slaves . . .
He replied: I want to take Persian sulphur to China,
Which, from what I have heard,
Fetches a high price in that country;
Then procure goods made in China
And take them to Rome (Rum);
And from Rome load a boat with brocades for India;
And with that trade for Indian steel (pulab) in Halib;
From Halib, I shall transport glass to the Yemen,
And take back Yemeni painted cloth to Persia.
Unlike Ibn Hauqal, the poet does not specify the origin of this travelling
merchant, who may not be Jewish. The Jews have never had a monopoly
over international trade. So Jewish traders were merely one of the
numerous links in this chain of transmission.
Whether they were or were not Jewish, these tradesmen used
numbers as often as they travelled or traded. And, like the various
languages they learnt in their business, they must also have become
acquainted with the different systems of arithmetic used by the peoples
they encountered.
As India was part of their route, they must surely have been obliged to
learn Indian numerals and arithmetic, and were thus one form of commu-
nication between India and the Maghreb.
FROM HINDI NUMERALS TO
GHUBAR NUMERALS A SIMPLE
QUESTION OF STYLE
To return to Arabic numerals, the Indian influence is evident, whether it be
on the Hindi symbols, or the Ghubar (Fig. 25.3 and 6).
Even a rapid comparison between the Indian Nagari numerals and the
Ghubar shows of course the presence of the Indian 1, but also 2, 3, 4 (with
a slightly different orientation in Arabic), 6, 7, 9 and 0, and even 5 and 8
(Fig. 25.5 and 7).
The Arabic numerals below (attested
in the early period of the Maghrebi
and Andalusian provinces)
Correspond to the Indian
numerals below (in a variety
of styles, from Brahmi to Nagari,
including others attested from
the beginning of the
CE to the eighth century)
1
f \ 1
— 'X.
l
2
•SL-* Z * *
= 2, ^ 3* V 3,
2
3
3
4
&
4
5
i 9 t £
b h r y m 8
5
6
e 6 a Q S
f f 7 Jr f £
6
7
1 S ?7 /) /i?
? 7<\?
7
8
3 3 # Z « 8
8
9
? <51 f <\
9
Fig.
FROM HINDI NUMERALS TO GHUBAR NUMERALS
5 39
In palaeographic terms, there is thus no difference between the Hindi
numerals of the Machreq and the Ghubar numerals of the Maghreb. Both
come from the same source. Any differences between them simply derive
from the habits of scribes and copyists in the two regions.
The history of Arabic writing styles helps us to understand these changes
more clearly (Fig. 25.8). From the beginning of Islam, two distinct forms of
writing evolved: a lapidary cursive style, derived from pre-Islamic inscrip-
tions; and an even more cursive style, from the earliest written Arabic
manuscripts, also dating to before the Hegira.
The lapidary cursive style produced the Kufic script, for inscriptions and
manuscripts, with its characteristic horizontal base line on which the rigid,
angular letters are set vertically. According to Ibn al-Nadim’s Fihrist (987),
this script derived from the early habits of the stone-carvers and scribes
from Kufa on the Euphrates, hence its name. (Founded in 638 CE, Kufa was
a centre of learning under the Omayyad caliphs until the foundation of
Baghdad in 762.) This script was also used, during the first centuries of
Islam, for legal and religious texts (in particular for the first copies of the
Koran, in mosques and on tombstones), which explains its hieratic nature.
It was then gradually replaced by the naskhi script, generally used by
copyists, and leading to the elegant calligraphy of the “Avicenna” Arabic
alphabet which is most commonly used today. Derived from ancient cur-
sive Arabic manuscripts, this style is marked by its smooth rounded forms,
broken up into small curved elements. It is also the source of the nastalik
script, used in Persia, Mesopotamia and Afghanistan, and the sulus script
of the Turkish Ottoman Empire. With certain exceptions, the form of the
letters remained very similar to Naskhi.
The difference between the two styles, at least at the beginning, was
really due to what they were used for and the material they were written on.
While the cursive manuscript style was used for everyday texts on papyrus
or parchment, the other one was reserved for inscriptions on stone, wood
or metal. The former was traced onto the papyrus, parchment or other
smooth surface with a quill or a reed (the famous qalam, or “calamus")
dipped in thick ink. But the latter was sculpted into stone, carved into
wood or engraved into copper. This naturally explains the former’s smooth
rounded forms, contrasting with the latter’s angular rigidity.
If we now return to the numerals and compare the signs contained in
Fig. 25.3 and 5, we can see that the cursive Hindi numerals are far more
rounded than those of the Maghreb, with the base line of the former break-
ing up into small curves. In other words, the eastern Arabs’ numerals
follow closely the rules of the Naskhi script.
On the other hand, the Ghubar numerals, while remaining cursive, are
nevertheless obviously more angular, stiff and rigid. A closer look reveals that
their curves, down-strokes and angles are absolutely identical to those used in
the Kufic script. This is, at least, what is revealed in the original of Kashf al
asrar ‘an ‘ilm al gobar, by the Andalusian mathematician al-Qalasadi. Its let-
ters and Ghubar numerals are written in a way which reflects the pure Kufic
tradition from the early centuries of Islam. This manuscript dates from the
fifteenth century and the Institut des Langues Orientales in Paris possesses a
copy of it from a lithograph made in Fez [see A. Mazaheri (1975)].
This is not surprising, for the Maghribi (or African) script which spread
across North Africa, Sudan and Muslim Spain after the ninth century is in
fact nothing more than a manuscript Kufic.
It should not be forgotten that the Maghrebi and Andalusians were
extremely attached to ancient Islamic traditions. This is particularly true of
the lapidary cursive style of the first conquerors of the region, the Abbasids
of Samara, which gave the Maghribi script its stiffness and rigidity.
By fixing their forms by means of the verses quoted above (Fig. 25.6),
they were made to adopt the characteristic shapes of Maghribi letters and
thus follow its cursive rules.
To sum up, whatever differences there may be between Hindi and
Ghubar numerals, their common source is demonstrably Indian.
But it was not in their Hindi form, but in the Ghubar style that Indian
numerals migrated from Spain to the Christian peoples of Western Europe,
before finally taking the shape they have today.
ARAB RESISTANCE TO INDIAN NUMERALS
It is tempting to think that the Indian system spread through the Islamic
world, replacing all other ways of representing numbers and, because of
their ingenious simplicity', the corresponding calculation methods were
rapidly accepted at all levels of Arab-Islamic society. The author humbly
admits that he was wrong in the first edition of the present work in which
he subscribed to that idea and neglected the following interesting details.
Of course, certain scholars such as al-Khuwarizmi and An Nisawi were suf-
ficiently astute to understand the superiority of this system. But there was
an equal number of Muslims who were, sometimes violently, opposed to
the use of numerals and even more so to their becoming generalised.
This means that, contrary to what is often believed, the domination of
the Indian system was a long, difficult process. Many arithmetical treatises,
for example, contain not a single Indian numeral, and sometimes no
numerals at all, because the numbers in each line are expressed by their
Arabic names. And if Indian numbers are to be found anywhere, then it is
most probably, or even one would think inevitably, in arithmetical works.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
NAME NUMERICAL in Naskhi in Maghribi
OF LETTER VALUE Arabic Arabic
alif 1 1
ba 2 V P
jim "C. 2
dal 4 a ►
ha 5 t 6
SHAPE OF LETTER
in Maghribi in Persian, in the
Arabic nasta'lik script
NASKHI STYLE
5
6
3
)
&
J or i.
V
J
e
J
j**
8
u>
i
j
j*
»
)
J or J
c
i
is
S
J
r
u
l" la . I.--* < j U a i i j yi« \ j Jjjj
r t - . • 'r .* * *.•+•** ?* •
0 w-SjiaJ' 0
- ^ / / ' 1 - ""
^ ^ ""l . r, ^
,-^j -C.— ■ ' * k w ^_L>w— ui Ou J J O ^a-»
O, . . ^ -/ • > . x ■ „ l _5 J >
'-IP Ip' ) P W — * W-l JjU*
^ ^ \ **•■'* *
j-- Ujkj ' ^1p
KUFIC STYLE
X,
prj 1 ♦ H- ,-jj L
i i.
U
M JL
u
U^-^O -li|j
LX,
o_J <vj
J>jLj
cLj&xi
jL jU>-*-IL
J>iLj
^jll[-JJ ,J*aA
■«< 4 ^jl!
ill
4b
Lji>&=»
MAGHRIBI STYLE
*6- _ _ ■* J . “ 0- Zi " o'
laula.^3 t 4 * c-^V 7 *£•
* A.t t.1.) W-^ Y < j^ULjV^
’ 1 , • 1 °i 3 - 0 .
^ y** laOUTk^> tuull ✓
« ^-jLa»Q^,q p— jIq^C. ft j # £flsS0 >4y4 »
„ . - 1 0 1 , ,.• - °(* = ^
4***j^^ ^^jlj 1>j >1 ^ laL <^0l / p.AA.Lo y -aD
Fig 2 5 . 8 b . Different styles of written Arabic (CPIN; see also de Sacy ; Sourdel in EIS )
Fig. 2 5 . 8 a . The Arabic alphabet in the Naskhi and Maghribi scripts
541
THE CONSERVATISM OF ARAB SCRIBES AND OFFICIALS
For Islam, like everywhere else, had its “traditionalists” bookkeepers
and accountants who remained deeply attached to previous practices and
vigorously opposed to scientific and technological innovations.
THE CONSERVATISM OF ARAB SCRIBES
AND OFFICIALS
One of the reasons for this opposition was the conservatism of Arab and
Islamic scribes and officials, who long remained attached to their ancestral
methods of counting and calculating on their fingers.
Thus, in his Kitab al mu’allimin (Schoolmasters’ Book), al-Jahiz gives this
advice, which provides a clear idea of the polemic that must have con-
fronted the users of Indian numerals and the ardent defenders of
traditional methods for several generations: “It seems better to teach pupils
digital calculation and avoid Indian arithmetic ( hisab al hindi), geometry
and the delicate problems of land measurement.” [British Museum Ms.
1129, f 13r].
This author, who scorned Indian numerals and arithmetic, thus recom-
mended teaching calculation using fingers and joints ( hisab al aqd) as
being, to his mind, more useful for the future official scribe of the period.
Some accountants even preferred manual calculation to the Arabs’ tradi-
tional means of calculation, the dust board.
This is, for example, revealed in Kitab al hisab bila takht bal bi’l yad
(“Treatise on calculation without the board, but with [the fingers of] the
hand”), written in 985 by al-Antaki [see A. Mazaheri (1975)].
In his Adab al kutab, destined for scribes and accountants, the Persian
writer As Suli (died 946) gives the reason for this preference for manual
calculation. After mentioning the “nine Indian characters” and “the
great simplicity of this system” when expressing “large quantities” he
then adds: “Official scribes nevertheless avoid using this system because
it requires equipment [i.e. a counting board] and they consider that a
system that requires nothing but the members of the body is more
secure and more fitting to the dignity of a leader." As Suli then eulogises
the official accountants of the Arab-Islamic world, with their supple
joints and movements “as fast as the twinkling of an eye” He quotes a
certain ‘Abdullah ibn Ayub who “compares the jagged lightning fork
with the rapidity of the accountant’s hand, when he says: ‘It seems that
its flash [of lightning] in the sky is made up of a scribe’s or accountant’s
two hands!’” Then he concludes: “That is why they content themselves
with just the iqd [i.e. counting on the fingers] and the system of joints”
[see J. G. Lemoine (1932)].
Officials always, of course, claim they are irreplaceable in order to keep
their privileged positions. They are thus never happy to see a new simple
system becoming generalised, which anyone can use without going through
their difficult and mysterious apprenticeship.
This is a universal tendency, which can be witnessed throughout
Antiquity, and in Western Europe from mediaeval times up until the
French Revolution. If Arab-Islamic scribes and officials violently opposed
the introduction of Indian numerals, it was because it could mean an end
to their monopoly.
But this traditionalism does not explain everything. We must also con-
sider the multiplicity and diversity of the peoples that made up the Muslim
empire. The heterogeneous nature of the cultures and populations of this
complex world, along with regional and individual habits, also played a part.
“Culture”, as E. Herriot put it, “is what remains when all else has been
forgotten.” It is the form of knowledge which enables the mind to learn
new things. Hence the idea of developing and enriching our various mental
faculties by intellectual exercises such as study and research.
But “culture”, in any given civilisation, is also the intellectual, scientific,
technological and even spiritual inheritance of its people. It is thus the sum
of knowledge, which its great minds have assimilated, and which greatly
adds to its enrichment.
In this way, Arab-Islamic civilisation was exceptional for its originality,
strong culture and deep insights of its thinkers, scholars, poets and artists.
And, to quote P. Foulquie, a culture is also the “collective way people
think and feel, the set of customs, institutions and works which, in any
given society, are at once the effect and the means of personal culture.”
Thus (to run Martin du Gard and Mead together), it is the set of virtues,
preconceptions, individual habits and works which make up a given nation
in its ways of behaving, acquired and transmitted by its members, who are
accordingly united by a shared tradition.
Like any other, Arab culture was also composed of varied customs,
countless details, endless habits and presumptions, characteristic of its
daily existence. Great minds thus coexisted with lesser, more ignorant souls
whose unthinking conservatism led them to clutch onto methods that had
been useful to their distant ancestors, but which had long since stopped
being appropriate to modern times and activities.
TRADITIONAL ACCOUNTANTS VERSUS USERS OF
OUTMODED SYSTEMS
When the Arab-Islamic civilisation found itself in contact with the
Christian West, some Arab accountants had the curious idea of adopting
Latin calculation methods using counters on a board, and thus set about
turning the clock back. This was the case with certain Syrian and Egyptian
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
accountants, presumably under the influence of their trading links with the
Genoans and Byzantines.
This was severely criticised by the Persian historian Hamdullah who, in
his 1339 Nuzhat al qolub (work of geography and chronology), says: “In the
year 420 [of the Hegira, thus 1032 CE], Ibn Sina invented the ‘calculation
knots’, thus freeing our accountants from the tedium of totting up counters
[ mishsara shumari] on instruments and boards, like the Latin abacus
[takhatayi jrenki ] and suchlike” [see A. Mazaheri (1975)].
As an accountant, Hamdullah had certainly been deeply impressed by a
calculation method called ‘uqud al hisab ("calculation knots”), recom-
mended for accountancy two centuries before by the famous Ibn Sina
(Avicenna), then the finance minister of Persia, under Buyid domination.
To gain a better understanding of this method, we must remember that
a “knot” (in Arabic ‘aqd or 'uqda, the singular of ‘uqud or ‘uqad) had at this
time not only its primary meaning, but also signified “class of numbers cor-
responding to the successive products of the nine units and any power of
ten” In other words, the “knot” stood for the decimal system. There was
the units knot, the tens knot, the hundreds knot and so on. This same term
can be found in al-Maradini [see S. Gandz (1930)] and in Ibn Khaldun’s
Prolegomena [see Muqaddimah, trans. Slane, I, pp. 243-4].
By extension, the expression ‘uqud al hisab came to mean “calculation
knots” in reference to an ancient way of recording numbers on knotted
cords, used by the Arabs in antiquity. The various places of consecutive
digits were marked by knots tied in predetermined positions. This system
was thus very similar to the South American Incas’ quipus and the ancient
Japanese ketsujo, used until recently in the Ryu-Kyu Islands (Fig. 25.9).
The Arabs (presumably before the advent of Islam) had long used these
knotted cords as a way of noting numbers for administrative records. The
numbers thus tied on the strings recorded accounts and various invento-
ries. This is reminiscent of the tradition, reported by Ibn Sa’ad, according
to which Fatima, Mohammed’s daughter, counted the ninety-nine attrib-
utes of Allah, and the supererogatory eulogies which followed the
compulsory prayers, on knotted cords, and not on a rosary. These cords
were also used as receipts and contracts. This is shown by the fact that, in
Arabic, the word 'aqd means both “knot” and “contract”
To return to the “calculation knots” which Avicenna is supposed to have
invented, it is highly probable that Hamdullah was referring to a means of
manual calculation.
The common Arabic expression for “hand counting” is hisab al yad
(from hisab, "counting, calculation”, and yad, “hand”). It can be found, for
example, in al-Antaki and As Suli lop. cit.), as well as al-Baghdadi in his
Khizanat al ‘adab.
542
But in many authors, the word ‘aqd or ‘uqda (“knot”) also means the
“join” between the finger and the hand, and by extension the “joints” of the
finger. For, this hisab al ‘uqud (“counting with knots”) is in fact “counting
on the joints of the fingers”, by allusion to the “knot” of the joints and the
“join” between the fingers and the hand.
There were several ways of counting on fingers in Islam. Although
Hamdullah is vague about Avicenna’s method, it is possible to work out
what it was by elimination. To Hamdullah’s mind, the word ‘uqud in the
expression ‘uqud al hisab (“calculation knots”) could in fact have meant the
“order of units” in an enumeration. And, as this concerns a manual
method, the “knots” in question could refer to units in a highly evolved
decimal system. What comes to mind is that “dactylonomy”, similar to deaf
and dumb sign language, which was used by the Arabs and Persians for
centuries, in which the units and tens were counted on the phalanxes and
joints of one hand, while the other one was symmetrically used for the hun-
dreds and thousands (see Chapter 3). This system was famously described
in a poem written in rajaz metre, called Urjuza Ji hisab al ‘uqud, composed
before 1559 by Ibn al-Harb and dealing with the science of “counting on
phalanxes and joints” [see J. G. Lemoine (1932)].
But this cannot be the method referred to by Hamdullah. As Guyard
explains: “the word ‘uqud, taken as a noun, stands for the shapes obtained
by bending the fingers and, by extension, the numbers thus formed.” That
is why the units in the manual systems already alluded to were called
“knots” But, this same word ‘uqud, taken as an action, means “bending the
fingers” [see JA, 6th series, XVIII (1871), p. 109], And, since he is discussing
arithmetic, what Hamdullah is talking about is definitely an action, not a
state. It is thus the science of calculating with what may be called “moving
knots” which is in question. For the other systems were mere static ways of
counting on the fingers and joints of the hand (just simple manual repre-
sentations of numbers), whereas the technique being envisaged here allows
calculations to be made by actively bending the fingers.
By opposing “calculation knots” to the Latin abacus, Hamdullah was
thinking of “knots” as an action, bending certain fingers and straightening
others, allowing arithmetical operations to be carried out in a much easier
way than on the abacus. That is why, according to this admirer of Avicenna,
these “moving knots” had freed “our accountants from the tedium of tot-
ting up counters” thrown down onto “the Latin abacus and suchlike.”
But Hamdullah is guilty of making an historical mistake. The method he
attributes to Avicenna had already existed in the Islamic world for a long time.
This is not our accountant historian’s only slip. For the method recom-
mended by the famous philosopher was only of use in operations on
common numbers. Hence Hamdullah’s error of judgment. He had not
understood that the Latin abacus, primitive though it was, allowed num-
543
THE NUMERICAL. NOTATION OF ISLAM’S OFFICIALS
bers to be reached that are far higher than can be obtained by any form of
manual calculation, no matter how elaborate. For the limits of the human
hand set the limits of the method.
Thus it was that, through ignorance of basic practical arithmetic, or per-
haps through sheer bloody-mindedness, users of a totally outmoded means
of calculation attacked other accountants with methods as primitive as
their own. The latter were, of course, to be upbraided for falling for a tech-
nique that came from a culture that was quite alien to Islam, and which the
former presumably held in disdain.
In this context, it is easy to imagine how both camps violently opposed
the introduction of Indian numerals and calculation methods, whose evi-
dent superiority over their archaic ways they would never admit.
Fig. 25 . 9 . Japanese ketsuj 0
This was a concrete accountancy method, used in ancient Japan and analogous to the quipus of the
Incas (Peru, Ecuador and Bolivia). Given the universal nature of this method, this Figure will provide
a good idea of how Arabs used knotted cords in the pre-lslamic era and probably also in the early
days of Islam (despite lack of evidence).
This ketsujo stands forfthe knots represent sums of money, as used in the Ryu-Kyu Islands, particu-
larly by workmen and tax collectors) [Frederic 1985, 1986, 1977-1987, 1994]:
A - cloth account given to the State, ora temple, from left to right:
- Yoshimoto family: 1 jo, 8 shaku, 5 sun and 7 bu;
- 1 jo, 4 shaku, 3 sun and 7 bu;
- Togei family: ibid.
B - Horizontal strand: 20 households.
Others, from right to left: 3 hyo, 1 to, 3 shaku and 2 sai.
THE NUMERICAL NOTATION OF ISLAM’S
OFFICIALS
In fact, the Indian system was introduced into the Islamic world in sev-
eral steps. As operators, and thus as a means of calculation, the numerals
were rapidly adopted by mathematicians and astronomers, soon followed
by an ever-increasing number of intellectuals, mystics, magi and sooth-
sayers. Meanwhile, others preferred to calculate by using the first nine
letters of the Arabic alphabet (from alef to ta). But as a way of represent-
ing numbers (i.e. when noting numerical values and not making
calculations), Indian numerals did not completely replace traditional
notation until a relatively recent date.
Thus it was that Arab, Persian and Turkish officials continued to favour
their own special notations, which had nothing to do with the Indian
numerals in public use, for official and diplomatic documents, bills of
exchange and administrative circulars until the nineteenth century.
This is shown in the Guide to the Writer's Art (1571-1572) [BN, Ms. ar.
4,441], which is a sort of handbook for professional writers. It gives a clear
idea of the plurality of the numerical systems used by the scribes, officials
and accountants of the Ottoman Empire at the end of the sixteenth century
(Fig. 25.10).
Among these varied forms, let us mention the Dewani numerals used in
Arab administrations, and the Siyaq numerals favoured by the accounts
offices in the Ottoman Empire’s finance ministry and in Persian adminis-
trations. These numerals were, originally, simply monograms or
abbreviations of the names of the numbers in Arabic, written in an
extremely cursive style. Later, they became so stylised and modified that
their origins were scarcely recognisable. It is easy to understand how they
were used to prevent fraudulent alterations to accounts, while at the same
time leaving the general public in the dark as to what amounts were being
described [see H. Kazem-Zadeh (1913); A. Chodzko (1852); L. Fekete
(1955); A. P. Pihan (1860); C. Stewart (1825)].
We should also like to mention the Coptic numerals, used since antiquity
by officials in the Arab administration of Egypt in their accounts, which
were in fact slightly deformed letter numerals from the ancient alphabet of
the Christian Copts of Egypt.
The Dewani numerals
These numerals were used in Arab administrative offices (called dewan,
hence their name).
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
544
Fig. 25 . 10 . Page from on Arobic work, entitled Murshida fi Sana at al katib ( Guide to the
Writer's Art"). Dated 1571-1572, it is a sort of handbook for professional writers.
It gives a very clear idea of the numerous different ways Arab-Muslim scribes, accountants and
officials wrote down their numbers at the end of the sixteenth century. It contains, countingfrom the
top down: the Ghubar numerals (2nd line) (Fig. 25.5): the Arabic letter numerals (5th line); the
Hindi numerals (6th line) (Fig. 25.3); then the Ghubar numerals again (7th line); the Dewani
numerals (8th line); the Coptic numerals (9th line); the Arabic letter numerals (10th line); the Hindi
numerals (11th line); the Ghubar numerals (12th line); two variants of the Coptic numerals (13th
and 14th lines); etc. IBN Paris, Ms. ar. 4441, f 22 j
They are abbreviations of the Arabic numerical nouns. Thus, number 1
is the letter alif, standing for ahad, “one”. Similarly, numbers 5, 10 and 100
correspond to the letters kha, ‘ayin and mim, standing for khamsa, “five”
‘ashara, “ten”, and mi’at, “hundred”
As for the number 1,000, it is a stylised form of the complete word alf,
meaning “thousand” Number 10,000 corresponds to a monogram of
‘ asharat alaf, “ten thousand” [A. P. Pihan (I860)].
Units
1
1
4
LuJ
7
Ijm
2
U
5
8
w
3
ill or HJ
6
9
\ju)
Tens
10
40
70
IdM
20
50
80
30
-Co
60
90
Xu
Hundreds
_D
O
O
400
700
1*4
200 J\o
500
U*
800
300 Uflu or
600
Isus
900
Thousands
1,000 'hJI or ljJI
4,000 cjJl*J
7,000 oJUt
2,000 (J slJI
5,000
8,000 uJLf
3,000
6,000
9,000
Ten Thousands
10,000 (jJlj,
40,000
70,000 ibut
20,000 Lh/y
50,000 iL*
80,000
30,000 ^
60,000
90,000 iLx *'
Hundred Thousands
100,000 oJMo
400,000
700,000 cjJHo Ljm
200,000 uJI I*
500,000 cjJJWk
800,000 uJM*
300,000 oJllotH.
600,000
900,000 lU
545
THE NUMERICAL NOTATION OF ISLAM’S OFFICIALS
Composite numbers
Units are always placed before tens and between the hundreds and tens,
as is done in spoken Arabic. Numerals are written from right to left, like the
Arabic words they represent in this same order for composite numbers.
11
17
206
14
21
3,478
15
9sr
24
62,789
The numerals of Egyptian Coptic officials
The Arab administration of Egypt employed Christian Copts, who had
their own special accountancy notation. These signs (which can be found in
several Arabic manuscripts from this region) are cursive derivatives of the
letter numerals of the Coptic alphabet, itself derived from Greek. Numbers
up to nine thousand are reached by using the units and underlining them.
For the ten thousands, the tens are underlined, as are the hundreds for the
hundred thousands. Finally, composite numbers are always topped by a
slightly curved line.
Units
Thousands
1,000 -v
4,000 2 ,
7,000 3
2,000 ^
5,000 i
8,000 t
3,000
6,000 C
9,000 £
Ten Thousands
10,000
l
40,000
*
70,000
O
s'
20,000
lu
50,000
V
80,000
*
30,000
J-
60,000
p
90,000
Ss
Hundred Thousands
100,000
400,000
c .
700,000
200,000
500,000
2
800,000
Ckj
X
300,000
X
600,000
5
900,000
z
Composite numbers
16 lr
803 ^
38,491
45
4,370 ?/- b
752,020
s'
The Persian Siyaq numerals
These numbers were used in Persian administrations, and were also
favoured by tradesmen and merchants. They are abbreviations of the
words for the numbers in Arabic (and not in Persian). They are written
from right to left, like the Arabic words they represent, as are the compos-
ite numbers [A. P. Pihan (1860); see also A. Chodzko (1852); H.
Kazem-Zadeh (1913); C. Stewart (1825)].
Units
Hundreds
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
546
Tens
10
r*
40
70
r"
20
50
r"
80
r’
30
r*
60
r
90
r’’
Composite numbers from 11 to 18
For these numbers, the final line of the units is rounded off and rises up
towards the top of number ten:
11
14 |-&TJ
17 r&i
12
i 5
t
oo
rH
Composite numbers from 21 to 99
The units and the other ten digits are linked together in the same way:
21
54
OO
Ir
43
76
99 r&
Hundreds
When written on their own, the hundreds have special signs, sometimes fol-
lowed by a sort of upturned comma and full stop, which are always omitted in
composite numbers. One sign calls for particular attention, because of possible
errors (if). With a line before, it stands for 400, and with no line, 700. The
same sign, with an additional curl to the right, stands for 900:
100
• ii
400
.<£/
700
.(if
200
.tf)
500
• iLzr
800
.(0
300
600
.(V
900
Composite numbers from 101 to 999
101
366
791
r"*
109
377
r&b'
820
P»*jc6
110
388
896
111
399
r&b
915
204
.-Alf)
472
999
r&V
Thousands
To form the multiples of 1,000, the characteristic patterns of the units are
used, with the final stroke lengthened from right to left. In this position,
and with a pronounced broadening, it is enough to indicate the presence of
the thousand in the combination:
1,000
»Ujl
4,000
7,000
_^l/l
2,000
5,000
. \S7
8,000
3,000
6,000
9,000
Composite numbers of four digits
The group •Uj! stands for the thousand, but only that exact value. For,
when followed by hundreds or tens, the group is used (abbreviation
of the Arabic word oUl , alf “thousand”):
1,050
ro/
1,200
d H>1
4,377
1,100
1,250
cZ&f
5,555
1,150
3,213
9,786
Ten and Hundred Thousands
After 10,000, the group l j t (abbreviation of the number 1,000) reap-
pears, and the final stroke of the ten thousands is lengthened below the
signs, instead of going down vertically:
10,000
99,112
25,072
110,100
'iL>f u
34,683
245,123
45,071
300,000
50,008
456,789
r&'&UsW
547
THE NUMERICAL NOTATION OF ISLAM’S OFFICIALS
Other variants of the Persian Siyaq numerals
Variants
noted by
Forbes
Variants
noted by
Stewart
1
/
2
H
&
4
S
5
A
**
6
L
1
S
S
8
jrV ord£-
9
n
22
33
44
55
<—
(U te
66
•— o
77
«-t Cr
88
99
Q-y>>
Variants
noted by
Forbes
Variants
noted by
Stewart
100
L
L
200
A
A
300
L or tr
t-
400
tS! or Ul
W
500
b
600
V
1/
700
if or U
U
800
0 or tf
V
900
8*oi
b
1,000
_J!
2,000
c/
t£fl
3,000
— K
40,000
^aJ
50,000
inn nnn
<4*
200,000
/ * / ) n
Note that the number 100,000 is none other than the Sanskrit word lakh ( ^ ), used by the
Indians for this amount.
The Siyaq numerals of the Ottoman Empire
These numbers were favoured by the accounts offices in the Ottoman
Empire’s finance ministry.
They are abbreviations of the words for the numbers in Arabic (and not in
Turkish). They are written from right to left, like the Arabic words they rep-
resent, as are the composite numbers. Also called Siyaq, they are analogous
to the Persian numerals of the same name, even though they differ in cer-
tain respects.
Note that the point (which stands for 6) normally replaces the other
sign ( L)for the same value in composite numbers. But when this point is
placed at the end of a number it is a mere punctuation mark, without any
numerical value. Finally, in composite numbers made up of tens and units,
the latter always come first, as in Arabic [A. P. Pihan (1860); see also L.
Fekete (1955)].
Units
1
J
4
jM
7
2
i
5
**
8
au or 4/
3
b
6
L' or .
9
3>
Tens
10
♦
40
70
'*0\
20
*-v/
50
80
‘-o
30
60
• -4/
90
•-(J 3
Hundreds
100
*L
400
700
*^6\
200
.jL
500
800
300
.1 V
600
.Leu?
900
Thousands
1,000
.-u JJ
4,000
7,000
2,000
•'-If
5,000
8,000
3,000
6,000
•*— !
9,000
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
Ten Thousands
10,000 ♦ <— CL
40,000 ♦
70,000 •
20,000 *
50,000 *
80,000
30,000 **~^**>
60,000 4
90,000 *
Composite numbers
Note that for composite numbers containing several digits, the Turks gen-
erally used the letter ** (sin), lengthening its horizontal stroke over the
group. This letter stood for the word & ** ( siyaq ).
*
641
1 1
L— L
168,875
347,592
* TjJ &
465,890
526,346
INDIAN NUMERALS’ MAIN ARAB RIVAL
Of all rival notations, with which Indian numerals were sometimes mixed
in Arabic writings, the most important was certainly Arabic letter numer-
als. These were known as hurufal jumal (literally “letters [for calculating]
series”) and also as Abjad (from its first four letters), because it does not use
the letters in the “dictionary” order, or mu’jama, (i.e. alif, ba, ta, tha.jim, ha,
kha, dal, dhal, etc.) but in a special order, called abajadi beginning alif, ba,
jim, dal, ha, wa, zay, ha, ta etc., attributed as follows: ‘a = 1, b = 2,j = 3, d = 4,
h = 5, w = 6, z = 1, h = 8,t=9 etc. This is not, of course, a simple number-
series (like one going from 1 to 26 by means of the Roman alphabet), but a
true place-value system, the first nine letters being the units, the next nine
the tens (y = 10, k = 20, / = 30, m = 40, n = 50, etc.), the following nine the
hundreds (q = 100, r = 200, sh = 300, ta = 400, etc.) and, finally, the twenty-
eighth letter standing for one thousand (gh = 1,000). Note that the al
abajadi order is very close to Hebrew, Greek and Syriac letter numerals and
548
is obviously the older order because it derives directly from the original
Phoenician alphabet (see Fig. 25. 8A). There were some differences between
East and West. In the former sin, sad, shin, dad, dha and ghayin stood for
60, 90, 300, 800, 900 and 1,000, but in the latter 300, 60, 1,000, 90, 800 and
900 respectively.
Islamic scholars and writers often preferred to use this system. One
example is the Kitab fi ma yahtaju ilahyi al kuttab min ‘ilm al hisab ( Book of
Arithmetic Needed by Scribes and Merchants), written by the geometer and
astronomer Abu’l Wafa al-Bujzani between 961 and 976.
The first two parts deal with calculating with whole numbers and frac-
tions, the third with surfaces of plain figures, the volumes of solid bodies
and the measurement of distances. The last four parts deal with various
arithmetical problems, such as in business transactions, taxation, units of
measurement, exchanges of currency, cereals and gold, paying and main-
taining an army, constructing buildings, dams etc. [A. P. Youschkevitch
(1976)]. Now, in this book, which was especially conceived for practical
use, the Indian decimal place-value system is never used. All numbers are
expressed by Arabic letter numerals.
A further significant example: the Kitab al kafi fi’l hisab (Summary of the
Science of Arithmetic), written by the mathematician al-Karaji towards the
end of the tenth century. It is rather similar to Abu’l Wafa’s work and, like
many later books, contains no mention of Indian numerals.
True, these works were especially for scribes, accountants and mer-
chants, and we know that this form of arithmetic was favoured not only by
scribes but also by officials and tradesmen. That is why this system stood
up for so long against the new Indian way, which was supported by al-
Khuwarizmi, An Nisawi and many others [A. P. Youschkevitch (1976)].
More surprisingly, the same phenomenon can be found in many Arabic
works dealing with algebra, geometry and geography, which also contain
only the letter system.
Works on astronomy
For astronomic treatises and tables, this was for a long time the only system
the Arabs used.
It may be useful to remind ourselves of certain points concerning the
sexagesimal system which the Arabs had inherited from the Babylonians,
via Greek astronomers.
Babylonian scholars used a place-value system with base 60 and, from
around the fourth century CE, they had a zero. These cuneiform numerals
were the vertical wedge for units and the slanting wedge for tens (see Fig.
13.41). As for zero, it was represented either by a double oblique vertical,
or by two superimposed slanting wedges.
549
INDIAN NUMERALS MAIN ARAB RIVAI.
This system was then adopted by Greek astronomers (at least from the
second century BCE), but only to express the sexagesimal fractions of units
(negative powers of 60). Otherwise, instead of using cuneiform signs, the
Greeks had their own letter numerals, from a to 9 for the first nine num-
bers, the next five (t to v) for the first five tens, with all the intermediate
numbers expressed as simple combinations of these letters. Influenced by
the Babylonians, they introduced a zero expressed either as sign written in
various different ways (presumably the result of adapting old
Mesopotamian cuneiform into a cursive script), or as a small circle topped
by a horizontal stroke (probably the letter omicron (o) the initial letter of
ouden, “nothing” and topped with a stroke to avoid confusion with the
letter o which stood for 70); or else as an upturned 2 (probably a cursive
variant of the above) (see Fig. 13.74A and *Zero).
Arab astronomers also took over the Greek sexagesimal system, adapt-
ing it to their own alphabet. Note that to express zero in their sexagesimal
calculations, the Arabs hardly ever used the Indian signs (the circle and
the point). Instead they used a sign written in a variety of different ways
(including the upturned 2 referred to above) which they had also
inherited from the Greeks. Woepcke has this to say about the Arabs’ sexa-
gesimal system:
Rather than [Indian] numerals, the Arabs preferred an alphabetic
notation for their astronomic tables. They apparently found it
more convenient. This use is confirmed in Arabic manuscripts con-
taining astronomic tables, in which Indian numerals are rarely met
with. The Arabs sometimes used them to express very large num-
bers, for example degrees over the circumference [see JA April-May
(I860)]. However, this exception was unnecessary. Sexagesimal cal-
culation, just as it had divided the degree into minutes, seconds,
thirds etc., also had higher values, superior to the degree so that it
was unnecessary to go higher than 59 in this notation. This is
revealing about the relationship between sexagesimal calculation
and alphabetic notation: it is after the number 60, useless in a rig-
orous sexagesimal system [i.e. based on place-value], that the
divergence between the African and Asian alphabetic notations
began [F. Woepcke (1857), p. 282],
The Arabs wrote an expression such as 0° 20' 35" as follows (reading from
right to left):
X
HL K 0
<
35 20 0
O being zero, K the letter kaf = 20 and the group FH, or lam-ha, the juxta-
position of lam ( = 30) and ha ( = 5).
To sum up, in their sexagesimal calculations and tables, Arab
astronomers generally used their alphabet in the way described above (see
Fig. 13.76). An exception to this rule was Abu’l Hasan Kushiyar ibn Fabban
al-Gili (971-1029), who wrote the Maqalatan fi osu’l hisab al hind ( Two
Books Dealing with Calculations Using Indian Numerals), the second book of
which is concerned with base 60. The “tables of sixty” (jadwal al sittini) are
expressed in the traditional Arabic letter numerals, but the operations are
made using Indian numerals [Aya Sofia Library, Istanbul, Ms. 4857, f 274
r and following; see A. Mazaheri (1975), pp. 96-141]. But, so far as I know,
this is the only author to break the rule stated above.
Books of magic and divination
The underlying reason for this preference is suggested in a work dealing
with Arab astrology and white magic, dating from 1631 CE [BN Paris, Ms
ar 2595, P 1-308]. The author, a certain al-Gili (not to be confused with the
mathematician cited above), uses a number of magic alphabets to name the
spirits and the seven planets and shows how to make talismans “using
Indian numerals” according to “the secret virtues of Arabic numerals”
When drawing up “judgements of nativities” (i.e. horoscopes) the writer
speculates about the numerals’ “magical properties” in what he calls hisab
al jumal (or “calculation of series”) in which the letters of the Arabic alpha-
bet are used, each with a number attached to it. He then draws up two lists
of numbers, one called the jumal al kabir (“large series”), the other jumal as
saghir (“little series”) [see P. Casanova (1922); A. Winkler (1930)]. The
author then explains how remarkable it is that “Arabic calculation {bi hisab
al ‘arabij is always used for the little series, and Indian calculation ( bi hisab
al hindi) for the large series” In other words, the “large series” is always
expressed in Indian numerals, the “little series” in Arabic letter numerals.
Why this difference? Large series were designed to give numerical
values, true arithmetical numbers, while the little series was compared with
it in order to give a name to each numerical value and determine its alleged
secret virtues. For the author is of course referring to letter numerals when
he mentions “the secret virtues of Arabic numerals” For him, the Indian
numerals had no hidden powers.
Thus, the Abjad system (also called Hurufal jumal) was considered by
the Arabs as “more their own than any other” [F. Woepcke (1857)]. They
even gave their own name to it: hisab al ‘arabi (“Arabic Calculation”).
Arab magi and soothsayers presumably wanted to make a clear distinc-
tion between a system which they considered to be typically Arabic and
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
part of Muslim traditions and practices, and another, the arithmetical
superiority of which they were willing to recognise, but which remained in
their eyes foreign and “not sacred”
A strange “machine" for thinking out events
To gain a clearer idea of the Arabs’ magical and divinatory practices, let us
listen to Ibn Khaldun who, in his Prolegomena, describes that strange
“machine” for thinking out events which is known as the za’irja. It inspired
Ramon Lull (died 1315) in his famous Ars Magna, and, even at the end of
the seventeenth century, Leibnitz was still one of its admirers.
It is claimed that by using an artificial system, we can know about the
contents of the invisible world. This is the za’irjat al ‘alam [“circular
chart of the universe”] supposedly invented by Abu’l Abbas as Sibti,
from Ceuta, one of the most distinguished of the Maghribi Sufi. Near
the end of the sixth century [of the Hegira = twelfth century CE], As
Sibti was in the Maghreb while Ya’qub al-Mansur, the Almohad
monarch, was on the throne.
The construction of the za'irja ["circular chart”] is a wondrous piece
of work. Many highly placed persons like to consult it to obtain useful
knowledge from the invisible world. They try to use enigmatic proce-
dures and sound out its mysteries in the hope of reaching their goals.
What they use is a large circle, containing other concentric circles,
some of which refer to the celestial spheres, and others to the ele-
ments, the sublunary world, spirits, all sorts of events and various
forms of knowledge. The divisions of each circle are the same as the
sphere they represent; and the signs of the zodiac, plus the four ele-
ments [air, earth, water and fire] are found within them. The lines
which trace each division continue as far as the centre of the circle and
are called “radii”
On each radius appears a series of letters, each with a numerical
value, some of which belong to the writing of records, that is to say to
signs which Maghribi accountants and other officials still use for writ-
ing numbers. [The author is of course referring to the monograms and
abbreviations of the Arabic names of the numbers, called Dewani
numerals].
There are also som egobar numerals [Fig. 25.5].
Inside the za'irja, between the concentric circles, can be found the
names of the sciences and various sorts of place name. On the other
side of the chart of circles, there is a figure containing a large number
of squares, separated by vertical and horizontal lines. This chart is
fifty-five squares high, by one hundred and thirty-one squares across.
550
[The author does not say that many of these squares are empty.] Some
of these contain numbers [written in Indian numerals], and others let-
ters. The rule which determines how the characters are placed in the
squares is unknown to us, as is the principle that determines which
squares are to be filled and which remain empty. Around the za’irja are
found some lines of verse, written in the tawil metre, rhymed on the
syllable la. This poem explains how to use the chart to obtain the
answer to a question. But its lack of precision and vagueness mean that
it is a veritable enigma.
On one side of the chart is a line of verse written by Abu Abdallah
Malik ibn Wuhaib [fl. 1122 CE], one of the West’s most distinguished
soothsayers. He lived under the Almoravid Dynasty and belonged to
the uleima of Seville. This line is always used when consulting the
za’irja in this way, or in any other way to obtain an answer to a ques-
tion. To have an answer, the question must be written down, but with
all the words split up into separate letters. Then, the sign of the zodiac
is located [in the astronomical tables] and the degree of that sign as it
rises above the horizon [i.e. its ascendant] coinciding with the
moment of the operation. Then, on the za’irja, the radius is located
which forms the initial boundary of the sign of the ascendant. This
radius is followed to the centre of the circle, and thence to the circum-
ference, opposite the place where the sign of the ascendant is
indicated, and all the letters found on this radius, from beginning to
end, are copied out.
Also noted are the numerical signs [Indian numerals] written
between the letters, which are then transformed into letters according
to the hisab al jumal system [the “series calculation”, used when replac-
ing Indian numerals by letters and vice versa]. Sometimes units must
be converted into tens, tens into hundreds, and vice versa, but always
under the rules drawn up for the za’ijra. The result is placed next to the
letters which make up the question. Then the radius which marks the
third sign from the ascendant is examined. All the letters and numbers
on this radius are written down, from its beginning to the centre, with-
out going to the circumference. The numbers are then replaced by
letters, according to the procedure already described, the letters being
placed one beside the other.
Then, the verse written by Malik ibn Wuhaib, the key for all opera-
tions is taken and, once it has been split into separate letters, it is put
to one side. After that, the number of the degree of the ascendant is
multiplied by what is called the sign’s ‘asas [literally “base” or “founda-
tion” an algebraic term for the index of a power, but here standing for
551
INDIAN NUMERALS MAIN ARAB RIVAL
the number of degrees between the end of the last sign of the zodiac
and the sign which is the ascendant at the time of the operation, the
distance being taken in the opposite direction from the normal order
of the signs].
To obtain this ‘asas, we count backwards, from the end of the
series of signs; this is the opposite of the system used for ordinary
calculations which starts at the beginning of the series. The product
thus obtained is multiplied by a factor called the great ‘asas and the
fundamental dur [“circuit” or “period” in astronomy used for
the time it takes a point to make a complete orbit of the earth. A
planet’s dur is thus either its orbit, or the time taken to return to any
given point in the heavens. But in the za’irja, dur also stands for
certain numbers used for selecting the letters which will give the
required answer].
The results are then applied to the squares on the chart, according
to the rules governing the operation, and after using a certain number
of dur. In this way, several letters are extracted [from the chart], some
of which are eliminated, while the rest are placed opposite Ibn
Wuhaib’s verse.
Some of these letters are also placed among the letters forming the
words of the question, which have already had others added to them.
Letters of this series are eliminated when they occupy places indicated
by the dur numbers. [Thus:] As many letters are counted as there are
digits in the dur, when the last dur figure is arrived at, the correspond-
ing letter is rejected; this operation is repeated until the series of letters
is exhausted.
It is then repeated using other dur. The isolated letters remaining
are put together and produce [the answer to the question asked by] a
certain number of words forming a verse, in the same metre and
rhyme as the key verse, composed by Malik ibn Wulaib. Many highly
placed persons have become absorbed in this pursuit and eagerly use
it in the hope of learning the secrets of the invisible world. They
believed that the relevance of the answers showed that they were
accurate. This belief is absolutely unfounded.
The reader will already have understood that the secrets of the
invisible world cannot be discovered by such artificial means.
It is true that there is some connection between the questions
and answers, in that the answers are intelligible and relevant, as in
a conversation.
It is also true that the answers are obtained as follows: a selection is
made between the letters in the question and on the radii of the chart.
The products of certain factors are applied to the squares on the chart,
whence some letters are extracted; certain letters are eliminated by
several selections using the dur, and the rest are then placed opposite
the letters making up the verse [of Malik ibn Wahaib].
Any intelligent person who examines the connections between
the various steps in this operation will discover its secret. For these
mutual connections give the mind the impression that it is in com-
munication with the unknown, and also provide the way of going
there. The faculty of noticing the connections between things is most
often found in people used to spiritual exercises, and practice
increases the power of reasoning and adds new strength to the fac-
ulty of reflection. This effect has already been explained on
numerous occasions.
This idea has resulted in the fact that almost everybody has attrib-
uted the invention of the za'irja to people [the Sufi, Muslim esoterics],
who had purified their souls by spiritual exercise.
Thus, the za’irja I have described is attributed to As Sibti [a Sufi]. I
have seen another one, invented, it is said, by Sahl ibn ‘Abdallah and
must admit that it is an astounding work, a remarkable production of
a profound spiritual application.
To explain why As Sibti’s za’irja gives a versified answer, I tend to
think that the use of Ibn Wuhaib’s verse as a starting point influences
the answer and gives it the same metre and rhyme.
To support this view, I have seen an operation made without this
verse as a starting point, and the answer was not versified. We shall
speak further of this later. Many people refuse to accept that this
operation is serious and that it can answer one’s questions. They
deny that it is real and look on it as something suggested by fancy
and imagination. If they are to be believed, people who use the za’irja
take letters from a verse they have composed as they see fit and insert
them among the letters making up the question and those from the
radii. They then work by chance and without any rules; finally they
produce the verse, pretending that it has been obtained by following
a fixed procedure.
Such an operation would only be an ill-conceived game. No one using
it would be capable of grasping the connections between beings and
knowledge, or of seeing how different the operations of perception are
from those of the intelligence. The observers would also be led to deny
anything they do not perceive.
To answer those who call the za’irja a piece of juggling, suffice it to
say that we have seen operations performed on it respecting the rules
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
and, according to our considered opinion, they are always carried out
in the same way and follow a genuine system of rules. Anybody pos-
sessed of a certain degree of penetration and attention would agree
with this, once one of these operations has been witnessed.
Arithmetic, a science producing absolutely clear results, contains
many problems which the intelligence cannot understand at
once, because they include connections which are hard to grasp and
elude observation.
How much more so, then, for the art of the za’irja, which is so extra-
ordinary and whose connections with its subject are so obscure?
We shall cite one rather difficult problem here, to illustrate this
point. Take several dirhams [silver coins] and, beside each coin,
place three fulus [copper coins]. With the sum of the fulus you buy
one bird, and with that of the dirhams several more at the same
price. How many birds have you bought? The answer is nine. We
know that there are twenty-four fulus to a dirham; so three fulus are
the eighth of a dirham. Now, since each unit is made up of eight
eighths, we can suppose that when making this purchase we have
brought together the eighth of each dirham with the eighths of the
other dirhams, and that each of these sums is the price of one bird.
With the dirhams we have then bought just eight birds; the number
of eighths in a unit; add to that the bird purchased with the fulus
and we have nine birds in all, since the price in dirhams is the same
as that in fulus.
This example shows us how the answer is hidden implicitly in the
question and is arrived at by knowing the hidden connections between
the quantities given in the problem.
The first time we encounter a question of this sort, we imagine that
it belongs to a category that can be solved only by applying to the
invisible world. But mutual connections allow us to extract the
unknown from what is known. This is especially true of things in the
sentient world and the sciences.
As for future events, they are secrets that cannot be known pre-
cisely because we are ignorant of their causes and have no certain
knowledge of them.
From what we have explained, it can be seen how a procedure
which, by using the za’irja, extracts an answer from the words of
the question is a matter of making certain combinations of letters,
which had initially been ordered to ask the question, appear in a
different form.
For anyone who can see the connection between the letters of the
question, and those of the answer, the mystery is now clear.
552
People capable of seeing these connections and using the rules we
have explained can thus easily arrive at the solution they require.
Each of the za’irja’s answers, seen under a different light, is like any
other answer, according to the position and combination of its words;
that is, it can either be negative or positive.
To return to the first point of view, the answer has another charac-
teristic: its indications are in the class of predictions and their
accordance with events [in other words, as Slane emphasises in
modern terms, these indications are part of the category of agreements
between discourse and the extrinsic].
But we shall never know [about future events] if we use procedures
such as the one just described.
What is more, mankind is forbidden to use it for these ends. God
communicates knowledge to whomsoever he wants; [for, as the Koran
says (sura 2, verse 216) God knows, but you know not.] [See Muqaddimah,
pp. 213-19; cf. Slane’s translation, pp. 245-53.]
We must salute, in passing, Ibn Khaldun’s eminently modern rationality,
categorically rejecting the rather strange practices of Arab astrologers and
soothsayers, which were in fact outlawed by Islam.
To this can be added the strange “revelation calculation” [hisab ‘an nim],
which soothsayers used in time of war to predict which of the two sover-
eigns would conquer or be conquered. Here is how Ibn Khaldun describes
it in his Prolegomena:
The numerical value of the letters in each sovereign’s name was
added up. Then each sum was reduced until it was under nine. The
two remainders were compared. If one was higher than the other, and
if both were odd or both even, then the king whose name had pro-
vided the lower figure would win. If one was even and the other odd,
the king whose name had provided the higher figure would win. If
both remainders were equal and even, then the king who had been
attacked would vanquish. But if both were equal and odd, then the
attacking king would be victorious [Muqaddimah, cf. Slane’s transla-
tion, I, pp. 241-2].
The underlying reasons for the preponderance of Arabic letter numerals
Thus, the system of Arabic letter numerals was favoured as a way of writing
numbers not only by scholars, mathematicians, astronomers, physicians
and geographers, but also by authors of religious works, mystics,
alchemists, magicians, astrologers, soothsayers, scribes, officials and
tradesmen, among both Arabs and Muslims.
553
NDIAN NUMERALS’ MAIN ARAB RIVAL
The system was so common in the Islamic world, that Arab poets even
invented a particular form of literary composition which used the letter
numerals. These ramz were versified according to the arithmetic equalities
or progressions of the numerical values of the letters in each line.
Even historians, and the lapidaries of North Africa, Spain, Turkey
and Persia, were (at least in later periods) fond of a technique called
tarikh, i.e. “chronograms” which consists in grouping a set of letters, the
numerical value of which when added together produces the date of some
past or future event, into one meaningful or significant word, or else into
a short phrase.
This shows how the representation of numbers was of vital importance
in the history of Islam. It was, of course, directly linked with both the
meanings and the characters of Arabic writing, since the “numerals” were
simply letters of the alphabet. This numerical notation was always written
from right to left, like words, and, as for ordinary letters, the characters
were generally joined up and slightly modified depending on whether they
were isolated, initial, medial or final.
Thus, for poets enamoured of the ramz, these “letter numerals” or
numerical letters were an integral part of their artistic expression, mirror-
ing the beauty of the language. For artists, they also harmonised with the
art of calligraphy, reflecting both their individual perspectives and the
emotional state in which the work was created. And for those with a mysti-
cal bent, these same “numerals” allowed them to produce graphic or
versified symbolic expressions, at once literal and numerical, of their quest
for Allah.
Meanwhile, the scribes, who adhered to their characteristic embellish-
ments, were able to give these numerical letters the same grace, balance
and rhythm as the ordinary letters in their miniatures and illuminations.
All of which confirms the perfect continuity between this system and
the purest of Arab and Islamic traditions, and the fundamental practices of
Muslim mysticism.
It must not be forgotten that the Arabic script is considered to embody
a Revelation and the spreading of the word of the Prophet; it is thus the
basic criterion for belonging to the Islamic community (the Umma). It is
this close connection between the Muslim religion and the Arabic script
which gives the Arabic alphabet its privileged, almost fundamentally
sacred, position. Tradition even has it that the reed pen, the famous qalam
(“calamus”), was the first of all Allah’s creations.
For the Hurufi (“Letterers”), sects based on beliefs attached to the
symbolic meaning of Arabic letters, a name was the essence of the
thing named.
And, as all names are supposed to be contained in the letters of the dis-
course, the entire universe was the product of Arabic letters. In other
words, from these letters proceeded the universe. Hence the association
between the “science of letters” (‘ ilm al hurufi, the “science of words” (‘/7m
al simiya) and the “science of the universe” ('ilm al ‘alam).
The mystic al-Buni was one follower of this belief. He established corre-
spondences between the Arabic letters and what he thought were the
elements of the visible world: the four elements (water, earth, air and fire),
the celestial spheres, the planets and the signs of the zodiac. And, as there
are twenty-eight letters, he associated them with the twenty-eight lunisolar
mansions [see E. Doutte (1909)].
God is a force translated by the Word; he acts through his voice and so,
by inference, through the very letters of the Arabic alphabet.
Thus the “sciences” of letters and words, once mastered, would reveal
the attributes of Allah as they are manifested in nature through the Arabic
letters. According to these doctrines:
The Arabic letter symbolises the mystery of being, through its fun-
damental unity derived from the divine Word and its countless
diversity resulting in virtually infinite combinations; it is the image of
the multitude of creation, and even the very substance of the beings it
names. Together, they are regarded as manifestations of the Word Itself,
inseparable attributes of the Divine Essence, as indestructible as the
Supreme Truth. Like the divine being, they are immanent in all things.
They are merciful, noble and eternal. Each of them is invisible (hidden)
in the Divine Essence [J. Chevalier and A. Gheerbrant (1982)].
This is why, according to the precepts of Islam, the Koran, as the
Revelation of the Prophet Mohammed, cannot be read in another language
than Arabic, nor can it be transcribed into a different script. For this Book,
seen as one of the expressions of the Word of Allah, is identified with the
Divine Essence. To quote Doutte:
This conception takes us back to ancient times, when the Romans, by
the word litterae, and the Nordic peoples, by the word “rune”, meant the
entirety of human knowledge. Nearer to the Arabs, in the Semitic world,
the Talmud teaches that letters are the essence of things. God created the
world by using two letters; Moses on going up to heaven met God who
was weaving crowns with letters. Ibn Khaldun has much to say about
these doctrines and gives a theory of written talismans [see
Muqaddimah, trans. Slane, II, pp. 188-95]: as the letters composing
them were formed from the elements which make up each being, they
could act upon them.
Such is the basis of ‘ilm al huruf and 'ilm as simiya, Islam’s mystical
“sciences” of letters and words.
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
One category of letters, whose magical powers have a religious
origin and are thus characteristic of Arab magic, are those at the begin-
ning of certain suras of the Koran, and whose meaning is totally
unknown (or else jealously guarded by Muslim mystics). For example,
sura II begins with alif, lam, mim\ sura III with alif, lam, mim, sad etc.
Orthodox Muslims call these letters mustabih, and say that their mean-
ing is impenetrable for the human mind; thus, unsurprisingly, they
have been adopted by magicians.
AI-Buni calls them al huruf an nuraniya; there are fourteen of
them, exactly half the number (28) of lunar mansions, from which he
draws further speculations. Each of them, he points out, is the initial
of one of the names of God. Two of these groups, which contain five
letters, have particularly attracted magicians. They are supposed to
have extraordinary virtues and many herz (“talismans”) have been
made using them.
If letters have magical powers, then these powers are increased
when they are written separately. In the Arabic script, individual let-
ters are more perfectly formed than when they are joined up.
But the letters’ most singular properties come from their numeri-
cal values. Two different words can have the same numerical total.
The mysticism of letters then says that they are equivalent. In the
Cabbala, this is the principle of “gematria” It is also a favourite of
Muslim magic. Not only are words linked together by the numbers
expressed by their letters, but these very letters can reveal their magi-
cal virtues through a numerical evaluation of the letters and words
[E. Doutte (1909)].
In other words, Arabic words have a numerical value. A reciprocal logic
even had it that numbers were charged with the semantic meaning of the
word or words they corresponded to. Hence, as with the Cabbala, ciphered
messages, “secret languages” and all sorts of speculations were cooked up
by mystics, numerologists, alchemists, magi and soothsayers. Their aim
was to stop laymen understanding and harmonising with these esoteric
meanings, which supposedly held a hidden truth, or else to compose cryp-
tographic texts wrapped in apparently indecipherable allegories and
puzzles, or to use them for a variety of interpretations, conclusions, prac-
tices and predictions (see Chapter 20).
It can thus be seen how a numerical value was added to the letters’ sym-
bolic, magical and mystical powers, thus giving them the broadest and
most effective range of meaning.
Words have always fascinated us, but numbers even more so. Since time
immemorial, numbers have been the mystic’s ideal tool. They do not
554
express only arithmetical values but, inside what was considered to be their
visible exteriors, numbers also contained magical and occult forces which
ran on an unseen current, rather like an underground stream. Such ideas
could be either for good or evil, depending on their inherent nature.
The magical and mystical character of numbers is a common human
belief. Their importance in Mesopotamia, ancient Egypt, pre-Columbian
America, China and Japan is beyond our scope. As are the theories and doc-
trines of the Pythagoreans and Neo-Platonists who, struck by the
importance of numbers and their remarkable properties, made them into
one of the bases of their metaphysics, believing that numbers were the
principle, the source and the root of all beings and things. But what should
be emphasised is the direct link between a belief in the magic of numbers
and the fear of enumeration, present among the Hebrews (see for example
Exodus, 30: 12 and II Samuel 24: 10), the Chinese and Japanese (who are
particularly superstitious about the number four), and also among several
African, Oceanian and American peoples, who find numbers repellent.
It should be said in passing, that the ancient fear of enumeration reveals
the difficulties humans have always had in assimilating the concept of
number, which they see, and rightly so, as highly abstract.
It is this very link between magic and the ancient fear of numbers which
forbids, for example, North African Muslims from pronouncing numbers
connected with people dear to them or personal possessions. For, accord-
ing to this belief, giving the number of an entity allows it to be
circumscribed. If you provide the number of your brothers, wives or chil-
dren, your oxen, ewes or hens, the sum of your belongings, or even your
age, you are giving Satan, who is ever on the lookout, the possibility to use
the hidden power of these numbers. You thus allow him to act upon you
and do evil to the people or things you so imprudently enumerated.
A sort of superstitious reciprocity led to the making of herz in the form
of magic squares: talismans with alleged beneficent powers, such as curing
female sterility, bringing happiness to a home or attracting material riches.
As a passing remark, Islamic religion and traditions see the number five as
a good omen in, for example, the five takbir of the Muslim profession of faith,
Allah huwa akbar (“God is Great”); the five daily prayers; the five days dedi-
cated to ‘Arafat; the five fundamental elements of the pilgrimage to Mecca,
the five witnesses of the pact of the Mubahala\ and the five keys to the mys-
tery in the Koran (6: 59; 31: 34). There is also Thursday, called in Arabic al
khamis (“the fifth”), which is a particularly sacred day. Then there are: the five
goods given as a tithe; the five motives for ablution; the five sorts of fasting;
the pentagram of the five senses and of marriage; the five generations that
mark the end of tribal vengeance; and so on. Naturally, there are the five fin-
gers of the hand, placed under special protection in memory of the five
555
INDIAN NUMERALS’ MAIN ARAB RIVAL
fingers of the “hand of Fatima” the daughter of Mohammed and Khadija,
and wife of ‘Ali, the Prophet’s cousin [see J. Chevalier and A. Gheerbrant
(1982); E. Doutte (1909); EIS; T. P. Hugues (1896)].
Even today if you foolishly ask Tunisians, Algerians or Moroccans how
old they are, or how much money they have, they will cast off the evil eye by
vaguely replying “a few” if they are polite, or else curtly say “five”, or even
brusquely slap the five fingers of their hand over your own "evil eye”
To sum up, each of the twenty-eight Arabic letters, as an ordinary letter,
was supposed to have its own symbolic meaning, magical power and cre-
ative force. But as a numeral or written in cipher, each was linked with a
number and, as such, was directly in touch with the supposed idea, power
and force contained in that number. A name is the outward sign of the
Word, considered to be one of the main magical and mystical forces. As it
is made up of letters, and thus of the corresponding numbers too, it is easy
to see why the Arabs’ alphabetic numbering (as a particular case among
their multiple ways of evaluating their letters and words) was for mystics,
magi and soothsayers a product of sound, sign and number, and hence had
powers that transcended the ordinary alphabet.
We can now see how important this system was at all levels of Islamic soci-
ety. And we can also see why the Indian place-value system was considered by
most authors to be something absolutely alien to their culture and traditions.
The direction it was written in added to its relative unpopularity. It ran
from left to right (one hundred and twenty-seven, for example, being writ-
ten as 127), the opposite way to Arabic script. And as the numerical letters
were written from right to left, from the highest digit to the lowest, and
obeyed the rules of the Arabic cursive script, they were favoured above any
other system.
The direction of Indian numerals had been highly practical for Indian
mathematicians and accountants, whose script went from left to right. But
this fact (which caused obvious problems for people accustomed to writing
from right to left) raised difficulties for Arab-Muslim scholars.
They would certainly have solved this problem if they had inverted the
original order of the Indian decimal system, by writing something like this;
8 7 6 3 2
when an Indian would have written:
23,678 (= 2 x 10 4 + 3 x 10 3 + 6 x 10 2 + 7 x 10 + 8).
They would thus have completely adapted the Indian system to their own
script. But this idea apparently never occurred to the Arabs, or else they
refused to break with the Indian tradition.
Another reason, the last we shall give here, for this opposition was
as follows.
During their relations with India, the Arabs were in contact with the
Hindi, but also with the Punjabi, the Sindhi, the Maharashtri, the
Manipuri, the Orissi, the Bihari, the Multani, the Bengali, the Sirmauri and
even the Nepali. A glance at Chapter 24 will confirm how much the writing
of numerals in India varied, not only from one period to another, but also
in different regions, and even with different scribes (Fig. 24.3 to 52). What
was a 2 for some became something like a 3, 7, or 9 for others, for palaeo-
graphic reasons. In other words, a lack of standardisation meant that the
written form of Indian numerals remained unstable. But for mathemati-
cians and astronomers numbers had to remain the same and be absolutely
consistent. How could one transmit the fundamental data of a work of
astronomy, for example, if the numerical value of observations and results
could be variously interpreted, depending on the time, place and habits of
the user? What is more, if a scribe or copyist made a mistake, it might
never be noticed. These numerals were therefore not sufficiently rigorous
for works dealing with mathematics, geography or astronomy in which
value was of prime importance. Hence the preference for numerical letters,
which did not present such a problem.
Need we add that, if the so-called “Arabic” numerals had really been
invented by the Arabs, then they would have been used more widely and
adopted by Muslims much more rapidly? There is also a good chance that these
numerals would have been written from right to left, like the Arabic script.
These important facts add to the indisputable evidence that our present
number-system comes from India.
Among other imperishable merits, the Arab-Islamic civilisation did cer-
tainly transmit our modern numerals and methods of calculation to
mediaeval Europe, which was at the time at a much lower scientific and cul-
tural level. In gratitude for this basic contribution, Europe then named
these numerals after the people who had provided them. But to say that
Islam was the cradle of these numerals would be to fall into the trap laid by
an erroneous term, which even Arab and Muslim scholars never used in
their writings or vocabulary.
DUST-BOARD CALCULATION
The time has come to discuss Indian calculation methods, which not only
played an important role in the transmission of Indian numerals throughout
the Islamic world, but also profoundly influenced how techniques evolved.
Many good reasons lead us to suppose that, from earliest times, Arab-
Muslim arithmeticians in the East and the West made their calculations by
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
sketching out the nine Indian numerals in loose soil, with a pointer, stick or
just with a finger. This was known as hisab alghubar (“calculating on dust”)
or hisab ‘ala at turab (“calculating on sand”).
But they did not always write on the ground; they also had other methods.
Their most common tool seems to have been the counting board, what is
called in the East takht al turab or takht al ghubar (from takht, “tablet” or
“board” turab, “sand” and ghubar, “dust”), which was also known in the
Maghreb and Andalusia as the luhat alghubar ( Mat being a synonym of takht).
Several Persian poets refer to it, at least from the twelfth century on,
such as Khaqani, in his eulogy for Prince Ala al-Dawla Atsuz (1127-1157)
[A. Mazaheri (1975)]:
The seven climates tremble with quartan fever;
And dust will cover the vaulted sky,
Like the accountant’s board (takht).
Or the mystical poet Nizami (died 1203) [Nizami (1313), cited by A.
Mazaheri (1975)]:
From the system of nine heavens
[Marked] with nine figures,
[God] cast the Indian numerals
Onto the earth board.
This counting board was favoured not only by professional Arab
accountants, mathematicians and astronomers, but also by magi, soothsay-
ers and astrologers.
In about 1155, Nizami told this story, which features the philosopher al-
Kindi (ninth century) [Nizami as above]: “Al Kindi asked for the dust
board, got up and [with his astrolabe] read the height of the sun, the hour
and traced the horoscope on the sand board ( takht al turab) . . .
It consisted of a board of wood, or of any other material, on which was
scattered dust or fine sand, so that the Indian numerals could be traced out
in it and calculations made. Powder, or sometimes even flour were also
used, as our sources indicate. The word ghubar in fact means “powder” or
“any powdery substance” as well as “dust”
This counting board was not unique to Arab arithmeticians. It was also
used long before Islam by the Indians (see *Patiganita).
TRACES OF THE OLD PERSIAN ABACUS FROM
THE TIME OF DARIUS
Old abacuses from time of Darius and Alexander were also used, at least in
Persia during the first centuries of the Hegira. Calculations were made by
556
throwing down pebbles or counters, and certain Persian accountants kept
up this method (see Fig. 16.72 and 73).
The following is, of course, just a hypothesis, but it is supported by
much of the evidence. The Persian verb “to count” “to calculate” is
endakhten, which also means “to throw” At this time, arithmetical opera-
tions were carried out on tables or rugs, divided by horizontal and vertical
lines, on which the counters were placed, their value changing as they
moved from one column to another.
It is also interesting to note that the action which corresponds to the verb
endakhten (“count” “calculate”) is endaza, which means three things: “throw-
ing” “counting” and “calculating” This is shown in this brief quotation from
Kalila wa Dimna, a famous Persian fable, here in a twelfth-century version by
Abu al Ma'ali [see A. Mazaheri (1975)]: “Having carefully listened to his
mother’s words / The lion threw them backwards (baz endakht) with his
memory.” This is so subtle that a commentary is necessary.
Even for a lion, “throwing words backwards” is meaningless. But if we
take the verb to mean “to calculate” or, by extension, “to measure” we can
then see that the king of the jungle had thought over, or “weighed”, his
mother’s words.
But let us not take etymology too far in order to explain something
which had already almost vanished from the old country of the Sassanids,
for these words had lost their numerical meaning by the thirteenth century.
And the instrument itself, rightly considered as cumbersome and impracti-
cal, had been rejected by the region’s professional accountants at an early
date. (Note also that they rejected the Chinese abacus, introduced by
Mongol invaders during the thirteenth century; but the unpopularity of
this excellent apparatus was due to the Persians’ hatred of Genghis Khan
and his successors.)
THE BOARD AS A COLUMN ABACUS
To return to calculations made on the ground, or else in dust scattered over
a flat surface or board, there were of course different ways of working. Here
is the most rudimentary.
The arithmeticians began by tracing several parallel lines on the surface
to be used, thus marking out a series of columns which corresponded to the
places of the decimal system. Then they drew the nine numerals inside each
one. In this way, they immediately acquired a place value.
The Arabs, like the ancient Indian arithmeticians, would write a
number such as 4,769 by tracing the number 9 in the units column, the
number 6 in the tens column, the number 7 in the hundreds column and
the number 4 in the thousands column.
557
Ten
thousands Thousands Hundreds Tens Units
So there was no need for zero. It was sufficient just to leave the column
empty, as in our next example which represents 57,040:
Ten
thousands Thousands Hundreds Tens Units
As for the calculations, they were carried out in the dust, then erased.
There is a clear trace of this in the etymology of the Sanskrit words gunara,
hanana, vadha, kshayam, etc., used by the Indians to mean “multiplication”
Literally, they mean “to destroy” or “to kill” in allusion to the successive
wiping out of intermediary products, as our example will now show.
Let us suppose that an accountant wants to multiply 325 by 28.
The first thing to do is trace out the four columns required. Then, inside
them, we place 325 and 28 as follows, with the highest place of the multipli-
cand in the same column as the lowest place of the multiplier.
We then multiply the upper 3 by the lower 2. As this equals 6, we place this
figure to the left of the upper 3:
Then we multiply the upper 3 by the lower 8. As this equals 24, we wipe out
the 3 and replace it with 4 (the unit column of 24, the partial product):
THE BOARD AS A COLUMN ABACUS
And, to the 6 we add 2 (the tens digit of 24):
The first step has now been carried out, both columns of the multiplier 28
having acted on the hundreds column of the multiplicand (the upper 3 of
the initial layout).
We then proceed to the second step by moving all the numbers of the
multiplier one place to the right:
8 4
2
5
2
8
>
Then, by using the tens digit of the multiplicand (the upper 2 of the initial
layout), we multiply 2 by 2. As this equals 4, we then add 4 to the 4 which
lies immediately to the left of upper 2:
We then multiply the same upper 2 by the lower 8. This makes 16, so we
replace, after erasing, the upper 2 with 6 (the units digit of the result):
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
We then add 1 (the tens digit of 16, as above) to the 8 just to the left of the
new 6:
V
8
OO
6
5
2
8
Then, after erasure, we have:
8
9
6
5
2
8
We have now finished the second step, since both digits of the multiplier 28
have operated on the tens digit of the multiplicand (the upper 2 of the ini-
tial layout).
We then begin the next step by moving the numbers of the multiplier
one column to the right again:
8 9 6
5
2
8
>
This time we multiply the units digit of the multiplicand (the upper 5 of the
initial layout) by the lower 2. This comes to 10, so we leave untouched the
upper 6 (there being no unit digit in the number 10), but add 1 (the tens
digit of 10) to the 9 immediately to the left of the 6:
V
OO
9
6
5
2
OO
But as this makes 10 again, we wipe out the 9, leave the space empty
(because of zero units in 10) and add 1 to the 8 just to the left of this blank
column:
Then, after erasure, we have:
558
We then multiply the upper 5 by the lower 8. As this makes 40, we wipe out
the upper 5, but leave the space empty because there is no unit in the prod-
uct found:
9
6
2 8
But we then add the 4 of the product to the upper 6:
V
9
6
2
8
As this again makes 10, we wipe out the 6, leave the space blank and add 1
to the number (zero) in the empty space immediately to the left:
And, as the lowest place of the multiplier is now in the lowest place of the
multiplicand (here, the units column of the abacus), we know that the mul-
tiplication of 325 by 28 has been completed.
All we have to do know is to read the number on the upper line, nine
thousand, one hundred, no tens, no units; so the result is 9,100:
559
THE BOARD AS A COLUMN ABACUS
This method thus consists in carrying out a number of steps corresponding
to the number of places in the multiplicand, each being subdivided into a
series of products of one of the digits of the multiplicand successively oper-
ated on by all the digits of the multiplier.
In this case, the procedure (now called the operation’s “algorithm”) has
three main phases, each subdivided into two simple steps consisting of cal-
culating a partial product; hence six simple steps in all:
Thousands
Hundreds
Tens
Units
3 . . x 28 =
First Step
(3x2) then (3x8)
(the hundreds of the multiplicand successively
multiplied by the digits of the multiplier, from
the highest down) y
Second Step
.2. X 28 =
(2 X 2) then (2 X 8)
(the tens of the multiplicand successively
multiplied by the digits of the multiplier,
from the highest down)
T
Third Step
.5 X 28 =
(5 x 2) then (5 x 8)
(the units of the multiplicand successively
multiplied by the digits of the multiplier,
from the highest down)
In other words, this “algorithm” works according to the following formula:
325 x 28 = (3 x 100 + 2 x 10 + 5) x (2 x 10 + 8)
= (3 x 2) x 1,000 + (3 x 8) x 100
(first step)
+ (2 + 2) x 100 + (2 x 8) x 10
(second step)
+ (5 x 2) x 10 + 5 x 8
(third step)
This counting board thus allows us to carry out calculations without using
zero, which explains why certain Arab manuscripts dealing with Indian
numerals and methods of calculation make no mention of it.
In certain parts of North Africa, this method continued to be used until
the end of the seventeenth century, which explains why the Ghubar numer-
als of the Maghreb generally come down to us in incomplete series, with
the zero missing (Fig. 25.5).
But in the East, it gradually disappeared after the tenth or eleventh century
and was replaced by more highly developed methods. It is true that this system
is long, tiresome and requires considerable concentration and practice.
In fact, very little distinguishes it from methods used in Antiquity. The
reason for this has less to do with the numerals themselves than with the
method used. It makes no difference whether we trace out the nine Indian
numerals, the first nine letters of the Greek or Arabic alphabet, or even
the first nine Roman numerals. The principle would still remain virtually
the same.
The Indians, as we have seen, certainly used such a system early in
their history. But they abandoned it as soon as they had developed their
own place-value system and their arithmetic allowed simpler rules to
be found.
To carry out arithmetical operations, the early Indians used whatever
was to hand. Like everybody else, they presumably began by using peb-
bles, or similar objects. Then, or perhaps at the same time, they carried
out operations on their fingers. But during the next stage, when they
developed their first written numerals, they conceived of the idea of draw-
ing several parallel columns, putting the units in the first one, the tens in
the second, the hundreds in the third, and so on. They thus invented the
column abacus, as others did before and after them. But instead of using
pebbles, counters or reeds, they preferred their own nine numerals,
which they traced out in dust with a pointer inside the appropriate
columns. This was the birth of their dust abacus, which they later
improved by working on a table or board covered with sand or dust,
instead of the ground.
But this system could not evolve further, so long as it continued to be a
column abacus; this concept in fact trapped the human mind for centuries,
preventing us from thinking out simpler and more practical rules.
This once again highlights the importance of the discovery of the place-
value system. This principle had, of course, long been present in the way
calculations were made, but without anybody noticing it. The creative
genius of the Indians then brought together all the necessary ideas for dis-
covering the perfect number-system. They had to:
• get rid of stones, reeds, knotted cords, manual techniques or, more
generally, any concrete method;
• eliminate any notions of ideogrammatic representation (writing
numerals as numbers of lines, points etc.), which certainly came later
than the previous system, but was just as primitive;
• eliminate any notation of numbers higher than or equal to the base
of the calculation system;
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
560
• keep only the nine numerals, in a decimal system, and apply place-
value to them;
• replace all existing systems by this group of nine numerals, indepen-
dent one from the other, and which visually represented only what
they were supposed to represent;
• get rid of the abacus and its now useless columns, and apply the
new principle to the numerals which were freed from any direct visual
intuition;
• fill the gap now created by this method when a place was not filled
by a numeral;
• think of replacing this gap by a written sign, acting as zero in the
strict arithmetical and mathematical sense of the term.
To sum up, it was by rejecting the abacus that Indian scholars discovered
the place-value system.
This raises a question concerning the arithmeticians of the Maghreb
and Andalusia, who continued to use the dust abacus and its associated
methods for several centuries: did the Western Arabs not know about zero
and the place-value system? The answer is no, because these arithmeticians
knew the Hindi numerals which, as we know, were based on the place-value
system and included zero.
In other words, they were aware that the numerals they used could also
be manipulated with zero and its associated rules. This is shown in certain
Maghrebi manuscripts, in which zero is drawn as a circle (Fig. 25.5).
Why, then, did they not use them for “written calculation” instead of
using a dust board? The answer seems to lie in the attachment the
Maghrebi and Andalusians always felt for traditions coming from the time
of the conquest of North Africa and Spain. Thus, the use of the dust
board/abacus has the same traditionalist explanation as their cursive
script, derived directly from the Kufic.
In fact, the Arabs inherited various arithmetical methods from the
Indians, ranging from the most primitive to the most highly developed. In
their thirst for knowledge, they presumably took from the Indians every-
thing they could find in terms of calculation methods, without realising
that certain things could well be left alone. We should not forget that India
is a veritable sub-continent, cut up into regions, peoples, practices, cus-
toms and traditions, and it has always been difficult, if not impossible, to
see it as a whole.
It is because they came into contact with people who used methods
already abandoned by the scholars, and decided to uphold this tradition,
that certain Arab-Muslim arithmeticians remained stuck in such a rudi-
mentary rut for several generations.
THE COLUMNLESS BOARD
But this was not, of course, the case for all the Arabs. Others were lucky or
bright enough to take up the dust board freed of its columns.
Among them was al-Khuwarizmi. In his Kitab al jama wa’l tafriq bi hisab
al hind ( Book of Addition and Subtraction According to Indian Calculations ) he
had not only explained the decimal place-value system when applied to
Indian numerals, but also recommended “writing the zeros so as not to mix
up the positions” [A. P. Youschkevitch (1976), p. 17]. There was also Abu’l
Hasan ‘Ali ibn Ahmad an Nisawi (died c. 1030), whose Al muqni'fi'l hisab al
hind ( Complete Guide to Indian Arithmetic) followed the same sources and
methods as the previous work.
Abu’l Hasan Kushiyar ibn Labban al-Gili (971-1029) also deserves a men-
tion. The first chapter of Book I of his Maqalatan fi osu’l hisab al hind ( Two
Books Dealing with Calculations Using Indian Numerals) begins as follows:
The aim of any calculation is to find an unknown quantity. To do this,
at least [one of these] three operations is necessary: multiplication [al
madrub], division [al qisma] and [extraction of] the square root [al
jadr] . . There is also a fourth operation, less often used, which is the
extraction of the side of a cube.
But before learning how to carry out these operations, we must
familiarise ourselves with each of the nine numerals [huruj], the posi-
tion [rutba] of each in relation to the others in the [place-value]
system [al wad 1 . . .
Here are the nine numerals [written in the Hindi style, but here
updated]:
98765432 1.
[Thus positioned], they represent a number and each stands in a
position [ martaba ].
The first is the image of one, the second of two, the third of three . . .
and the last of nine. What is more, the first is in the position of the
units, the second in the tens, the third in the hundreds, the fourth in
the thousands . . .
As for the number formed by these numerals, it must be read: nine
hundred and eighty-seven million six hundred and fifty-four thousand
three hundred and twenty-one.
[When writing] a number [containing several place-values] we
must put a zero [sifr, literally “void”] in each place where there is no
numeral. For example, to write ten, we put a zero in the place of the
units; to write a hundred, we put two zeros, one in the place of the
missing tens and one in the place of the units.
561
Here are these two figures:
THE COLUMNLESS BOARD
Ten: 10
Hundred: 100
There are no exceptions to this rule.
For any of the nine numerals under consideration, the one immedi-
ately to its left stands for tens, the next one to the left for hundreds,
and the next one to the left for thousands.
In the same way, any of the nine numerals under consideration
stands for the tens of the numeral immediately to its right, for the
hundreds of the next numeral to its right, for the thousands for
the following one, and so on [P 267v and 268r; A. Mazaheri (1975),
pp. 75-76],
These scholars had thus understood that the place-value system and
zero removed the need for columns on a counting board.
So, like the Indians, they entered into the era of modern “written
calculation”
But they now had to know off by heart the tables giving the results of
the four basic operations on these numerals. This is what the Persian
mathematician Ghiyat ad din Ghamshid ibn Mas ‘ud al-Kashi explains
in his Miftah al hisab ( Key to Calculation), in which he reproduces one of
these tables: “Here is the table for multiplying numbers inferior to ten.
The arithmetician should learn it by heart and know it perfectly, for it
can also be used for the multiplication of numbers superior to ten ” [see
A. Mazaheri (1975)].
Calculating on a columnless board by erasing intermediate results
Our first example of this method comes from the work of Kushiyar ibn
Labban al-Gili, cited above [P 269v to 270v]:
We want to multiply three hundred and twenty-five by two hundred
and forty-three.
We put them on the board as follows:
3 2 5
2 4 3
the first numeral [on the right] of the bottom number being always
under the last numeral [on the left] of the top number.
We then multiply the upper three by the lower two; this makes six,
which we place above the lower two, to the left of the upper three,
thus:
6 3 2 5
2 4 3
If the six had contained tens, these would have been placed to its left.
Then we multiply the upper three again by the lower four; this
makes twelve, of which we place the two above the four and add the one
[which represents the tens] to the six of sixty, obtaining seventy, thus:
7 2 3 2 5
2 4 3
Then we multiply the upper three by the lower three; this makes nine,
which replaces the upper three:
7 2 9 2 5
2 4 3
We then advance the bottom number one place towards the right, thus:
7 2 9 2 5
2 4 3
And we multiply the two above the lower three by the lower two; this
makes four which, added to the two above the lower two, makes six:
7 6 9 2 5
2 4 3
Then we multiply the upper two again by the lower four; this makes
eight, which we add to the nine above the four:
7 7 7 2 5
2 4 3
Then we multiply the upper two again by the lower three; this makes
six, which replaces the upper two above the lower three:
7 7 7 6 5
2 4 3
We then advance the bottom number one place [towards the right], thus:
7 7 7 6 5
2 4 3
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
Finally, we multiply the upper five by the lower two; this makes ten,
which we thus add to the tens position above the lower two:
7 8 7 6 5
2 4 3
Then we multiply the five again by the lower four; this makes two
[tens]. Added to the tens [in the position above] the four, [these two
numbers] together make nine:
7 8 9 6 5
2 4 3
Finally, we multiply the five by the lower three; this makes fifteen,
thus leaving the five alone, we just add one [the tens digit] to the
tens, thus:
7 8 9 7 5
2 4 3
The [upper] number is the one we wanted to calculate.
This method thus consists in applying the same number of steps as
there are places in the multiplicand, each being subdivided into as many
products of one of its numbers and the successive digits of the multiplier.
The same method, with some variants, can be found in, for example
al-Khuwarizmi and An Nisawi, as well as numerous Indian mathematicians
such as Shridharacharya (date uncertain), Narayana (1356), Bhaskaracharya
(1150), Shripati (1039), Mahaviracharya (850), etc. [B. Misra (1932)
XIII, 2; H. R. Kapadia (1935), 15; B. Datta and A. N. Singh (1938),
pp. 137-43].
THE DUST BOARD SMEARED WITH A TABLET OF
MALLEABLE MATTER
Despite being freed of columns, this approach remained primitive. It was
merely a written imitation of older methods and could hardly develop fur-
ther because of the limitations imposed by the medium.
The dust board was certainly very practical for calculation methods with
or without the abacus columns, and especially for the technique of wiping
out intermediate results, as this passage from Psephophoria kata Indos
shows (by Maximus Planudes (1260-1310), a Byzantine monk):
It would perhaps not be superfluous to show another multiplication
method. But it is extremely inconvenient when done with ink and
paper, while it is suited for use on a board covered with sand. For it is
562
necessary to wipe out certain numbers, then replace them with others;
when using ink, this leads to much inextricable confusion, but with
sand it is easy to wipe out a number with one’s finger and replace it
with others. This method of writing numbers in sand is especially
useful, not only for multiplication, but for other operations as well . . .
[BN Paris. Ancien Fonds grec, Ms 2381, P 5v, 11. 30-35; Ms 2382, P 9r,
11. 13-25; Ms 2509, P 105v, 11. 2-10] [see A. Allard (1981); H.
Waeschke (1878); F. Woepcke (1857), p. 240].
But the dust board became increasingly impractical as the numerals began
to resemble one another more and more.
Just take a wooden board, sprinkle it with dust or flour, then draw num-
bers on it in the usual way. Then try to carry out one of the operations we
have seen, following the same method. You will immediately see how hard
it is to replace one number with another. If you sprinkle the number to be
removed, or use a flat instrument to wipe it out, the very nature of the pow-
dery matter means you risk wiping out all the adjacent numbers as well.
Attempts were made to get round this problem by leaving a large space
between the different numbers. But there are limits to the size of the board,
and this means that longer, more complicated calculations would require a
larger space. What is more, by wiping out intermediate results, this
method limits the contribution of the human memory and makes spotting
intermediate mistakes extremely difficult. Hence an obvious block on find-
ing out simpler and more practical methods.
It is possible to guess what replaced sand calculation and the use of the
dust board in certain Islamic countries.
As we have seen, in Persian and Mesopotamian provinces, the preceding
method of calculation was also called takht al turab (or in Persian takhta-yi
khak), literally “board of sand” This expression is found, for example in the
Jami’ al hisab bi’t takht wa’t turab, by the mathematician and astronomer
Nasir ad din at Tusi (1201-1274). This work’s title can be translated liter-
ally as “Collection of arithmetic using a board and dust” [A. P.
Youschkevitch (1976), p. 181, n. 71],
But the Arabic word turab, and its Persian equivalent khak, means not
only “sand” or “dust”, but also “earth” “clay” and “cement”. Hence the dif-
ficulty in precisely translating this author’s ideas: for Persian and
Mesopotamian arithmeticians, did this word mean only “sand” and “dust”
or did it also cover a wad of clay? We can, in fact, suppose that for reasons
linked to climate and the nature of the soil in different regions, these arith-
meticians were led to use clay for carrying out their calculations, rather
than a board scattered with sand. This hypothesis is reasonable, given the
limited number of material solutions. It becomes even more probable when
we remember that, in these regions, clay tablets had been used for writing
563
for thousands of years. It is sufficient to remember the Sumerians, the
Elamites, the Babylonians, the Assyrians and the Acheminid Persians, the
distant precursors of these Persian and Mesopotamian arithmeticians, to
support the idea that, even under Islam, these peoples had not forgotten
their ancient writing materials.
According to this hypothesis, these arithmeticians would then have
smeared soft clay over their boards and traced numbers on them with a
stylus, pointed at one end and flattened at the other. This is why the Arabic
expression takht al turab, and its Persian equivalent takhta-yi khak, as in At
Tusi’s book cited above, could be translated by “board smeared with clay"
This hypothesis can be applied to the regions of Persia, Mesopotamia
and Syria, but less so to other Muslim provinces.
If we return to the “board”, the Arabic word luha, used by the Maghrebi
and Andalusians for this article had, and always has had, as broad a range
of meaning as its Eastern equivalent takht (which comes from the Persian
takhta, itself derived from the Sassanid takhtag). Both words mean “table”
but also “board” “plank”, “tablet” and “plate” or “plaque”, be it of wood,
leather, metal, earth or even clay.
At a certain time, it is not impossible that wax came to replace the dust
or flour used on the board in the Maghreb, and elsewhere. In other words,
it can be supposed that the Maghrebi and other Islamic peoples calculated
on tablets covered with wax, like those of the ancient Romans, using a
stylus with a flattened tip for rubbing out.
All of these techniques perhaps coexisted, each being favoured at differ-
ent times, in different regions and according to local customs. It is
extremely unlikely that people living in such a vast and varied world as
Islam would have all used the same method.
CALCULATING WITHOUT INTERMEDIATE
ERASURES
What is certain is that the Arab arithmeticians’ next step was to “calculate
without erasures, by crossing out and writing above their intermediate results”
This method is found, for example, in the Kitab al fusul fi’l hisab al hind
{Treatise on Indian Arithmetic), written in Damascus in 952 (or 958) by Abu’l
Hasan Ahmad ibn Ibrahim al-Uqlidisi. It can also be found in works by An
Nisawi (1052), al-Hassar (c. 1175), al-Qalasadi (c. 1475), etc., in which it is
described as the a’mal al hindi (“method of the Indians”) or else as tarik al
hindi (literally “way of the Indians”) [see A. Allard (1976), pp. 87-100; A.
Saidan (1966); H. Suter BMA, II, 3, pp. 16-17; F. Woepcke (1857), p. 407].
CALCULATING WITHOUT INTERMEDIATE ERASURES
Here are the rules, applied to the product of 325 and 243:
As before, we begin by placing the multiplicand above the multiplier,
thus:
3 2 5 <— Multiplicand
2 4 3 <— Multiplier
We then multiply the upper 3 by the lower 2; this makes 6, which we place
on the line above the multiplicand, in the same column as the 2 of the
multiplier:
6
3 2 5 <— Multiplicand
2 4 3 <— Multiplier
And we cross out the 2 of the multiplier:
6
3 2 5 <— Multiplicand
Ti 4 3 <— Multiplier
Then we multiply the upper 3 by the lower 4; this makes 12, we carry for-
ward the 1 and place the 2 on the same line as the 6, above the 4:
6 2
3 2 5 4— Multiplicand
•if 4 3 4— Multiplier
Then we add the carried-forward number to the 6; so we cross out 6 and
write 7 on the line above, just over the crossed-out number:
7
2
3 2 5 4— Multiplicand
T, 4 3 <— Multiplier
And we cross out the 4 of the multiplier:
7
j6 2
3 2 5 4— Multiplicand
-2/4 3 4— Multiplier
We then multiply the upper 3 by the lower 3; this makes 9, which we write
in the same column as the 3 of the multiplier, but on the line above the
multiplicand:
NDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
7
A 2 9
3 2 5 <— Multiplicand
Z A 3 4— Multiplier
And we cross out the 3 of the multiplier:
7
A 2 9
3 2 5 4- Multiplicand
Z A A 4 — Multiplier
The first step of the operation has now been completed, so we write the
multiplier 243 again on the line below, but moving one column to the right,
after having crossed out the 3 of the multiplicand:
7
A 2 9
A 2 5 4— Multiplicand
Z A A
2 4 3 4— Multiplier
Then we multiply the 2 of the multiplicand by the 2 of the multiplier; hence
4, which we add to the 2 to the right of the already crossed-out 6 on the line
above the multiplicand; we thus cross out this 2, and write 6 on the line
above, in the same column:
7 6
6 % A
A 2 5
AAA
2 4 3
And we cross out the 2 of the multiplier:
7 6
A Z A
A 2 5 4— Multiplicand
Z A A
Z 4 3 4— Multiplier
We then multiply the 2 of the multiplicand by the 4 of the multiplier; this
makes 8, which we add to the 9 in the same column in the line above the
multiplicand; this makes 17, we carry forward 1 and place 7 on the line
above (just over the 9), after crossing out the 9:
4— Multiplicand
4— Multiplier
564
7 6 7
A Z Z
A 2 5 4— Multiplicand
Z A A
,2 4 3 4— Multiplier
Then we add the carried-forward 1 to the 6 on the top line; we thus cross
out this 6 and write a 7 on the line above, in the same column:
7
7 A 7
A Z Z
A 2 5 4— Multiplicand
Z A A
,2 4 3 4— Multiplier
And we cross out the 4 of the multiplier:
7
7 A 7
A Z Z
A 2 5 4— Multiplicand
Z A A
Z A 3 4— Multiplier
Then we multiply the 2 of the multiplicand by the 3 of the multiplier; this
makes 6, so we write 6 in the same column as the 2 in the line just above:
7
7 A 7
A Z Z 6
A 2 5 4— Multiplicand
Z A A
Z A 3 4 — Multiplier
And we cross out the 3 of the multiplier:
7
7 A 7
A Z A 6
A 2 5 4— Multiplicand
Z A A
Z A A <— Multiplier
The second step has now been completed, so we write the multiplier 243
once again on the line below, moving one column to the right, after having
crossed out the 2 of the multiplicand:
565
7
7 0 7
0 2 0 6
0 0 5 <— Multiplicand
2 0 0
2 0 0
2 4 3 <— Multiplier
Then we multiply the 5 of the multiplicand by the 2 of the multiplier; this
makes 10, we carry forward 1, but add nothing to the 7 in the same column
as the 2 on the second line above the multiplicand. We then add the
carried-forward 1 to the 7 on the top line; we cross out this 7 and write 8
on the line above:
8
0
7
0
7
0
2
0
6
Z
2 5
<— Multiplicand
2
A
z
2
A
0
2
4 3
<— Multiplier
And we cross out the 2 of the multiplier:
8
77
10 7
0 2 0 6
0 0 5 <— Multiplicand
0 A Z
2 A 0
2 4 3 <— Multiplier
Then we multiply the 5 of the multiplicand by the 4 of the multiplier; this
makes 20, we carry forward the 2, but add nothing to the 6 in the same
column as the 2 on the line just above the multiplicand. Then we add the
carried-forward 2 to the 7 in the column just to the left; we cross out this 7
and write 9 on the line above:
CALCULATING WITHOUT INTERMEDIATE ERASURES
8
4 9
7 4 0
4 4 4 6
4 4 5
4 A 0
AAA 3
^43
And we cross out the 4 of the multiplier:
8
4 9
7/6/7
4 4 A 6
4 A 5 <— Multiplicand
4 A 0
AAA
4 A 3 <— Multiplier
Finally, we multiply the 5 of the multiplicand by the 3 of the multiplier; this
makes 15, so we write a 5 above the 5 of the multiplicand:
8
77 9
7
4
77
0
2
A
6
5
0
A
5
Multiplicand
A
A
0
A
A
A
2
A
3
<— Multiplier
Then we add the carried-forward 1 to the 6 immediately to the left on the
same line; we cross out this 6 and write 7 on the line above:
8
4
9
7
4
7!
7
4
A
A
4
5
4
A
5
<— Multiplicand
4
A
0
A
A
A
2
A
3
<— Multiplier
<— Multiplicand
<— Multiplier
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
566
And we cross out the 3 of the multiplier and the 5 of the multiplicand:
8
7l 9
7
z
z
7
Z
z
z
Z
5
z
Z
/5
<— Multiplicand
a
A
z
z
A
z
Z
A
Z
<— Multiplier
As the operation has now been completed, all we have to do is read the
uncrossed-out numerals, from left to right, to obtain the result:
8
A 9
7 Z A 7
0 Z Z Z 5
A Z Z
Z A Z
1 A Z
Z A Z
V V V V V
325 x 243 7 8 9 7 5
The advantage of this method over the preceding one is the possibility to
check the operation and so spot any errors. This is why it was used by many
Muslim arithmeticians for some time; and that is also why it survived in
Europe until the end of the eighteenth century.
The disadvantage was to make the writing of calculations extremely
crowded and their progression difficult to follow.
This can be seen in the following example of division “a la frangaise”, as
explained in F. Le Gendre’s Arithmetique :
A 1
Z Z
0 A Z
AZIZ
Z Z Z 0
Z 0 X Z 2
A Z Z Z Z Z
Z Z Z Z 0 0
0 0 0 A Z Z 1
AZZZ1ZZZ
ZZZZAZZZZ
0 000ZZZZZ
AZZZZA0000A
1 9 9 9 9 3 0
(quotient) ZZZZZZZZZZ (remainder)
ZZZZZZZZ
z z z z z z
z z z z
z z
We will not weary the reader by explaining this extraordinarily complex
system. Suffice it to say that this represents 19,999,100,007 divided by
99,999 [Le Gendre, Arithmetique en sa perfection (Paris 1771), p. 54].
It can thus be understood that division, even when written down, long
remained beyond the scope of the average person.
It is also true that this work was not meant for a large public. As the
author himself makes clear, the “perfection” of this arithmetic was based
on "the usage of financiers, experienced people, bankers and merchants”
FROM THE WOODEN BOARD TO PAPER OR
BLACKBOARD
However, long before Le Gendre, several Arab and Indian arithmeti-
cians embarked on a far better way, omitting intermediate results and
thus dropping the technique of constant erasure. But this method, and
its consequent change of writing medium, called for a greater applica-
tion of memory.
The changes in methods and the changes in materials affected
each other, long before they resulted in our present day techniques. (See
*Pati, *Patiganita.)
It can be supposed that, as in India, Muslim arithmeticians at some time
started using a blackboard, or else a wooden board painted black, with
567
FROM THE WOODEN BOARD TO PAPER OR BLACKBOARD
chalk to write and cross out their numbers, and a cloth for erasing them.
(See *Patiganita.)
By using chalk and keeping or, even better, rubbing out intermediate
results, certain Indian and Arab arithmeticians were able to free their imag-
inations and work towards the methods we now use.
CT03I10 peto ehe fu intmdi cbe fono aim 1110 A' ve
niolnplicare per (cackieroth qiuli la (Taro n! frudr
o tuotmetmicfo It epempli fot fotamente in fbjma*
come po:ai oedere qnt fotto
O2 togli w fare To p?edi: to fcacbtero.joe.3 14.
fts.9 3 4.e noca yz farloper U quatro modi come-
qm os fotto.
3 * 3
9 3 4/*
3 4
T7+
9 3 4
nmi
I I <?l 3 I
614
I i*b!4l «
Z9 3 t 7 6
ta
* 9
5
z
4
9 5 z 7 6
\ 1 /
x V n
/ 9 \/z\
f,
1° X|
9 \/ 9 \
W7
*71
/4|
1
1*
l r s /XT7
5\/6\/z
XI
[4
Somma-
9 5 4
X 6 l\z |\6 |
* V 1 \f i \|4
X 9 |\s l\4f
oXfo \|o \| i
s;*7'i\9ixn
z \1 o \| i \f T
* 9 5
6
7
z
Fig. 25 . 11 . Page of an anonymous Italian arithmetical treatise, published in Treviso in 14 78. It
contains various forms oj “jealousy" multiplication (per gelosiaj. (Document in the Palais de la
DecouverteJ
Fig. 25 . 12 . Page of an Arabic treatise dealing with written calculation using Indian numerals,
explaining a method of multiplication “by a table ” (known in the West as per gelosiaj. To the left we
have the product of 3 and 64 and bottom right the product of 534 and 342. Sixteenth-century copy of
Kashf al mahjub min 'ilm al ghubar (see below). Paris, BN, Ms. ar. 2473, f 9)
“jealousy” multiplication
Here follows an example of a highly developed technique, which the Arabs
must have invented in around the thirteenth century. At the end of the
Middle Ages it was transmitted to Europe, where it was known as multiplica-
tion per gelosia (“by jealousy”), an allusion to the grid used in the operation
which is reminiscent of the wooden or metal lattices through which jealous
wives, and especially husbands, could see without being seen. It is described
in an anonymous work published in Treviso in 1478 (Fig 25.11) and in the
Summa de arithmetica, geometria, proporzioni di propoaionalita by Luca Pacioli,
an Italian mathematician (Venice, 1494). The Arabs called this system “multi-
plication on a grid” {al darb bi'l jadwal ) and it was described in about 1470 by
Abu’l Hasan ‘Ali ibn Muhammad al-Qalasadi, in his Kashf al mahjub min ‘ilm
al ghubar ( Revelation of the Secrets of the Science of Arithmetic), the word ghubar
here being used for “written arithmetic” in general and not in the original
NDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
568
sense of “dust” (Fig. 25.12). But there is a much earlier version, from about
1299, by Abu’l ‘Abbas Ahmad ibn Muhammad ibn al-Banna al-Marrakushi in
his Talkhis a’mal al hisab {Brief Summary of Arithmetical Operations) [see A.
Marre (1865); H. Suter (1900-02), p. 162]. But in India there is no trace of it
before the middle of the seventeenth century. It was described there for the
first time in 1658 by Ganesha in his Ganitamahjari [see B. Datta and A. N.
Singh (1938), pp. 144-5].
The layout is quite simple, and the final result is arrived at, rather as in
our present-day system, by adding together the products of various num-
bers contained in the multiplier and multiplicand.
Let us multiply 325 by 243. There are three digits in the multiplier and
three in the multiplicand, we thus draw a square grid with three columns
and three lines.
Above the grid we write the numbers 3, 2 and 5 of the multiplicand
from left to right; then we write the 2, 4 and 3 of the multiplier up the right-
hand side of the grid:
<— Multiplicand
Multiplier inverted
We then divide each square of the grid in half by drawing a diagonal from
the top left-hand corner to the bottom right-hand corner. Then, in each
square we write the product of the number on the same line to the right
and the number in the same column at the top. This product must, of
course, be inferior to 100.
We then write the tens digit in the left-hand triangle of the square and
the units digit in the right-hand triangle. If either of these digits is missing,
then we must write zero.
In the first upper right-hand square we thus write the product of 5 and
3, i.e. 15, by placing the 1 in the left-hand triangle and the 5 in the right-
hand triangle.
And so on, thus:
Outside the grid, we then add up the numbers contained in each oblique
strip, beginning with the 5 in the top right-hand corner. We then proceed
from right to left and from the top to the bottom. When necessary we carry
forward any tens digits and add them onto the next strip and we thus
obtain all the digits of the final result outside the grid, thus:
▼
(=1 + 6 + 0 )
¥
(=0 + 2 + 0 + 4 + l
+ number carried over)
The result is then obviously read from left to right, following the arrow,
therefore 78,975:
569
3 2 5
8
Note that the Arabs often wrote the resulting digits along an oblique seg-
ment, perpendicular to the main diagonal, to the left of the grid; the result,
of course, still reads from left to right:
This method may seem long in comparison with the one we now use. But its
advantage is that the final result is grouped together at the end whereas in
our modern system it is produced gradually during the intermediate steps.
Other ways of proceeding
Instead of following a falling diagonal, we follow a rising one; and instead
of writing the multiplier backwards, we write it the correct way round.
“jealousy” multiplication
Hence the following arrangement, which is used in the same way, except
that the additions appear to the left of the grid:
3 2 5
Other possibilities also exist, of course, by placing the digits of the multi-
plier to the left rather than to the right of the grid.
Simplified techniques
In the anonymous work cited above, we also find another layout alongside
the preceding ones. Instead of noting down all the details of the operation,
we simply give the results, which certainly requires a greater effort of
memory, especially when it comes to carrying numbers forward during the
intermediary steps (Fig. 25.11).
As both the multiplicand and multiplier have three digits, we draw a
rectangular grid with four columns and three lines, the extra column being
used for noting partial results with more digits than in the multiplicand.
Then we place the digits of the multiplicand and multiplier thus:
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
570
We then calculate the products at the intersections of the columns and
lines. But, as there are here no diagonals, only one digit must be written in each
square, with the tens digit being added onto the following square on the left.
On the first line, to the right, the first square thus gives 10; we note the 0
and carry forward the 1 to the next square on the left. Its own result is 4, to
which the 1 is added, making 5; and so on:
3 2 5
And, finally, on the third:
2 5
The result is then obtained by adding together the numbers along each line
parallel to the rising diagonal from right to left; it is then read from left
to right:
3 2 5
Note that we can work the other way round, by writing the digits of the
multiplier backwards on the left. But we must then follow the falling diago-
nal from left to right to obtain the result:
571
jealousy" multiplication
3 2 5
An even simpler technique
At the end of the fifteenth century, the following more highly developed
variant could be found in Europe.
We draw a line and write the digits of the multiplicand above it then, below
to the right, the digits of the multiplier obliquely rising from left to right:
3 2 5
3
4
2
To the left of the 3 of the multiplier, we then write its products with the
digits of the multiplicand:
Then with the 2:
3 2 5
9 7 5 3
1 3 0 0 4
6 5 0 2
We then add up the products, which gives us:
3 2 5
9 7 5 3
1 3 0 0 4
6 5 0 2
7 8 9 7 5
This method is thus as highly developed as our own (which we append here
to facilitate comparison), the only difference being the position of the
multiplier:
3 2 5
2 4 3
9 7 5
13 0 0
6 5 0
7 8 9 7 5
NASIR AD DIN AT TUSl’s METHOD
3 2 5
9 7 5
3
4
2
Then we do the same with the 4:
3 2 5
9 7 5
13 0 0
Here now is a multiplication technique, already existing in the thirteenth cen-
tury, particularly in the Miftah al Hisab ( Key to Calculation) by Ghiyat ad din
Ghamshid ibn Mas’ud al-Kashi, from the Persian town of Kashan; this work
was completed in 1427 [see A. P. Youschkevitsch (1976), p. 181, n. 67].
But it was known and used two centuries earlier. It can be found, with a
slight variation, in Nasir ad din at Tusi’s Jami’ al hisab bi't takht wa’t turab
(Collection of Arithmetic Using a Board and Dust), which dates back to 1265
and was copied by his disciple Hasan ibn Muhammad an Nayshaburi in
1283 [translated by S. A. Akhmedov and B. A. Rosenfeld; see A. P.
Youschkevitsch 1976, p. 181, n. 71].
Let us multiply 325 by 243. The multiplicand and multiplier are placed
as follows:
3 2 5
2 4 3
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
572
We multiply the 5 of the multiplicand by the 3 of the multiplier and place
the result beneath the line, being careful to respect the place values:
3 2 5
2 4 3
1 5
Then we multiply the 3 of the multiplicand (not the 2, which is for the
moment ignored) by the 3 of the multiplier and we place the product
beneath the same line, on the left of the previous one:
3 2 5
2 4 3
9 15
We now return to the 2 of the multiplicand, which we multiply by 3 and
this time place the result on the line below, one step to the left:
3 2 5
2 4 3
9 1 5
6
We draw another horizontal below these results and then multiply the 5 of
the multiplicand by the 4 of the multiplier, placing the result one step to
the left:
3 2 5
2 4 3
9 15
6
2 0
Then we multiply the 3 of the multiplicand by the 4 of the multiplier and
write the product on the same line on the left of the preceding result:
3 2 5
2 4 3
9 15
6
12 2 0
Then we return to the 2 of the multiplicand, which we multiply by the 4 of
the multiplier and place the result on the line below the preceding ones,
one step to the left:
3 2 5
2 4 3
9 1 5
6
12 2 0
8
Then we draw another line and carry out the preceding operations with the
2 of the multiplier, placing the first product one step towards the left. With
the 5 of the multiplicand we obtain:
3 2 5
2 4 3
9 1 5
6
12 2 0
8
1 0
Omitting the 2, with the 3 we obtain (on the same line):
3 2 5
2 4 3
9 15
6
12 2 0
8
6 10
And with the 2 of the multiplicand, on the line below and one step to the
left, we have:
3 2 5
2 4 3
9 1 5
6
12 2 0
8
6 10
4
The intermediate steps are thus over. All we have to do now is draw another
line and add up the partial results, position after position, from the right to
the left, and so easily obtain our result:
573
THE INDIAN MATHEMATICIAN B H A S K A R A C H A R Y a’s METHODS
3 2 5
2 4 3
9 1 5
6
12 2 0
8
6 10
4
7 8 9 7 5
THE INDIAN MATHEMATICIAN
bhaskaracharya’s METHODS
In his Lilavati, Bhaskharacharya (c. 1150) often uses a more highly devel-
oped method than the preceding one, which he called, in Sanskrit,
sthanakhanda (literally “separation of positions”). There are several vari-
ants, the main ones being as follows [see J. Taylor (1816), pp. 8-9; B. Datta
and A. N. Singh 1938, p. 147]:
To multiply 325 by 243, we begin by setting out the operation like this,
separating the three digits of the multiplier and copying the digits of the
multiplicand three times:
243 243 243
3 2 5
We begin multiplying with the 5. First we take the product of 5 and 3 and
write the full result below the line, without carrying anything forward; we
then move to the product of 5 and 2 (skipping the product of 5 and 4) and
write the result on the same line, again without carrying forward, just to
the left of the first result:
243 243 243
3 2 5
10 15
Then we take the product of 5 and the 4 we had omitted and write the
result below the others, one step to the left:
243 243 243
3 2 5
We draw a line below these results and add them up:
243 243 243
3 2 5
10 15
2 0
12 15
We then multiply using the 2, in the same way, placing the sum of the par-
tial results one step to the left from the first one:
243 243 243
3 2 5
4 6 10 15
8 2 0
12 15
4 8 6
Then we multiply by 3 in the same way, placing the sum of the partial
results one step to the left from the previous one:
243 243 243
3 2 5
6 9 4 6 1015
1 2 8 2 0
12 15
4 8 6
7 2 9
We draw a final line, add up the totals and obtain the result:
243 243 243
3 2 5
6 9 4 6 1 0 1 5
1 2 8 2 0
12 15
4 8 6
7 2 9
7 8 9 7 5
10 15
2 0
Another method
One variant of Bhaskaracharya’s method uses a layout like this:
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
574
3 2 5
2 4 3
We then multiply the 3 of the multiplicand (the highest place) by each of
the numbers in the multiplier (this time from the lowest first):
3 2 5
2 4 3
7 2 9
Then we multiply the 2 of the multiplicand by each of the numbers of the
multiplier, placing the result on the line below, one step to the right from
the previous result:
3 2 5
2 4 3
7 2 9
4 8 6
Finally, we carry out the same procedure with the 5 of the multiplicand and
move one step more to the right to note the result:
3 2 5
2 4 3
7 2 9
4 8 6
12 15
We draw a line and add up the partial results to obtain the final answer:
3 2 5
2 4 3
7 2 9
4 8 6
12 15
7 8 9 7 5
THE INDIAN MATHEMATICIAN
Brahmagupta’s methods
Long before Bhaskaracharya, Brahmagupta, in his Brahmasphutasiddhanta
(628 CE) described four even more highly developed methods, which he
called gomutrika, khanda, bheda and isa [S. Dvivedi (1902), p. 209; H. T.
Colebrooke (1817); B. Datta and A. N. Singh (1938), p. 148].
Here, as an example, is the method called gomutrika (which, in Sanskrit, lit-
erally means “like the trajectory of a cow’s urine” an allusion to the
zigzagging of the arithmetician’s eyes as he carries out the operation).
To multiply 325 by 243, we begin with the following layout, copying the
multiplicand onto three successive lines, moving one step to the right as we
go down. We place the digits of the multiplier vertically from top to bottom
starting on the top line:
2 3 2 5
4 3 2 5
3 3 2 5
On the first line we then mentally multiply the 2 of the multiplier by the 5
(the lowest digit) of the multiplicand; this makes 10, we write 0 on a lower
line in the same column as this 5 and carry forward the 1 which will be
added to the next product:
2 3 2 5
4 3 2 5
3 3 2 5
0
Then we multiply the same 2 by the 2 of the multiplicand, which makes 4,
and which is added to the carried-forward 1. The result is placed under the
line, to the left of the 0:
2 3 2 5
4 3 2 5
3 3 2 5
5 0
Then we multiply the same 2 by the 3 of the multiplicand; this makes 6,
which we place under the line to the left of the 5:
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
We then move to the line with the multiplier 4 and carry out the same
steps, this time placing the results on a line below the 650, one step to the
right, thus:
- with the product of 4 and 5:
575
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
0
- then with 4 and 2 (adding the 2 carried forward):
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
0 0
- and with 4 and 3 (adding the 1 carried forward):
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
13 0 0
We then go down to the line with the multiplier 3, carrying out the same
steps, this time with the partial results on a line below the 1,300, one step
to the right, thus:
- with the product of 3 and 5:
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
13 0 0
5
- then with 3 and 2 (adding the 1 carried over):
3 2 5
3 2 5
3 2 5
6 5 0
13 0 0
7 5
2
4
3
THE INDIAN MATHEMATICIAN BRAHMAGUPTA’S METHODS
- and finally with 3 and 3:
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
13 0 0
9 7 5
All we have to do now is add up the partial results to obtain the final
answer:
2 3 2 5
4 3 2 5
3 3 2 5
6 5 0
13 0 0
9 7 5
7 8 9 7 5
Other variants of this method
Another layout Brahmagupta used was as follows, with the multiplicand
copied three times on three successive lines, each moving one step to the
left in comparison with the line above, and with the multiplier placed on
the right, from the bottom to the top:
3 2 5 3
3 2 5 4
3 2 5 2
9 7 5
13 0 0
6 5 0
7 8 9 7 5
Brahmagupta’s method was, thus, highly developed. There was just one
more small step to be taken for it to become as efficient as our present-day
technique. This was, in fact, what happened as is shown in Brahmagupta’s
works, which contain the following extremely interesting variant.
Instead of copying the multiplicand three times, Brahmagupta wrote it
just once in the layout below, in which the multiplier is written as in the
INDIAN NUMERALS AND CALCULATION IN THE ISLAMIC WORLD
preceding example, i.e. from the bottom to the top and below the initial
line, each partial result being noted opposite and to the left of the number
that produces it:
3 2 5
9 7 5 3
1 3 0 0 4
6 5 0 2
7 8 9 7 5
This is exactly the same as the method which Italian mathematicians in the
second half of the fifteenth century (Luca Pacioli, etc.) had deduced from
simplifying the pergelosia, and laid out as follows (Fig. 25.11):
3 2 5
9 7 5
3
576
1 3 0 0 4
6 5 0 2
7 8 9 7 5
In other words, from as early as the beginning of the seventh century,
Indian mathematicians had a way of multiplying that was far simpler than
the “jealousy” method; a procedure which, with a mere change in the layout
of the numbers, was to give rise to our present-day technique.
It can now be seen just how advanced the Indians and their Arab succes-
sors were in this field.
577
RENAISSANCE ARITHMETIC
CHAPTER 26
THE SLOW PROGRESS OF
INDO-ARABIC NUMERALS IN
WESTERN EUROPE
All that is now left to tell is the story of how India’s discoveries reached the
Christian West through Arabic intermediaries. As is well known, this trans-
mission did not happen in a day. Quite the contrary!
When they first encountered numeral systems and computational methods
of Indian origin, Europeans proved so attached to their archaic customs, so
extremely reluctant to engage in novel ideas, that many centuries passed
before written arithmetic scored its decisive and total victory in the West.
RENAISSANCE ARITHMETIC:
AN OBSCURE AND COMPLEX ART
I was borne and brought up in the Countrie, and amidst husbandry: I
have since my predecessours quit me the place and possession of the
goods I enjoy, both businesse and husbandry in hand. I cannot yet cast
account either with penne or Counters [Montaigne, Essays, Vol. II
(1588), p. 379].
These words were written by one of the most learned men of his day:
Michel de Montaigne, born 1533, was educated by famous teachers at the
College de Guyenne, in Bordeaux, travelled widely thereafter, and came to
own a sumptuous library. He was a member of the parlement of Bordeaux
and then mayor of that city, as well as a friend of the French kings Francois
II and Charles IX. And he admits without the slightest embarrassment, that
he cannot “cast account” - or, in modern language, do arithmetic!
Could he have been aware of the fabulous discoveries of Indian scholars,
already over a thousand years old? Almost certainly not. Cultural contacts
between Eastern and Western civilisations had been very limited ever since
the collapse of the Roman Empire. Montaigne might have known of two
ways, at most, of doing sums: with “Counters” on a ruled table or abacus;
and using written Arabic numerals (“with penne”). The first operating
method stands in the highly complicated tradition of Greece and Rome; the
second, which Montaigne would no doubt have ascribed to the Arabs, was
in fact the invention of Indian scholars. But no one had thought of teaching
it to him; Montaigne, like most of his contemporaries, no doubt viewed it
with mistrust and suspicion.
The following anecdote gives a good picture of the arithmetical state of
Europe in the fifteenth and sixteenth centuries. A wealthy German merchant,
seeking to provide his son with a good business education, consulted a learned
man as to which European institution offered the best training. “If you only
want him to be able to cope with addition and subtraction,” the expert replied,
“then any French or German university will do. But if you are intent on your
son going on to multiplication and division - assuming that he has sufficient
gifts - then you will have to send him to Italy.”
It has to be said that arithmetical operations were not in everyone’s
grasp: they constituted an obscure and complex art, the specialist preserve
of a privileged caste, whose members had been through a long and rigorous
training which had allowed them to master the mysterious and infinitely
complicated use of the classical (Roman) counter-abacus.
A student of those days needed several years of hard work as well as a
long voyage to master the intricacies of multiplication and division - some-
thing not far short of a PhD curriculum, in today’s terms.
The great respect in which such scholars were held provides a measure
of the difficulty of the operational techniques. Specialists would take several
hours of painstaking work to perform a multiplication which a child could
now do in a few minutes. And tradesmen who wanted to know the total of
the week’s or the month’s takings were obliged to employ the services of
such counting specialists (Fig. 26.1).
Fig. 26.1. Arithmetician performing a calculation on a counter-abacus. From a fifteenth -century
European engraving, reproduced from Beauclair, 1968
THE SLOW PROGRESS OF INDO-ARABIC NUMERALS IN WESTERN EUROPE
578
This situation did not alter in the conservative bureaucracies of the
European nations throughout the seventeenth and eighteenth centuries.
Samuel Pepys, for example, became a civil servant after taking a degree at
Cambridge, and after a time in the Navy, became a clerk to the Admiralty.
From 1662, he was in charge of naval procurement. Though thoroughly
well-educated by the standards of the day, Samuel Pepys was nonetheless
quite unable to make the necessary calculations for checking the
purchases of timber made by the Admiralty. So he resolved to educate
himself afresh:
Up at 5 a-clock. . . By and by comes Mr Cooper, Mate of the Royall
Charles, of whom I entend to learn Mathematiques; and so begin with
him today. . . After an hour’s being with him at Arithmetique, my first
attempt being to learn the Multiplication table, then we parted till
tomorrow; and so to my business at my office again. . . [Pepys, Diary,
(1985), p. 212],
He eventually mastered the techniques, and was so proud of himself that
he sought to teach his wife addition, subtraction and multiplication. But he
didn’t dare launch her into the subtleties of long division.
It is now perhaps easier to understand why skilled abacists were long
regarded in Europe as magicians enjoying supernatural powers.
THE EARLIEST INTRODUCTION OF “ARABIC”
NUMERALS IN EUROPE
All the same, even before the Crusades, Westerners could have made full
and profitable use of the Indian computational methods which the Arabs
had brought to the threshold of Europe from the ninth century CE. A
channel of transmission existed, and it was by no means a paltry one.
A French monk with a thirst for knowledge, named Gerbert of Aurillac,
could indeed have played the same role in the West as had the learned
Persian al-Khuwarizmi, in the Arab-Islamic world. In the closing stages of
the tenth century CE, Gerbert - who was to become Pope in the year 1000
- could have broadcast in the West the discoveries of India which had
reached North Africa and the Islamic province of Andalusia (Spain) some
two centuries earlier. But he found no followers in this respect.
In order to understand the circumstances attendant on the first arrival
of Indian numerals in Western Europe, we have to remember the long-
drawn-out sequels of the collapse of the Roman Empire and the ensuing
Barbarian invasions.
From the end of the Roman Empire in the fifth century until the end of
the first millennium, Western Europe was continually laid waste by
epidemics, by famine, and by warfare, and suffered centuries of political
instability, economic recession, and profound obscurantism. The so-
called “Carolingian renaissance” in the Benedictine monasteries of the
ninth century may have revitalised the idea and structure of education
in the era of Charlemagne and also laid the bases of mediaeval philosophy,
but it actually brought only minor and temporary relief to the general
situation.
Scientific knowledge available at that time was very elementary, if not
entirely deficient. The few privileged men who received any “education”
learned first to read and to write. They went on to grammar, dialectics and
rhetoric, and sometimes also to the theory of music. Finally, they received
basic instruction in astronomy, geometry and arithmetic.
“Theoretical” arithmetic in the High Middle Ages was drawn from a
work attributed to the Latin mathematician Boethius (fifth century CE) who
had himself drawn handsomely on a second-rate work by the Greek
Nicomacchus of Gerasa (second century CE). As for “practical” arithmetic,
it consisted mainly in the use of Roman numerals, and in operations with
counters on the old abacus of the Romans; it also included the techniques
of finger-counting transmitted by Isidore of Seville (died 636 CE) and by
Bede (died 735 CE).
In these almost completely “dark ages”, even the memory of human arts
and sciences was almost lost. But a sudden reawakening occurred in the
eleventh and twelfth centuries:
A massive demographic explosion brought many consequences in its
wake - the development of virgin lands, the growth of towns and of the
monastic orders, the crusades, and the construction of ever larger
churches. Prices rose, the circulation of money accelerated, and, as
sovereign states began to control feudal anarchy, trade also began to
prosper. An increase in international contacts created a favourable
environment for the introduction of Arabic science in the West [G.
Beaujouan (1947)].
Gerbert of Aurillac was certainly one of the most prominent scientific
personalities of this whole period. Born in southwest France c. 945 CE, he
took holy orders at the monastery of Saint-Geraud at Aurillac, where his
sharp mind and passion for learning soon marked him out. He learned
mathematics and astronomy from Atton, the Bishop of Vich, and then,
probably as a result of a visit to Islamic Spain from 967 to 970, he absorbed
the lessons of the Arabic school. He learned to use an astrolabe, he learned
the Arabic numeral system, as well as the basic arithmetical operations in
the Indian manner. From 972 to 987, Gerbert was in charge of the Diocesan
school of Reims, and then, after a period as abbot of Bobbio (Italy), he
became an adviser to Pope Gregory V and became successively Archbishop
of Reims, Archbishop of Ravenna, and finally Pope Sylvester II, from
2 April 999 until his death on 12 May 1003.
579
Legend has it that Gerbert went as far as Seville, Fez and Cordoba to
learn Indo-Arabic arithmetic and that he disguised himself as a Muslim
pilgrim in order to gain entrance to Arab universities. Though that is not
impossible, it is more likely that he remained in Christian Spain at the
monastery of Santa Maria de Ripoll, a striking example, according to
Beaujouan, of the hybridisation of Arabic and Isidorian traditions. The
little town of Ripoll (near Barcelona, in Catalonia) had indeed long served
as a meeting point for the Islamic and Christian worlds.
One thing is nonetheless quite certain: Gerbert brought back to France
all the techniques necessary for modern arithmetic to exist. His teaching at
the diocesan school at Reims was highly influential and did much to
reawaken interest in mathematics in the West. And it was Gerbert who first
introduced so-called Arabic numerals into Europe. Arabic numerals,
indeed - but alas, only the first nine! He did not bring back the zero from
his Spanish sojourn, nor did he include Indian arithmetical operations in
his pack.
So what happened? Gerbert’s initiative actually met fierce resistance:
his Christian fellows clung with conservative fervour to the number-
system and arithmetical techniques of the Roman past. Most clerics of the
period, it has to be said, thought of themselves as the heirs of the “great
tradition” of classical Rome, and could not easily countenance the superi-
ority of any other system. The time was simply not ripe for a great
revolution of the mind.
A Victorian howler
The mediaeval forms of the Arabic numerals are found in a manuscript
entitled Geometria Euclidis (Euclidian Geometry) which for a long time
was attributed to Boethius, a Roman mathematician of the fifth century
CE. The text itself claims that the nine numeral symbols shown and their
use in a place-value system had been invented by Pythagoreans and
derived from the use of the table-abacus in Ancient Greece. For this
reason, the shapes of the nine Indo-Arabic numerals used in the Middle
Ages came to be called “the apices of Boethius”, even though, as we have
seen, there is no possibility whatsoever that a Roman of the fifth century,
let alone Greek Pythagoreans, could have known or understood Indo-
Arabic arithmetic.
The solution of this conundrum is very simple. As modern analyses
have shown, Geometria Euclidis was put together by an anonymous
compiler in the eleventh century, and its attribution to Boethius is
entirely apocryphal.
THE EARLIEST INTRODUCTION OF “ARABIC” NUMERALS IN EUROPE
Early forms of Arabic numerals in the West
The earliest actual appearances of Arabic numerals in the West are to be
found in the Codex Vigilanus (copied by a monk named Vigila at the
Monastery of Albelda, Spain, in the year 976) and the Codex Aemilianensis,
copied directly from the Vigilanus in the year 992 at San Millan de la
Cogolla, also in northern Spain (see Fig. 26.2).
These nine figures are clearly integrated in the cursive script of “full
Visigothic of the Northern Spanish type” (in the terms of R. L. Burnam,
1912-25), but their Indian origin is nonetheless quite manifest. Both manu-
scripts give the numerals shapes that are very close to the Ghubar figures of
the Western Arabs.
From the early eleventh century, the nine figures appear in a whole
variety of shapes and sizes in a great number of mss copied in more or less
every corner of the European continent. The variations in shape and style
are the result of palaeographic modifications occurring in different periods
and places, as can be seen in Figure 26.4.
However, contrary to first appearances, “Arabic” numerals did not first
spread through the West by manuscript transmission, but through a piece of
calculating technology called Gerbert’s abacus. In other words, the numerals
were disseminated not by writing but by the oral transmission of the knowl-
edge necessary to learn how to operate the entirely new kind of abacus that
Gerbert of Aurillac had promoted from around 1000 CE, and thereafter by
his numerous disciples in Cologne, Chartres, Reims and Paris.
Let us recall that for many centuries the Christianised populations of
Western Europe had expressed number almost exclusively through the
medium of Roman numerals, a very rudimentary system of notation whose
operational inefficacy lay at the root of all the difficulties experienced in
Fig. 26.2. Detail from the Codex Vigilanus (976 CE, Northern Spain). The first known
occurrence of the nine Indo-Arabic numerals in Western Europe. Escurial Library, Madrid, Ms. lat.
d.I.2,fi9v. See Burnam (1920), II, plate XXIII
THE SLOW PROGRESS OF INDO-ARABIC NUMERALS IN WESTERN EUROPE
580
calculation throughout that long period of the “dark ages”. First-millennium
mediaeval arithmeticians made their calculations just like their Roman
predecessors, through a complicated game of counters placed on tables
marked out with rows and columns delimiting the different decimal orders.
On a Roman abacus, you place as many unit counters in a column for a
specific decimal order as there are units in that order. But just before 1000 CE,
it occurred to Gerbert of Aurillac, who had seen Arabic counting methods
during his time in Andalusia, to reduce the number of counters used and so to
simplify the material complexity of computation on an abacus.
Gerbert ’s system involved jettisoning multiple unit counters and replac-
ing them with single counters in each decimal column, the new horn
“singles” being marked with one of the nine numerals he had brought back
from the Arabs. These number-tokens were called apices ( apex in the singu-
lar) and were each dubbed with a specific name that has nothing to do with
the number shown (though a few of them seem to hark back to Arabic and
Hebrew number-names).
The apex for 1 was called Igin
for 2 was called Andras
for 3 was called Ormis
for 4 was called Arbas
for 5 was called Quimas
for 6 was called Caltis
for 7 was called Zen is
for 8 was called Temenias
for 9 was called Celentis
Fig. 26.3.
So the one, two, three, four, five, six, seven, eight or nine unit-counters
in each column of the Roman abacus were replaced by a single apex bearing
on it the corresponding numeral in “Arabic” script.
When a decimal order was “empty”, the abacist simply put no apex in
the corresponding column. So to represent the number 9,078, you put the
apex for 8 in the unit column, the apex for 7 in the tens column, and the
apex for 9 in the thousands column, leaving all the other columns empty.
DATES
SOURCES
1
2
3
4
5
6
7
8
9
0
976
Spain: Codex Vigilanus. Escurial, Ms, lat.
d.1.2, P 9v
I
Z
z
Y
V
L.
1
8
?
992
Spain: Codex Aemilianensis. Escurial, Ms.
lat. dl.l, P 9v
I
z
l
Y
V
h
1
S
y
Before
1030
France (Limoges). Paris, BN Ms. lat. 7231,
P 85v
i
V
1*
&
b
of
b
8
1077
Vatican Library, Ms. lat. 3101, P 53v
/
5
z
f
7
b
A
8
2
XlthC
Bernelinus, Abacus. Montpellier, Library of
the Ecole de Medecine, Ms. 491, P 79
i
la
k
9
h
h
8
?
1049?
Erlangen, Ms. lat. 288, P 4
I
V
£
£
9
la
A
i
9
XlthC
Montpellier, Library of the Ecole de
Medecine, Ms. 491, P 79
7
■zr
9 s -
9
p
h
8
?
XlthC
France: Gerbertus, Raaones numerorum
Abaci. Paris, BN Ms. lat. 8663, P 49v
1
z
9*
4
p
A
8
P
Xlth/
Xllth C
Lorraine: Boecius, Geometry. Paris, BN, Ms.
lat. 7377, P 25v
I
z
tfl
V
4
h
r-*
8
?
Xlth C
Boecius, Geometry. London, BM, Ms. Harl.
3595, P 62
i
z
5
r
y
p
r
8
2
XlthC
Germany (Regensburg). Munich,
Bayerische Staatsbibliothek, Clm 12567, P 8
I
Ta
13
9
b
r
8
6
XlthC
Boecius, Geometry. Chartres, Ms. 498. P 160
I
z
•H*
■B
9
L>
A
5
a
®
Early
Xllth C
Bernelinus, Abacus. London, BM Add.
Ms. 17808, P, 57
I
z
3-
v-
4
A
8
2
Late
Xlth C
Bernelinus, Abacus. Paris, BN Ms. lat.
7193, P 2
I
V
k
%
4
V
%
&
Late
Xlth C
France (Chartres?): Anon., Arithmetical
tables. Paris, BN Ms. lat. 9377, P 113
I
rs
UK
%
4
h
A
8
<3*
Late
XlthC
Bernelinus, Abacus. Paris, BN Ms. lat.
7193, P 2
t
z
z
—
P
h
8
Cs>
Xllth C
Rome, Alessandrina Library, Ms. 171, P 1
i
£
1*1
S
b
b
V
8
(9
i
Xllth C
Paris, Saint Victor. Gerlandus, De Abaco.
Paris, BN, Ms. lat. 15119, P 1
l
T
ib
b
h
V
8
b
Xllth C
Boecius, Geometry. Paris, BN, Ms. lat. 7185,
P70
E
*
it-
<i
la
<?
8
4
Xllth C
France, Chartres(?):Bemelinus, Abacus.
Oxford, Bodley, Ms Auct. F. 1. 9, P 67v
1
C
H
4
t>
S
2
Xllth C
Gerlandus, De Abaco. London, BM Add.
Ms. 22414, P 5
i
z
/ft
h
Lx
V
8
6
Xllth C
Gerlandus, De Abaco. Paris, BN Ms. lat. 95,
P 150
t
z
rb
50
b
Isr
Y
8
b
Early France (Chartres): Anon. Paris, BN Ms. lat.
Xlllth C Fonds Saint-Victor, 533, P 22v
r
z
*
(a
A
8
f
Fig. 26 . 4 . Mediaeval apices. Sources: BSMF X (1877), p. 596: Burnam (1920), II, plates XXIII
& XXIV; Folkerts (1970) Friedlein (1867) p. 397: Hill (1915) Smith and Karpinski (1911) p. 88
581
THE EARLIEST INTRODUCTION OF “ARABIC.'" NUMERALS IN EUROPE
Supiuf u hoc*
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refyondh unrein. tjt>awr birutrt 0 ..’&
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fexti ^inano ^ia'rnar^^ccTCtv^ morfri
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JUi aur in but modi op 1 apu^nawraU nume
ro 4s tnicrtptw tantum (orrtn s
toofcnim apac^ttn uarttrccupuiucr© dd£
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no udictxjfcamcDc^a^uTMcrKlo lacar^"
m i »« < a
naui tpia difprt najeererrr • pnmu jut
numeruide btnanu umtaf rnf urm
Fig. 26.5. Apices in an eleventh-century Latin manuscript. Berlin, Ms. lat. 8° 162 (n),f 74.
From Folkerts (1970)
1
■
Si
poj
I
ia
B
C ak
ri<
ar
U*f
B
li
1
1
1
Q
i
1
a
S
S
a
i
a
I
1
|
Tail
1
1
c
X
1
1
1
1
H
1
I
1
H
§
a
1
1
1
1
1
1
|
1
1
a
m
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a
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B
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Fig. 26.6. Apices and the columns of Gerbert's abacus in an eleventh-century Latin manuscript.
Berlin, Ms. lat. 8° 162 (n),f 73v. From Folkerts (1970)
THE SLOW PROGRESS OF INDO-ARABIC NUMERALS IN WESTERN EUROPE
582
So at this early stage the Arabic numerals introduced by Gerbert served
only to simplify the use of an abacus identical in structure to that of classi-
cal Rome. Indeed, some mediaeval arithmeticians continued to use Roman
numerals - or even the letters of the Greek numeral alphabet (a = 1, P = 2,
... 9 = 9) - on their apices.
Hundreds of thousands
tens of thousands
thousands
hundreds
... tens
.... units
1
T
?
T
▼
T
c
X
I
c
X
I
®
®
©
Fig. 26.7.
To multiply 325 by 28
1. Place the apices for multiplicand (325) on the bottom row of the
abacus, putting the “5” counter in the units column, the "2” counter in
the tens column, and the “3” counter in the hundreds column:
Fig. 26.8a.
c
X
i
c
X
I
®
©
©
Multiplicand
2. Then place the apices for the multiplier (28) on the top row of the
abacus, putting the “2” and “8” counters in the tens and units columns
respectively:
ARITHMETICAL OPERATIONS ON
gerbert’s abacus
The following example shows how sums can be done on Gerbert’s abacus
without zero. The fact that it is possible to complete these operations
correctly explains why mediaeval manuscripts of the eleventh and twelfth
centuries contain no symbols for zero, nor ever even mention the concept.
The nine symbols of Indian origin spread around Europe, but only in very
restricted circles, since the whole business of counting was in the hands of
a tiny elite of arithmeticians, appropriately called abacists.
Fig. 26.8b.
Multiplier
Multiplicand
583
ARITHMETICAL OPERATIONS ON GERBERT’s ABACUS
3. Now find the product of the 8 x 5 in the units column. Since the
product is 40, place a “4” counter in the upper part of the central
rows of the abacus, in the tens column, leaving the units column
empty, thus:
5. Now multiply the same unit 8 by the apex in the hundreds
column, in other words 3x8. The product being 24, place a “2”
counter in the thousands column and a “4” counter in the hundreds
column:
Fig. 26.8c.
c
X
i
c
1
X
I
©
©
©
®
©
©
Multiplier
Multiplicand
Fig. 26. 8e.
Multiplier
Multiplicand
4. Now find the product of the counter in the units column by the
one in the tens columns, in other words 2x8. The product being 16,
place a “1” counter in the hundreds column and a “6” counter in the
tens column, still leaving the units column empty, thus:
6. Since the multiplying of the “8” is now complete, remove the “8”
token from the abacus before turning attention to the “2” in the
multiplier:
Fig. 26. 8d.
c
X
i
c
X
I
...j
©
©
©
© ©
.. j
©
©
©
Multiple
Multiplicand
c
X
i
c
X
I
©
©
©©
© ©
...j
©
©
©
Multiplier
Multiplicand
Fig. 26. 8f.
THE SLOW PROGRESS OF INDO-ARABIC NUMERALS IN WESTERN EUROPE
584
7. Now multiply the 2 by the 5 in the multiplicand. Since the 2 is
in the tens column, the product (10) requires us to place a “1” counter
in the hundreds column, thus:
9 Now multiply the same 2 by the 3 in the hundreds column of the
multiplicand. The product, 6, means six tens of hundreds, so we place
a “6” counter in the thousands column, thus:
8. Now multiply the 2 of the multiplier by the 2 in the multipli-
cand, giving the answer 4. Since both factors are in the tens columns,
the result (four tens of tens) is registered by placing a “4” in the
hundreds column, thus:
c
X
i
c
X
I
©
©
© © © ©
© ©
Multiplier
Fig. 26. 8h.
® ® © Multiplicand
Fig. 26. 8j.
® © © Multiplicand
585
11. As all the multiplications of the highest number in the multi-
plier are now also complete, all that remains is to sum the partial
products on the board, replacing counters whose total is more than 10
by a unit counter in the next-leftmost column. Since the “4” and “6” of
the tens column total 10, they are taken off the board and replaced by
a “1” counter in the hundreds column, thus:
Remainder
Product
Fig. 26.8k.
12. Now sum the tokens in the hundreds column. As they total 11,
remove all counters bar the “1”, and place a “1” in the thousands
column, thus:
Remainder
Product
c
X
i
c
X
I
© © ©
©
Multiplier
Fig. 26. 8 l .
Multiplicand
ARITHMETICAL OPERATIONS ON GERBERt’s ABACUS
13. Finally, sum the tokens in the thousands column, which gives 9,
so remove all the tokens and replace them by a “9” in the thousands
column, thus:
14. The result of the operation is therefore 9,100 (since the tens and
units columns are empty). This example shows how Gerbert’s abacus
made arithmetical operations long and complicated; its use presup-
posed lengthy training and a high degree of intelligence.
FROM “ARABIC” NUMERALS TO
EUROPEAN APICES
The shapes of the Arabic numerals brought back from Spain by Gerbert of
Aurillac were represented with the most fantastical variations on European
horn apices. Consider the following versions of “4” found over the first two
hundred years of the second millennium CE:
Archetype |
t X cfr
pc.
e*
*
Archetype, Limoges
Fleury
Lorraine
Auxerre
Regensburg
Chartres
Spain, (France),
(France),
(France),
(France),
(Bavaria),
(France),
Xth C Xlth C
XlthC
Xllth C
Xllth C
Xllth C
XUIth C
Fig. 26.9.
Styles obviously varied from one region to another, from one school to
another, even from one engraver to another, in a period that had no concept
of standardisation. Indeed, what we can see happening in these examples is
the adaptation of the Ghubar forms of the Arabic numerals to the very
different styles of writing practised in different parts of Europe. So in Italy
we see numerals assimilated to the round shapes and wide openings
of Italic script, in England to the narrower and more angular shapes of
English script, in Germany to the thicker and squarer writing style
of German script, and in France and Spain we see them being shaped in
harmony with the dominant styles of Carolingian script.
A similar phenomenon has already been observed in India and in the
Indie civilisations of Southeast Asia. Scribes and stone-carvers adapted the
basic nine symbols to their own indigenous writing styles and applied their
own aesthetic sense to the shapes, so that there quickly resulted widely
differing sets of numerals that at first glance seem quite unrelated.
Similar diversity has been seen in the Arabic world too, where scribes
and copyists adapted the same basic figures to the different scripts used in
different areas of the Arabic-speaking world.
So there is no reason for the Western world not to have also generated a
range of distinctive variations on the numeral set. However, as Beaujouan
has pointed out, there was a supplementary factor in the West. All the
different shapes found, he insists, are virtually superimposable on each
other provided they are rotated by some degree. That is particularly notice-
able for the 3, 4, 5, 6, 7 and 9 (see Fig. 26.4).
The reason is that the apices were often placed on the abacus without
any particular regard for the original orientation of the shape. In some
schools, for example, the apices were placed upside-down, so the 5 was
sometimes found with its “tail” at the bottom. The 9 was sometimes placed
on its right side, sometimes on its left side, and sometimes placed upside
down so that it looked like today’s 6.
Some scribes and stone-carvers simply replaced the original shape of the
numeral with the shape that they had grown accustomed to, or which
seemed more “logical” in their eyes. Confusion became generalised, and
even mathematical course-books often taught the numbers upside-down
and back-to-front.
The obvious solution would have been to mark the top or bottom of
each horn apex with a dot, but people were content merely to distinguish
the two figures that could most easily be confused by writing the 6 with
sharp, angular lines and the 9 with curved and flowing lines.
However, mediaeval apices did not actually give rise directly to our
current numerals. After the Crusades, these early forms of the numerals
were simply abandoned, and shapes closer to the original Arabic forms were
re-introduced - and it is these later arrivals, which eventually stabilised into
standard forms, that ultimately gave rise to modern “Arabic” numerals.
THE SECOND INTRODUCTION OF ARABIC
NUMERALS IN EUROPE
We might have expected Pope Sylvester II to have opened the millennium
onto a new era of progress in the West, thanks to the numerals and opera-
tional techniques he had brought back from the Arabic-Islamic world. But
such expectations would be vain: the ignorance and conservatism of the
Christian world blocked the way.
Although modern numerals and number-techniques were in fact avail-
able from the late tenth century, they were used only in the most
rudimentary ways for over two hundred years. They served solely to
simplify archaic counting methods and to give rise to rules of procedure
which, according to William of Malmesbury, “perspiring abacists barely
comprehended themselves”.
Some arithmeticians even put up a solid resistance to the new-fangled
figures from the East by inscribing their apices with the Greek letter-
numerals from a = 1 to 0 = 9, or the Roman figures I to IX. Anything
was better than having recourse to the “diabolical signs” of the “satanic
accomplices” that the Arabs were supposed to be!
Gerbert of Aurillac also suffered at the hands of the rearguard. It was
rumoured that he was an alchemist and a sorcerer, and that he must have
sold his soul to Lucifer when he went to taste of the knowledge of the
Saracens. The accusation continued to circulate for centuries until finally,
in 1648, papal authorities reopened the tomb of Sylvester II to make sure
that it was not still infested by the devil!
The dawn of the modern age did not really occur until Richard
Lionheart reached the walls of Jerusalem. From 1095 to 1270, Christian
knights and princes tried to impose their religion and traditions on the
Infidels of the Middle East. But what they actually achieved was to bring
back to Europe the cultural riches they encountered in the Holy Land. It
was these campaigns - or rather, their secondary consequences - that
finally allowed the breakthrough which Gerbert of Aurillac, for all his
knowledge and energy, had failed to achieve at the end of the tenth century.
For the wars implied a whole range of contacts with the Islamic world, and
a number of clerks travelling with the armies learned the written numerals
and arithmetical methods of the Indo-Arabic school.
Gerbert’s abacus thus slowly fell into disuse. Gradually, numerals
written on sand or dust, instead of being engraved on horn-tipped apices,
led to the disappearance of the columns on the abacus. This allowed much
587
simpler, much faster and more elegant operations, which now came to be
called algorisms, after al-Khuwarizmi, the first Islamic scholar who had
generalised their application.
Xllth C Toledo (Spain): Astronomical Tables.
Munich, Bayerische Staatsbibliothek,
Clm 18927, P lr, lv
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After England: Algorism. London, BM Ms.
1264 Add. 27589, P 28
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Add. 24059, P 22r
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Add. 8784, P 50r-51
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THE SECOND INTRODUCTION OF ARABIC NUMERALS IN EUROPE
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lat. 7413, part 11. Facsimile in the Ecole des Chartes, AF 1113
So the first European “algorists” were born at the gates of Jerusalem. But
unlike the “abacists”, the new European counting experts were obliged to
adopt the zero, to signify missing orders of magnitude, otherwise compu-
tations written in sand would lead to confusing representations of number
and mistaken operations. At last, then, true “Arabic” numerals including
zero, and the arithmetical tradition that had been born long before in India,
were able to make their way into Europe.
There were of course other contacts with the Islamic world on the other
side of the Mediterranean, by way of Sicily and most especially through Spain
and North Africa. It was in Spain that a huge wave of translations began in the
twelfth century, bringing into Latin works written in Arabic, and even more
importantly Greek and Sanskrit texts already translated into Arabic. Thanks
to translators like Adelard of Bath and to centres of scholarship at Cordoba
and Toledo, the resources available for acquiring knowledge of arithmetic,
THE SLOW PROGRESS OF INDO-ARABIC NUMERALS IN WESTERN EUROPE
588
mathematics, astronomy, natural sciences and philosophy swelled almost
by the day; and it was by means of translation from the Arabic that the
West eventually became familiar with the works of Euclid, Archimedes,
Ptolemy, Aristotle, al-Khuwarizmi, al-Biruni, Ibn Sina, and many others.
Between them, the Crusaders at Jerusalem and the scholars of Toledo
were ensuring the more or less rapid death of the abacus and of abacism.
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Script numerals in the style of the capitals
of Trajan’s column in Rome
S Samples of numeral faces from the
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1204307000 Pa 1850
Fig. 26.12. The development of printed numerals since the fifteenth century
The spread of “algorism” was given renewed impetus from the start of
the thirteenth century by a great Italian mathematician, Leonard of Pisa (c.
1170-1250), better known by the name of Fibonacci. He visited Islamic
North Africa and also travelled to the Middle East. He met Arabic arith-
meticians and learned from them their numeral system, the operational
techniques, the rules of algebra and the fundamentals of geometry. This
education was what underlay the treatise that he wrote in 1202 and which
was to become the algorists’ bible, the Liber abaci ( The Book of the Abacus).
Despite its title, Fibonacci’s treatise (which assisted greatly the spread of
Arabic numerals and the development of algebra in Western Europe) has
no connection with Gerbert’s abacus or the arithmetical course-books of
that tradition - for it lays out the rules of written computation using both
the zero and the rule of position. Presumably Fibonacci used “abacus” in his
title in order to ward off attacks from the practical abacists who effectively
monopolised the world of accounting and clung very much to their coun-
ters and ruled tables. At all events, from 1202 the trend began to swing in
favour of the algorists, and we can thus mark the year as the beginning of
the democratisation of number in Europe.
Resistance to the new methods was not easily overcome, however, and
many conservative counting-masters continued to defend the archaic
counter-abacus and its rudimentary arithmetical operations.
Professional arithmeticians, who practised their art on the abacus,
constituted a powerful caste, enjoying the protection of the Church. They
were inclined to keep the secrets of their art to themselves; they necessarily
saw algorism, which brought arithmetic within everyone’s grasp, as a threat
to their livelihood.
Knowledge, though it may now seem rudimentary, brought power and
privilege when it represented the state of the art, and the prospect of seeing
it shared seemed fearful, perhaps even sacrilegious, for its practitioners. But
there was another, more properly ideological reason for European resis-
tance to Indo-Arabic numerals.
Even whilst learning was reborn in the West, the Church maintained a
climate of dogmatism, of mysticism, and of submission to the holy scrip-
tures, through doctrines of sin, hell and the salvation of the soul. Science
and philosophy were under ecclesiastical control, were obliged to remain in
accordance with religious dogma, and to support, not to contradict, theo-
logical teachings.
The control of knowledge served not to liberate the intellect, but to
restrict its scope for several centuries, and was the cause , of several
tragedies. Some ecclesiastical authorities thus put it about that arithmetic
in the Arabic manner, precisely because it was so easy and ingenious,
reeked of magic and of the diabolical: it must have come from Satan
589
himself! It was only a short step from there to sending over-keen algorists
to the stake, along with witches and heretics. And many did indeed suffer
that fate at the hands of the Inquisition.
The very etymology of the words “cypher” and “zero” provides evidence
of this.
Fig. 26.13. Written arithmetic using “Arabic" numerals. European engraving, sixteenth century.
Paris, Palais tie la Decouverte
When the Arabs adopted Indian numerals and the zero, they called the
latter sifr, meaning “empty”, a plain translation of the Sanskrit shunya. Sifr
is found in all Arabic manuscripts dealing with arithmetic and mathemat-
ics, and it refers unambiguously to the null figure in place-value numbering.
(See for example the manuscripts in the Bibliotheque nationale, Paris, shelf-
marks Ms. ar. 2457, P 85v-86; Ms. ar. 2463, P 79v-80; Ms. ar. 2464, P 3v;
Ms. ar. 2473, P 9; Ms. ar. 2475, P 45v-46r; and University of Tunis Ms.
10301, P 25v; Ms. 2043, Pl6v and 32v.) Etymologically, sifr means “empty”
and also “emptiness” (the latter can also be expressed by khala or faragh).
THE SECOND INTRODUCTION OF ARABIC NUMERALS IN EUROPE
The stem SFR can also be found in words meaning “to empty” ( asfara ), “to
be empty” (safir) and “have-nothing” (safr alyadyn, literally “empty hands”,
that is to say, “he who has nothing in his hands”.
When the concept of zero arrived in Europe, the Arabic word was assim-
ilated to a near-homophone in Latin, zephyrus, meaning “the west wind”
and, by rather convenient extension, a mere breath of wind, a light breeze,
or - almost - nothing. In his Liber Abaci, Fibonacci (Leonard of Pisa) used
the term zephirum, and the term remained in use in that form until the
fifteenth century:
The nine Indian figures [figurae Indorum ] are the following: 9, 8, 7, 6, 5,
4, 3, 2, 1. This is why with these nine figures and the sign 0, called
zephirum in Arabic, all the numbers you may wish can be written
[Fibonacci, as reproduced by B. Boncompagni (1857)].
However, in his Sefer ha mispar (Number Book), Rabbi Ben Ezra (1092-1167)
used the term sifra [see M. Silberberg (1895) p. 2; D. E. Smith and
Y. Ginsburg (1918)]. In various spellings, the Arabic term sifra ( cifra , cyfra,
cyphra, zyphra, tzyphra. . .) continued to be used to mean “zero” by some
mathematicians for many centuries: we find it in the Psephophoria kata
Indos ( Methods of Reckoning of the Indians ) by the Byzantine monk Maximus
Planudes (1260-1310) [A. L. Allard, (1981)], in the Institutiones mathemati-
cae of Laurembergus, published in 1636, and even as late as 1801 in Karl
Friedrich Gauss’s Disquisitiones arithmeticae (Gauss must have been one of
the very last scholars to write in Latin).
In popular language, words derived from sifr soon came to be associated
not with figures in general but with “nothing” in particular: in thirteenth-
century Paris, a “worthless fellow” was called a cyfre d’angorisme or a cifre en
algorisme, i.e. “an arithmetical nothing”.
However, it was Fibonacci’s term, zephirum, which gave rise to the
modern name of zero, by way of the Italian zefiro (zero is just a contraction
of zefiro, in Venetian dialect). The first known occurrence of the modern
form of the word occurs in De arithmetica opusculum by Philippi Calandri
and which, despite its Latin title, was written in Italian, and published in
Florence in 1491. There is absolutely no doubt that zero owes its spread to
French (zero) and Spanish (cero) (and later on to English and other
languages) to the enormous prestige that Italian scholarship acquired in the
sixteenth century.
Meanwhile, Arabic sifr had also developed into the French word chiffre,
the English cipher, German Zijfer, Spanish cifra. To begin with, the Latin
Items figuris and numero were used to refer to the set of number-symbols (in
English they still are called figures or numerals, more or less interchangeably);
but from about 1486 in French, we find chiffre being used not to mean zero,
THE SI.OW PROGRESS OF IN DO- ARABIC NUMERALS IN WESTERN EUROPE
5 9 0
but to mean a figure or numeral; and a similar development can be found in
sixteenth- and seventeenth-century mathematical texts written in Latin,
such as those by Willichius (1540), Conrad Rauhfuss Dasypodius of
Strasbourg ( Institutionum Mathematicarum, 1593), and the Chronicle of
Theophanes (1655).
Fig. 26.14. The Quarrel of the Abacists (to the left) and the Algorists (to the right). Adapted from
an illustration in Robert Recorde (1510-1558), The Grounde of Artes (1558)
Why did the original name of zero come to be used for the whole set of
Indo-Arabic numerals? The answer lies in the attitude of the Catholic
authorities to the counting systems borrowed from the Islamic world. The
Church effectively issued a veto, for it did not favour a democratisation of
arithmetical calculation that would loosen its hold on education and thus
weaken its power and influence; the corporation of accountants raised its
own drawbridges against the “foreign” invasion; and in any case the Church
preferred the abacists - who were most often clerics as well - to keep their
monopoly on arithmetic. “Arabic” numerals and written calculation were
thus for a long while almost underground activities. Algorists plied their
skills in hiding, as if they were using a secret code.
All the same, written calculation (on sand or by pen and ink) spread
amongst the people, who were keenly aware of the central role played by
zero, then called cifra, or chifre, or chiffre, or tziphra, etc. By a very common
form of linguistic development, known as synecdoche, the name of the part
(in this case, zero) came to be used for the whole, as in a kind of shorthand,
so that words derived from sifr came to mean the entire set of numerals or
any one of them. Simultaneously, it also came to mean “a secret”, or a secret
code - a cipher.
So the history of words for zero also tell the history of our culture: each
time we use the word “cipher”, we are also reviving a linguistic memory of
the time when a zero was a dangerous secret that could have got you burned
at the stake.
It is now easier to understand why in the mid-sixteenth century
Montaigne could not “cast account” either “with penne or with Counter”. For
even with the introduction of written arithmetic, multiplication and division
long remained outside the grasp of ordinary mortals, given the complicated
operating techniques that were used. It was not until the end of the eigh-
teenth century that simpler techniques were generalised and brought basic
arithmetical operations even to those with little taste for sums.
The quarrel between the abacists (the defenders of Roman numerals
and of calculations done on ruled boards with counters) and the algorists,
who supported the written calculation methods originally invented in
India, actually lasted several centuries. And even after the latter’s victory,
the use of the abacus was still so firmly entrenched in people’s habits that
all written sums were double-checked on the old abacus, just to make sure.
Until relatively recently, the British Treasury still used the abacus to
calculate taxes due. And because the reckoning-board was called an exche-
quer (related to the words for chess and chess-board in various European
languages), the Finance Minister of the United Kingdom is still called the
Chancellor of the Exchequer.
Even long after written arithmetic with Arabic numerals had become the
sole tool of scientists and scholars, European businessmen, financiers,
bankers and civil servants - all of whom turned out to be more conservative
than men of learning - found it hard to abandon entirely the archaic
methods of the bead and counter-abacus.”
Only the French Revolution had the strength to cut through the muddle
and to implement what many could see quite clearly, that written arith-
metic was to counting-tokens as walking on a well-paved road was to
wading through a muddy stream. The use of the abacus was banned in
schools and government offices from then on.
Calculation and science could thenceforth develop without hindrance.
Their stubborn and fierce old enemy had finally been put to rest.
* Translator’s note: my lather was trained as an accountant in the City of London in the late 1920s. Although
he had of course learned modern arithmetic at school, he was required to learn how to tally sums on a bead
abacus before being allowed to draw a wage, (db)
591
THE SECOND INTRODUCTION OF ARABIC NUMERALS IN EUROPE
Fig. 26 . 15 . Wood-block engraving from Gregorius Reisch, Margarita Philosophica (Freiburg,
1503). Lady Arithmetic (standing in the centre) gives her judgment by smiling on the arithmetician
(to our left, her right) working with Arabic numerals and the zero (the numerals also adorn her
dress). The quarrel of the abacists and algorists is over, and the latter have won.
BEYOND PERFECTION
592
CHAPTER 27
BEYOND PERFECTION
That then was how numerical notation was brought to its full completion,
democratised, and universalised: after a long history of twists and turns,
with leaps forward and steps backward, ideas lost and found again, and
with the friction between different systems used in conjunction ultimately
generating the flash of genius on which it is all based: the decimal place-
value system.
Is the story really at an end? After such a long and eventful history, could
there not be more adventures to come? No, there could not. This really is
the end. Our positional number-system is perfect and complete, because it
is as economical in symbols as can be and can represent any number,
however large. Also, as we have seen, it is the most efficacious in that it
allows everyone to do arithmetic.
True, the development of computers and of electronic calculators with
liquid crystal displays in the last half century has brought some changes in
the graphical representation of the “Arabic” numerals. They have taken on
more schematic shapes that would no doubt have horrified the scribes and
calligraphers of yesteryear. In reality, however, these changes have had no
effect whatever on the structure of the number-system itself. The numerals
have been redesigned to meet the physical constraints of the display media,
while also meeting the requirement to be readable both by machines and
by the human eye.
Of course, as we have seen many times, a different base could have been
used for our number-system. The base 12, for example, is in many ways
more convenient than our decimal base; and the base 2 is well adapted to
electronic computers which usually can recognise only two different states,
symbolised by 0 and 1, of a physical system (perforation of a tape, or direc-
tion of magnetisation or of a current, etc.). But a change of base would
change nothing in the structure of the number-system: this would continue
to be a positional system and would continue to possess a zero, and its
fundamental rules would be identical to those which we know already for
our decimal system.
In short, the invention of our current number-system is the final stage in
the development of numerical notation: once it was achieved, no further
discoveries remained to be made in this domain.
The difficulties encountered on the road to a fully finished number-
representation bear witness, on a limited front though one rich in possibil-
ities, to true progress in human affairs.
From the beginning, human beings have shown the unique characteris-
tic of harnessing the forces of nature to their development, their survival
and their domination over other species, through discovering the laws of
nature by means of observing the effects of their actions on their environ-
ment. Instead of following immutably programmed instinct, they act, seek
to understand the “why” of things, ponder, and create.
In his novel Les Animaux denatures, Vercors recounts a telling story. A
tribe of “primitive” people share a valley with a colony of beavers. The
valley is swept by a flood. The beavers, driven by their hereditary instincts,
build a dam and thus protect their dens. The humans, on the other hand,
guided by their grand wizard, climb the sacred hill and meditate, begging
mercy from their gods; this, however, does not prevent their village from
being devastated by the flood.
At first sight, the behaviour of the humans seems stupid. But on reflec-
tion we see something really profound in it, for it is the germ of all future
civilisation. They were certainly wrong to attribute the disaster to super-
natural forces but, despite appearances, their reaction leapt beyond the
mere instinct of the beavers, since they sought to understand the true cause
of their misfortune. Humanity has surely passed through such phases: we
know how far our tribulations have brought us.
This is not the place to retrace the evolution of the human race since the
time of the first hominids. We must rather recall that human beings are
characterised above all, not by what is innate and does not need to be
learned, but by the predominance of what they can adjoin to their nature
from learning, experience and education.
In other words, humankind is universally an intelligent social animal,
and is differentiated from other higher animals by, above all, the predomi-
nance of what is acquired over what is inborn.
That fundamental truth has not always been, nor indeed yet is, obvious
to everyone. For reasons ranging from the political to the criminal, this ques-
tion has been subjected to systematic mystification in order that irrelevant
criteria, such as the colour of the skin or the shape of the face, may be used
to demonstrate the supposed superiority of one race over others.
The principal motivations and the basic ideas of racist and segregation-
ist philosophies are directed towards maintaining great confusion between
the notion of race and the ideas of a people, of a tribe, of an ethnicity and
of a linguistic group, and towards cultivating a belief that there are so-called
superior races who have a kind of natural right to exploit or even to
suppress so-called inferior ones.
These indefensible racist mystifications, which the Nazis elevated to
political ideology during Germany's Third Reich and which throughout the
Second World War gave rise to the greatest barbarity of all time and led
593
BEYOND PERFECTION
millions of innocents to slaughter, reflected an appalling eugenic mentality
whose spirit still haunts the world decades after Nazism was crushed. All
those who may have forgotten it, or who would wish that it should be forgot-
ten, need to be reminded that “one man is not the same as another” but at
the same time “one race is not unequal to another, still less is one people
unequal to another” (J. Rostand, Heredite et Racisme, p. 63).
As to the colour of the skin, this in fact (according to Francois Jacob)
depends on the intensity of sunlight or, as the Arab philosopher Ibn
Khaldun expressed it around 1390: “The climate gives the skin its colour.
Black skin is the result of the greater heat of the South” [Muqaddimah,
Prolegomena, p. 170; see V. Monteil (1977), p. 169],
The concept of race, in fact, is strictly biological, while that of people is
historical. We talk, therefore, not of the French race but of the French
people, which is made up of a mixture of several races. Nor is there a Breton
race, but there is a Breton people; no Jewish race, but the Jewish people; no
Arab race but Arab culture; no Latin race but Latin civilisation; and neither
Semitic nor Aryan races, but Semitic and Aryan languages.
According to R. Hartweg (GLE Vol. 8, p. 976) the concept of race is
“one of the categories of zoological classification. It denotes a relatively
broad grouping within a species, a kind of sub-species, a collection of
individuals of common origin which share a number of sufficiently mean-
ingful biological characters.” It therefore "rests on genetic, anatomical,
physiological and pathological criteria. The difficulty with attempting
to apply a racial classification to humankind therefore arises from: 1.
the choice of criteria; 2. the fact that there are at present very few races
which might be considered relatively pure', because of inter-breeding; 3.
the transitory nature of the definition of any given race since races, like
humanity itself, undergo continual evolution.” D. L. Julia (1964) has the
following view of this question:
From the biological point of view, the notion of race as applied to
humans is very imprecise. Features such as skin colour or facial
structure are definite morphological characters, but they are
biologically vague. Even if we suppose that different races exist,
criteria such as physical strength, or intelligence (as measured by
IQ tests), show no systematic variation. Though the people of
industrialised nations may have weaker constitutions than those of
African nations, for example, and although culture and education
may seem less prevalent among the latter than among Western
peoples, nonetheless this has no bearing on the physical potential
of the former nor on the intellectual potential of the latter. On the
other hand, differences of character - whereby we traditionally
contrast the intellectual strictness of the “whites” with the intuitive
mind and generous spirit of the "blacks”, or the openness of both of
these with the feline suppleness and deep capacity for dissimulation
of the “yellow” peoples - bear no relationship to a scale of values.
Differences of character should not be a source of conflict, but an
occasion for learning and therefore of enrichment: in coming to
understand other people, any persons of any race will come to
better understand themselves as individuals, and learn wisdom for
the conduct of their own lives.
In short, "racist theories are gratuitous constructs, based on tendentious
and immature anthropological ideas” (J. Rostand, Heredite et Racisme,
p. 57). "The truth is, that there is no such thing as a pure race, and to base
politics on ethnographic analysis is to base it on a chimera” (E. Renan,
Discours et Conferences, pp. 93-4).
In the domain of the history of numbers, at least, we have seen that
human intelligence is universal and that the progress has been achieved in
the mental, cultural and collective endowment of the whole of humankind.
From the Cro-Magnon to the modern period, no fundamental change in
the human brain has in fact occurred: only cultural enrichment of mental
furnishings. This means that all human beings, whether white, red, black
or yellow, whether living in the town, the country or the bush, have
without exception equal intellectual potential. Individuals will develop the
possibilities of their intelligence, or not, according to their needs, their
environments, their social circumstances, their cultural heritage and their
diverse individual aptitudes. These strictly personal individual differences
are what determine whether one mind will be more or less enlightened,
more or less inventive, than another.
As was stated in the Preface, number and simple arithmetic nowadays
seem so obvious that they often seem to us to be inborn aptitudes of the
human brain.
This was no doubt why the great German mathematician Kronecker said
“God created the integers, the rest is the work of Man”, whereas in fact the
whole is an invention, the pure creation of the human mind; as the German
philosopher Lichtenberg said: "Mankind started from the principle that
every magnitude is equal to itself, and has ended up able to weigh the sun
and the stars.”
And the invention is of purely human origin: no god, no Prometheus, no
extra-terrestrial instructor, has given it to the human race.
The actual history of numbers serves also, incidentally, to refute all
those popular stories of extra-terrestrials who came to Earth to civilise the
human race. Had we been visited by a scientifically and technologically
advanced civilisation from outer space, we would not first have learned
BEYOND PERFECTION
594
from it mysterious methods of erecting megaliths, but a number-system
based on the principle of position and endowed with a zero. There is abun-
dant documentation which proves that these were of late appearance, and
that historically there was a great variety of number-systems in use. Quite
sufficient to disprove any extra-terrestrial source for arithmetic - and there-
fore for everything else.
This profoundly human invention is also the most universal of inven-
tions. In more than one sense, it binds humanity together. There is no
Tower of Babel for numbers: once grasped, they are everywhere understood
in the same way. There are more than four thousand languages, of which
several hundred are widespread; there are several dozen alphabets and
writing systems to represent them; today, however, there is but one single
system for writing numbers. The symbols of this system are a kind of visual
Esperanto: Europeans, Asiatics, Africans, Americans or Oceanians, inca-
pable of communicating by the spoken word, understand each other
perfectly when they write numbers using the figures 0, 1, 2, 3, 4. . . , and
this is one of the most notable features of our present number-system. In
short, numbers are today the one true universal language. Anyone who
thinks that number is inhuman would do well to reflect on this fact.
The invention and democratisation of our positional number-system
has had immeasurable consequences for human society, since it facilitated
the explosion of science, of mathematics and of technology.
This in its turn gave rise to the mechanisation of arithmetical and math-
ematical calculations.
Yet all the elements needed to construct a true calculating engine had
already been in existence, known and utilised since ancient times by schol-
ars and engineers such as Archimedes, Ctesibius or Hero of Alexandria -
such devices as levers, the endless screw, gears, toothed wheels, etc. But
when we look at the numerical notations which they used at the time we
can see that it would have been out of the question for them to conceive of,
let alone to construct, such machines.
Nor did the technology of the time permit their actual construction: not
until the start of the seventeenth century, when clockwork mechanisms
underwent enormous development, would the first implementations of such
devices be seen. Without a positional number-system with a zero, Schickard
and Pascal would have been unable to imagine the components of their
calculating machines. Pascal, for example, would not have thought of the
transferrer (a counter-balanced pawl which, when one counting-wheel
advanced from “9" to “0” after completing a revolution, advanced the next
wheel through one step), nor of the totalisator (a device which, for each
power of ten, had a cylinder bearing two enumerations from “0” to “9”, in
opposite directions, one used for additions and the other for subtractions).
To sum up: if the positional number-system with a zero had not existed,
the problem of mechanising the process of calculation would never have
found a solution; still less would it have been conceivable to automate the
process. This, however, is another story, the story of automatic calculation,
which begins with the classical calculating or analytical engines, passes on
to machines for sorting and classifying data, and culminates in the emer-
gence of the computer.
These powerful developments would never have seen the light of day,
had the Indian discovery of positional notation not influenced the art of
calculation itself. Since, however, it evidently did, we are led to look far
beyond the domain of mere figures into the universe of number itself.
Note first that, unlike almost all earlier systems, our modern number-
system allows us to write out straightforwardly any number whatever, no
matter how large it may be. But modern mathematicians have introduced a
simplification in the representation of very large numbers by means of so-
called "scientific" notation which makes use of the powers of ten. For
example, 1,000 may be written as 10 3 , a million as 10 6 and a billion as 10 9 ,
the small number in the exponent denoting the number of zeros in the
standard representation of the number. For a billion, for example, we write
down three figures instead of ten.
As it stands this is no more than an abbreviated notation, which
effects no change in the number-system being used. Nonetheless, it is
more than mere shorthand, since it lends itself to the procedure known
as exponentiation (“raising to the power”) which stands to multiplication
as multiplication stands to addition, since we can write:
a m x a" = a m+n ; a m /a n = a m_n ; (a m ) n = a mn
Using this notation, a very large number such as
72,965,000,000,000,000,000,000,000,000,000 (27 zeros)
can be written more economically as
72,965 x 10 27
which simply indicates that by adjoining 27 zeros to 72,965 the complete
representation of the number is obtained. We can also use “floating-point
notation, and express the first number as a decimal fraction followed by the
appropriate power of ten, as in
7.29 65 x 10 31
which indicates that the decimal point is to be moved 31 places to the right
in order to obtain the complete representation.
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BEYOND PERFECTION
Most pocket calculators and electronic computers have a facility of this
kind which allows them to show numerical results which exceed the decimal
capacity of the display (or at least to show their approximate values).
The positional number-system gave rise to great advances in arithmetic,
because it showed the properties of numbers themselves more clearly.
It similarly enabled mathematicians of recent times to unify apparently
distinct concepts, and to create theories which had previously been
unthinkable.
Fractions, for example, had been known since ancient times, but owing
to the lack of a good notation they were for long ages written using nota-
tions which were only loosely established, which were not uniform, and
which were ill adapted to practical calculation.
Originally, remember, fractions were not considered to be numbers.
They were conceived as relations between whole numbers. But, as methods
of calculation and arithmetic developed, it was observed that fractions
obeyed the same laws as integers, so that they could be considered as
numbers (an integer, therefore, being a fraction whose denominator was
unity). As a result, where numbers had previously served merely for count-
ing, they now became “scales” which could be put to several uses.
Thereafter, two magnitudes would no longer be compared “by eye”; they
could be conceived as subdivided into parts equal to a magnitude of the
same kind which served as a unit of reference. Despite this advance,
however, the ancients, with their inadequate notations, were unable to
unify the notion of fraction and failed to construct a coherent system for
their units of measurement.
Using their positional notation with base 60, the Babylonians were the
first to devise a rational notation for fractions. They expressed them
as sexagesimal fractions (in which the denominator is a power of sixty)
and wrote them much as we now write fractions of an hour in minutes
and seconds:
33m 45s (= 33/60h + 45/3600h).
They did not, however, think of using a device such as the “decimal point”
to distinguish between integers and sexagesimal fractions of unity, so that
the combination [33; 45] could as well mean 33h 45m as Oh 33m 45s. They
had, so to speak, a “floating notation” whose ambiguities could only be
resolved by context.
The Greeks then tried to make a general notation for vulgar fractions,
but their alphabetic numerals were ill-adapted for the purpose and so
they abandoned the attempt. Instead, they adopted the Babylonian sexa-
gesimal notation.
Our modern notation for vulgar fractions is due to the Indians who,
using their decimal positional number-system, wrote a fraction such as
34/1,265 much as we do now:
34 (numerator)
1,265 (denominator).
This notation was adopted by the Arabs, who brought it into its modern form
by introducing the horizontal bar between numerator and denominator.
Then, following the discovery of “decimal” fractions (in which the
denominator is a power of ten), people gradually became aware of the
importance of extending the positional system in the other direction, i.e. of
representing numbers “after the decimal point", and this is what finally
allowed all fractions to be written without difficulty, and which showed the
integers to be a special kind of fraction, in which no figures appear after the
decimal point.
The first European to make the decisive step towards our modern
notation was the Belgian Simon Stevin. Where we would write 679.567,
he wrote:
679(0) 5(1) 6(2) 7(3)
which stood for 679 integer units, 5 decimal units of the first order (tenths),
6 of the second order (hundredths) and 7 of the third (thousandths).
Later on, the Swiss Jost Biirgi simplified this notation by omitting the
superfluous indication of decimal order, and by marking the digit repre-
senting the units with the sign 0 :
679567
At the same time, the Italian Magini replaced the ring sign by a point placed
between the units digit and the tenths digit, creating the decimal-point
notation which is still the standard usage in English-speaking countries:
679.567
In continental Europe, a comma is commonly used instead of the point,
and this was introduced at the start of the seventeenth century by the
Dutchman Snellius:
679.567
This rationalisation of the concept and of the notation of fractions had
immeasurable consequences in every domain. It led to the invention of the
“metric system", built entirely on the base 10 and completely consistent: in
1792, the French Revolution offered “to all ages and to all peoples for their
greater good” this system which replaced the old systems of arbitrary,
BEYOND PERFECTION
596
inconsistent and variable units. We know full well the fantastic progress
that this brought in every practical domain, by virtue of the enormous
simplification of every kind of calculation.
Once established, positional decimal notation opened up the infinite
complexity of the universe of number, and led to prodigious advances in
mathematics.
In the sixth century BCE the Greek mathematicians, following
Pythagoras, discovered that the diagonal of a square “has no common
measure” with the side of the square. It can be observed by measurement,
and deduced by reason, that the diagonal of a square whose side is one
metre long has a length which is not a whole number of metres, nor of
centimetres, nor millimetres. ... In other words, the number V 2 (which is
its mathematical magnitude) is an “incommensurable” number. This was
the moment of discovery of what we now call “irrational" numbers, which
are neither integers nor fractions.
This discovery greatly perturbed the Pythagoreans, who believed that
number ruled the Universe, by which they understood the integers and
their simpler combinations, namely fractions. The new numbers were
called “unmentionable”, and the existence of these “monsters” was not to
be divulged to the profane. According to the Pythagorean conception of the
world, this inexplicable error on the part of the Supreme Architect must be
kept secret, lest one incur the divine wrath.
But the secret soon became well known to right-thinking people who
were prepared to mention the unmentionable, to name the unnameable,
and who delivered it up to the profane world. That perfect harmony
between arithmetic and geometry, which had been one of the fundamental
tenets of the Pythagorean doctrine, was seen to be a vain mystification.
Once we are free of these mystical constraints, we can accept that there
are numbers which are neither integers nor fractions. These are the “irra-
tional” numbers, of which examples are V2 , V 3, the cube root of 7 and of
course the famous rt.
Nevertheless, this class of numbers remained ill defined for many
centuries, because the defective number-systems of earlier times did not
allow such numbers to be represented in a consistent manner. They were in
fact designated by words, or by approximate values which had no apparent
relation to each other. Lacking the means to define them correctly, people
were obliged to admit their existence but were unable to incorporate them
into a general system.
Modern European mathematicians, with the benefits of effective numer-
ical notation and continual advances in their science, finally succeeded
where their predecessors had failed. They discovered that these irrational
numbers could be identified as decimal numbers where the series of digits
after the decimal point does not terminate, and does not eventually become
a series of repetitions of the same sequence of digits. For example: ^2 =
1.41421356237. . . This was a fundamental discovery: this property charac-
terises the irrational numbers.
Of course, a fraction such as 8/7 also possesses a non-terminating
decimal representation:.
8/7 = 1.142857142857142857. . .
but its representation is periodic: the sequence “142857” is indefinitely
repeated, with nothing else intervening: we can therefore, for instance, easily
determine that the 100th decimal digit will be “8” , since 16 repetitions will
take us to the 96th place, and four more digits will give the digit “8”.
On the other hand, the irrational numbers do not follow such a pattern.
Their decimal expansion is not periodic, and there is no rule which allows
us to determine easily what digit will be in any particular place. This is
precisely the respect in which the vulgar fractions (what we today call
“rational numbers” ) differ from the irrational numbers.
However, nowadays this is not how irrational numbers are defined.
Instead, an algebraic criterion is used, according to which an irrational
number is not the solution of any equation of the first degree with integer
coefficients. The number 2, for example, is the solution of x -2 = 0, and the
fraction 2/3 is the solution of 3x - 2 = 0. On the other hand, it can be proved
that the number V2 cannot be the solution of any equation of this kind, and
so it is irrational.
Nonetheless, the concept of such numbers would not have been fully
understood without the introduction of a further extension of the notion of
number: the “ algebraic” numbers. This concept was discovered in the nine-
teenth century by the mathematicians Niels Henrik Abel of Norway, and
Evariste Galois of France. An algebraic number is a solution of an algebraic
equation with integer coefficients. Clearly this holds for any integer or
fractional number, but it also holds for any irrational number which can
be expressed by radicals. For example, V 2 is a solution of the equation
x 2 -2 = 0, and the cube root of 5 is a solution of the equation x 3 - 5 = 0. The
set of algebraic numbers, therefore, includes both the set of rational
numbers (which itself includes the integers) and the set of all numbers that
can be expressed by the use of radicals.
However, even this is inadequate to contain all numbers. After the discov-
eries of Liouville, Hermite, Lindemann and many others, we know that there
are additional “real numbers”, which are not integers, or fractions, or even
algebraic irrational numbers. These are the so-called “transcendental’
numbers, which cannot occur as a solution of any algebraic equation with
integer or fractional coefficients. They are, of course, irrational; but they
597
BEYOND PERFECTION
cannot be expressed by the use of radicals. There are infinitely many of them;
examples include the number “tt” (the area, and also half the circumference, of
a circle with unit radius), the number “e” (the base of the system of natural
logarithms invented in 1617 by the Scottish mathematician John Napier), the
number “log 2” (the decimal logarithm of 2) and the number “cos 25°” (cosine
of the angle whose measure is 25 degrees). However, we cannot here let
ourselves be carried away into the further reaches of the theory of numbers.
Now, if it is possible as we have seen to write any number whatever in a
simple and rational way, no matter how large or strange it may be, then we
negative
real numbers
►
positive
zero
integers
rational numbers
algebraic numbers
real numbers
Fig. 27.1. The successive algebraic extensions of the concept of number
may well ask if there is a last number, greater than all the others. We can
directly see from the positional notation that this cannot be so, since if we
write down the decimal representation of an integer then all we have to do
is to add a zero at the right-hand end, to multiply this number by 10.
Proceeding indefinitely in this way, we readily see that the sequence of inte-
gers has no limit. All the more so for the fractions and the irrational
numbers, for which we can demonstrate that there exist “several infinities”
between any two consecutive integers.
From the dawn of history, people came up against the dilemma of the
infinite (see the article ‘Infinity, in the Dictionary). Since then, however,
the concept of infinity has been made perfectly precise and objective, and
presents no fundamental obscurity - at least, not such as the common
mind attributes to it. Infinity has its own symbol: like a figure 8 on its
side, called “lemniscate” by some and introduced quite recently into math-
ematical notation by the English mathematician John Wallis who first
employed it in 1655. But we can hardly prove the existence of infinity - the
impossibility of counting all numbers - since infinity, nowadays, is taken as
an axiom, a mathematical hypothesis, on which the whole of contemporary
mathematics is based.
It is but one step from infinity to zero, and it is a step which leads us on
to algebra, since the null is the opposite of the unlimited.
For thousands of years, people stumbled along with inadequate and
useless systems which lacked a symbol for "empty” or “nothing” . Similarly
there was no way of conceiving of “negative” numbers (-1, -2, -3, etc.), such
as we nowadays use routinely to express, for example, sub-zero temperatures
or bank accounts in deficit. Therefore a subtraction such as “3 - 5” was for a
long time considered to be impossible. We have seen how the discovery of
zero swept away this obstacle so that ordinary (“natural”) numbers were
extended to include their “mirror images” with respect to zero.
That inspired and difficult invention, zero, gave rise to modern algebra
and to all the branches of mathematics which have come about since the
Renaissance (see the article ‘Zero in the Dictionary).
Algebra would not however have blossomed as it did if, as well as the
zero, there had not also been another, equally important discovery made by
Franciscus Vieta in 1591 and brought to perfection by Rene Descartes in
1637: this is the use of letters as mathematical symbols, which inaugurated
a completely new era in the history of mathematics.
Algebra, in fact, is a generalisation of arithmetic. An x or ay, or any other
letter, is a new sort of “number”: it stands for any number, whose value is
unknown. One might say that it is a sign in wait for a number, holding the
place for one or more figures yet to come, just as the zero sign itself filled the
place of a digit corresponding to a missing decimal order of magnitude.
BEYOND PERFECTION
598
But this is no merely formal artifice. Using a letter to stand for a para-
meter or an unknown value finally freed algebra from enslavement to
words, leading to the creation of a kind of “international language” which is
understood unambiguously by mathematicians the world over.
In its turn, literal notation underwent a further liberation from certain
restrictions acquired in its everyday usage. The symbol x or y did not simply
represent a number: it could be considered in itself, independently of what
kind or size of thing it represented. Thus the symbol itself transcends what
it represents and becomes a mathematical object in its own right, obeying
the laws of calculation. Mathematical arguments and calculations could
therefore be abbreviated and systematised, and abstraction became directly
accessible. Leibnitz wrote that “This method spares the work of the mind
and the imagination, in which we must economise above all. It enables us
to reason with small cost in effort, by using letters in place of things in
order to lighten the load on the imagination.” In turn, the spread of algebra
throughout Europe brought about great scientific progress, and led to
substantial refinement of operational symbolism in its widest sense.
Taking a very rapid overview of the history of mathematics, this science
arose in Ancient Greece when her philosophers and mathematicians
brought a decisive advance into human thought: that combination of
abstraction, generalisation, synthesis and logical reasoning which had
previously lain hid in shadow. The Greeks, however, were enamoured of
what is beautiful and simple and, consequently, of what is divine. They
thereby cut themselves off from the world of reality and therefore from
applied mathematics. The epic Graeco-Latin era was succeeded in the West
by the long dark night of the Middle Ages, feebly lit up from time to time
by a few individuals of no great stature.
It was the Arabs who took over. They were well placed to assimilate the
whole of the Ancient Greek legacy, together with Indian science, saving the
essentials from oblivion, and they developed and propagated it according
to “scientific reasoning”.
In due course, the first great European universities were founded and the
pursuit of knowledge was resumed: the Western world once again awoke
and initiated the study of nature based on independence of thought. This
great dawning derives above all from the work of Fibonacci, Liber abaci
(1202) which, over the next three centuries, was to prove a rich source of
inspiration for the development of arithmetic and algebra in the West. But
the West also established numerous contacts with Arabic and Islamic
culture from the eleventh century onwards, whereby European mathemati-
cians came to know not only the works of Archimedes, Euclid, Plato,
Ptolemy, Aristotle and Diophantus, but also became acquainted with the
work of Arab, Persian and Indian thinkers and learned the methods of
calculation which had been invented in India.
The true renaissance - or rather the true awakening - of mathematics
in Europe would not take place until the seventeenth century, first of all in
the work of Rene Descartes who made full use of the new knowledge in his
invention of algebraic and analytic geometry. Pascal later opened new
questions in considering the problems of mathematical infinity, followed
in this by Newton who also, with Leibnitz, ushered in the era of the infin-
itesimal calculus.
During the eighteenth century the spirits of Greek and of Cartesian
mathematics were sustained together, leading on to a synthesis which,
continuing into the nineteenth century, gave rise to the invention of deter-
minants and matrices and the development of vector calculus.
In the nineteenth century, Gauss, Cauchy and Picard completed the
Graco-Cartesian edifice. Lobachevsky questioned the foundations of
Euclidean geometry and invented non-Euclidean geometry. On the last
night of his all too short and dramatic life, the young Evariste Galois, a
political revolutionary, left for the world his creation of the first abstract
algebraic structures. George Boole laid the foundations of mathematical
logic and Georg Cantor worked out the fundamentals of the theory of sets
and of modern topology. The century closed with Hilbert’s publication of
his axiomatisation of geometry, which became the model for the modern
axiomatic study of mathematics.
Since then, the explosion of modern mathematics has been charac-
terised by an ever more pronounced algebraic approach: unlike the ancient
mathematics which was based on very specific concepts of line and of
magnitude, its basis is the universal and very abstract concept of a set. This
recent unification in terms of logic and the theory of sets has made mathe-
matics, for the first time in its history, an undivided subject.
And, finally, this unity in abstraction of modern mathematics laid the
foundation of the computer science which is being developed today.
Therefore we must pay tribute to all the mathematicians, be they
English, French, American, Italian, Russian, German, Japanese or any
other, who have brought mathematics to its present extraordinary flower-
ing, for which the words of Arthur Cayley in 1883 are still a beautifully apt
description: “It is difficult to give an idea of the vast extent of modern math-
ematics. The word ‘extent’ is not the right one: I mean extent crowded with
beautiful detail - not an extent of mere uniformity such as an objectless
plain, but a tract of beautiful country to be rambled through and studied in
every detail of hillside and valley, stream, rock, wood, and flower. But, as for
everything else, so for mathematical beauty - beauty can be perceived but
599
BEYOND PERFECTION
not explained.” We may not, however, omit from this roll of honour to the
glory of Western mathematics the Indian civilisation which invented the
modern number-system in which the later great discoveries are rooted. Nor
should we omit the Arabic and Islamic civilisation which carried the flame
whilst the West slept.
There is a last great question. Could modern mathematics, in all its
rigour, and in all its principles, with its theoretical extensions and practical
applications which have revolutionised the way we live - could mathemat-
ics have possibly occurred in the absence of a positional numerical notation
so perfect as the one we have? It seems incredibly unlikely. Modern science
and technology may have their roots in antiquity, but they could only flour-
ish as they have in the context of the modern era and in the framework of a
number-system as revolutionary and efficient as our positional decimal
system, which originated in India. To move mountains, the mind requires
the simplest of tools.
And so our history of numbers is now completed. However, it is itself
but a chapter in another history, the history of the representations of the
world, and that history, beyond doubt, will never be completed.
601
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THE UNIVERSAL HISTORY OF NUMBERS
616
INDEX
Aba, tomb of 52
abacus 125-133, 207-211, 333-334, 366, 556-562; abacists
and algorists 578, 590-591; Akkadian 139-141; Assyro-
Babylonian 140; Chinese 125, 211, 283-294, 556; French
290; Gerbert’s 579, 581-586, 588; Greek 200-203, 208;
Inca 308; Indian 434, 559; Latin 542-543; Liber Abaci
361-362, 588-589; Mesopotamian tablet 155; multi-
plication 208-209, 557-559; Persian 556, 562-563; Roman
187, 202-207, 209-211, 577, 579-580, 582; Russian 290;
suan pan 288-294; Sumerian sexagesimal 126-133, 140;
Table of Salamis 201-203; wax or sand 207-209, 563
Ibn Abbad 527
al-Abbas 521
ibn ’Abbas, ’Ali 524
’Abbas, Caliph Abu’l 520
Abbasid caliphs 512, 514, 520
ibn ’Abdallah, Abu ’Amran Musa ibn Maymun
see Maimonides
ibn ’Abdallah, Ahmed ibn ’Ali 252
ibn ’Abdallah, ’Ali 252
ibn ’Abdallah, Sahl 551
Abel, Niels Henrik 596
Abenragel 356, 363, 524
Abjad numerals (ABC) 244, 248, 250, 261-262, 548-555
aboriginals, Australian 6, 18
Abraham 73, 253-254, 257, 364
Abrasax 259
absolute quantity 21
abstraction 16; counting 10, 19-20, 76; numbers 5, 23 see also
calculation; model collections; place-value system; zero
Abulcassis 522
Abusir 390
Abyssinia 96, 246, 387
Academy of Sciences (France) 42
accounting 101-120, 187, 541-543; balance sheet 109-111;
Cretan 178; Elamite 102-107; Japan 288-289; Jews 236;
Mayan 304-305; Mesopotamian 132; pocket calculator
209-211; Roman 187, 209-211; Sumerian 122-124, 131
see also bullae; calculi; quipus; tablet; tally sticks
Acor 406
acrophonic number systems 186, 214, 387
Adab 81
Adad 161
Adam (first man) 254
al-Adami, Ibn 523, 529-530
addition: abacus 127, 204-206, 285; calculi, Sumerian use
122; Egypt, Ancient 174; suan pan 291-292
additive principle 231, 325-329, 333-336, 347-351; Americas
306, 308; India 397, 434; Roman numerals 187; Sheban 186
Adelard of Bath 207, 362, 587
ibn ’Adi, Yahya 513, 523
Afghanistan 377, 386, 522-523, 528; and Arabs 512, 520, 523;
counting 94, 290; numerals 228, 368, 534; writing 376, 539
Aflah, Jabir ibn 526
Africa: Arabian provinces 521; base five in 36; Central 5, 22;
counting 10, 96, 125; East 72; Maghribi script 539; number
mysticism 93-94, 554; South 5; West 19, 24-25, 70, 74
Africa, North: and Arab-Islamic world 520, 528, 587;
calculation 559; counting 47, 214; Goths 226; Morra 51;
number mysticism 248, 250, 262, 553; numerals 242, 244,
356, 534-537; writing 248, 539
Agade see Akkadian Empire
ages of the world 426
Aggoula, B. 335
Ah Puch, god of death 312
Aharoni, Y. 236
Ahmad, Abu Hanifa 522
Ahmad, Ali ibn 523
ibn Ahmad, Khalil 58, 520
ibn Ahmad, Maslama 524
Ahmed 363, 511, 519
Ainus 36, 305
Akhiram 213
al-Akhtal 520
Akkad 135
Akkadian Empire 81, 134-146; bullae 100-101; counting
139-141; Mari 74, 81, 134, 142-146, 336; number
mysticism 93, 159-160; numbers 90, 134, 136-139,
142-146; writing 130-133, 135-136, 159-160 see also
Assyrian; Babylonian civilisation
Aksharapalli numerals 388
Aksum 246, 387
al Shamishi system 248
al Tadmuri system 248
Albania 33-35, 528
Albategnus 522
Albright, W. F. 142
alchemy 518-519, 553-554
Alexander the Great 135, 256, 386, 407
Alexandria 515, 522
Algazel 525
algebra 588, 597-598; Arabs and 521, 524, 527-528, 531;
Brahmagupta 419, 439, 530
algebraic numbers 596-597
Algeria 248, 521, 528; counting 49, 66, 555
algorists 587-590
algorithms 559, 587; al-Khuwarizmi 531, 587
Alhazen 524
’Ali, Abu’l Hassan 58
’Ali, caliph 519-520, 555
ibn ’Ali, Hamid 522
ibn ’Ali, Sanad 364, 521
Ali (language) 22
alien intervention theory 593-594
Allah 47, 59, 214, 514-515, 553; attributes of 11, 50-51, 71,
261-262, 542, 553
Allah, Abu Sa’id ’Ubayd 525
Allah, Sa’id ibn Hibat 525
Allard, A. L. 365, 434, 533, 562-563, 589
Alleton, V. G. 266-268, 272
almanacs 195-196
alphabet 212-214; Greek 190; Hebrew 215-218; palaeo-
Hebrew 212, 233; Samaritans 212
alphabetic numerals 156-157, 212-262, 329, 483-484; Arabic
158, 241-246, 516, 548-555; Aryabhata’s 432; Ethiopian
246-247; Greek 218-223, 227, 232-233, 238-239, 329,
333, 360; Hebrew 158, 227, 233-236, 238-239, 346, 362;
Indian 389; Syriac 240-242, 329; Varnasankhya’s 388
see also mysticism
Alpharabius 523
Alphonsus VI of Castile 525
Americans, native: counting 10, 64, 70, 72, 125, 196; number
mysticism 93-94, 554; use base five 36 see also Maya
amicable numbers 522
Amiet, P. 80, 101-102
’Ammar 525
Ammonites 212
Amon 164
Amorites 39, 135
amp 43
al-’Amuli, Beha ad din 363, 528
Anaritius 522
Anatolia 75-76, 97-98 see also Hittite; Ottoman Empire;
Turkey
Anbouba 511
al-Andalusi, Sa’id 515-516
Andalusia 525-526; abacus 556, 560, 563; and Arabs 512;
numerals 534-539
Andhra numerals 397-398
Andromeda nebula 524
Anglo-French, word for money 72
Anglo-Saxon, number names 33-35
animals, counting abilities 3-4
anka (numerals) 368, 415-416
Annam 272-273 see also Vietnam
al-Ansari 528
al-Antaki 541-542
Antichrist 260
Anu 93-94, 161
Anushirwan, King Khosroes 512
Anuyogadvara Sutra 425
Anwari 58, 525
Api language 36
apices 580-586; of Boethius 579
Apollonius of Perga 221-222, 361, 513, 523
apostles, New Testament 258
Apuleus 55-56
617
INDEX
Aquinas, St Thomas 515
ibn ’Arabi 527
Arabic numerals 25, 56, 392-396, 534-539, 592; in Europe
577-591; origins 356-359, 385
Arab-Islamic civilisation 52, 82, 157-158, 185-187, 389,
511-576
language 135-137, 212-213, 513, 517-518
number systems 58, 157, 228, 349, 438, 543-548, 595;
alphabetic numerals 241-246, 516, 552-555; counting
39, 47, 49, 66, 70, 96, 428-429; Indian numerals 368,
511-541
science 512, 514-515, 520 see also Baghdad; Muslims
Arad 213, 236
Aramaean Indian writing see Kharoshthi writing
Aramaeans 134, 236, 376; number system 39, 137, 227-231,
331, 335, 351
Aramaic script 212-213, 236, 240, 376-377, 387, 390;
cryptography 248; Jews adopt 233, 239
Aranda people 5, 72
Arawak, base five in 36
archaic numerals 85, 87-90, 92-93; accounting tablets 107,
121; bullae 104; calculi 125; Sumerian 77-79, 83-84,
92-93, 99-100, 107, 117
Archimedes 207, 222, 361, 518, 522-524, 594, 598; Arabic
translation 513, 588; high numbers 333
Ardha-Magadhi 383
are (unit of measurement) 42
Argos 219
Aristarch of Samos 221
Aristophanes 47
Aristotle 20, 512-515, 517, 588, 598
arithmetic 5, 10, 96, 206, 528; early 76, 96-97; during
Renaissance 577-578; systems 185, 220-222, 248, 442
Arithmetic 207
Arithmetic, Lady 205, 591
Arithmetical Introduction 43
Arjabhad 427
Arjuna 423-424
Arkoun, M. 514, 519
Armenia 139, 290, 519, 528; alphabet 212; numerals 33-35,
224-225, 329
Arnaldez, R. 512, 514-519
Arnold, Edwin 421, 538
Artaxerxes, King 55
Aryabhata 388, 419-420, 427, 432, 447-451, 530
Aryans 385-386
as (Roman unit) 92, 210-211
Asankhyeya 451
Asarhaddon, King 146
al-Ash’ari, Abul Hasan ’Ali ibn Ismail 522
Ashtadhyayi 389
Asia 81, 402-407, 512; counting 36, 48-49; number systems
94, 412-413 see also China; India
Asia Minor 76, 81, 180, 219, 524
Asianics 134
al-Asma’i 521
Asoka, Emperor 375-377, 379, 386-387, 420, 433, 435; edicts
390-391, 397
Assemblee constituante 42
Assurbanipal, library of 160
Assyrian Empire 134-135, 180; counting 39, 99; language 135;
number systems 92, 139, 141, 231
Assyro-Babylonian civilisation 135, 140; number system 9,
137, 141-142, 331-332, 351
astrology 159, 549, 553, 556; “Greek” 420; Indian 417, 463;
and Koran 514
Astronomie Indienne 443
astronomy 92, 522, 524, 529, 551; Arabic 243, 530, 548-549;
Babylonian 153, 156-158, 407; Chinese 277-278; Greek
156- 157, 408, 549; Indian 409-411, 416-417, 431-432,
443, 463-464, 513-514; Mayan 297-298, 308-313,
315-316, 321-322; sexagesimal system 91-92, 95, 140,
157- 158, 548-549; tables 146, 157-159, 198, 521; Ikhanian
Tables 527; trigonometrical 420 see also lunar cycle
al-Asturlabi, Ali ibn ’Isa 521
al-Asturlabi, Badi al-Zaman 518, 526
al-Atahiya, Abu 521
Athens 182-183, 219, 233
’Attar, Farid ad din 527
Attica 183, 219
Atton, Bishop of Vich 578
Auboyer, J. 389
Augustine 257
aureus (Roman money) 210
Aurillac, Gerbert of 362, 518, 578-579, 581-586
Australia 5-6, 18, 72, 93-94
Austria 66
Autolycus 522
Avempace 526
Avenzoar 526
Averroes 514-515, 519, 526
Avestan 32-35
Avicebron 514
Avicenna 363, 513-515, 517, 519, 525, 527, 542; “Avicenna”
Arabic alphabet 539
Avigad 234
Awan 81
Axayacatl 301
Ayala, Guaman Poma de 69, 308
Aymard, A. 389
Aymonier, E. 403
Azerbaidjan 519, 528
Aztecs 301-303, 315-316; monetary system 72-73, 302-303,
306; number system 36, 44, 47, 305-308, 326, 348-349;
writing 302-303, 305-307
Aztlan 301
al-Ba’albakki, Qusta ibn Luqa 518, 522
Babel, tower of 159
Babur 528
Babylonian civilisation 81, 134-161, 180-181; arithmetic 40,
99, 139, 154-156; cryptograms 158-160; number system
92-93, 139-154, 231, 337-342, 345, 353-354, 407-408;
writing 134, 138, 153, 158
Bachelard, Gaston 443
ibn Badja (Avempace) 526
Baghdad 527-528; House of Wisdom 512-514, 516, 520-523,
525-528, 530-531
al-Baghdadi, *’Abd al-Qadir 528
al-Baghdadi, Muwaffaq al din Abu Muhammad 363, 527, 542
Bahrain 49
bakers, counting methods 65-66, 70
ibn Bakhtyashu’, Jibril 513, 521-522
Baki 528
Bakr, Abu, caliph 519, 522
Bakr, Muhammad ibn Abi 527
al-Bakri 525
balance sheet 109-111
Balbodh writing 380
Bali 405-407, 421; numerals 375, 383-385
Balkans 528
Balmes, R. 21
Baltic 33
Bamouns, decimal counting 39
Banda 36, 44
Banka Island 404
banzai 275
Baoule 39
al-Baqi, Muhammad ibn ‘’Abd 525
al-Baqilani, Abu Bakr Ahmad ibn ”Ali 524
Baqli, Ruzbehan 526
al-Barakat, Abu 526
barayta 254
Barguet, P. 176
Barmak 513
Barnabus 257
barter 72-75
Barton, G. 88
Baru Musa ibn Shakir brothers 518, 522
base numbers 23-46, 96; auxiliary 426-429; eleven 41; five
36, 44-46, 62, 192-193; ’m’ 355; six 142 see also binary;
decimal; duodecimal; sexagesimal system
Basil I 522
Basilides the Gnostic 259
Baskerville face numerals 588
Basques 38-39
Basra 519
al-Basri, Hasan 520
Bastulus 523
Batak numerals 383, 385
Bath see Adelard
ibn Batriq, Yahya 513, 521
THE UNIVERSAL HISTORY OF NUMBERS
618
al-Battani (Albategnus) 514, 522
Ibn Battutua 527
Bavaria 585
al-Bayasi, ’Abu Zakariyya Yahya 518, 526
Bayer 359
Bayley, E. C. 61, 386-387
al-Baytar, Ibn 527
Beast, number of 260-261
Beauclair, W. 577
Beaujouan, G. 356, 578
Bebi-Hassan 52
Becker, O. 91, 519
Bede, Venerable 49-50, 52-56, 200, 223, 578
Beg, Tughril 525
Behar 251
Bek, Ulugh 528
Belgium 31, 65
Belhari 401
Belize 299
Bengal 49, 390, 526
Bengali numerals 370, 381, 384, 421, 438
Beni Hassan 51-52
Benin, Yedo 305
Bequignon sisters 26
Berbers 39, 512
Bereshit Rabbati 253
Bergamo, Gnosticism 260
Bernelinus 580
Beschreibung von Arabien 48
Bessarabia 528
Bete 36
Bettini, Mario 356
Bhadravarman, King 407
Bhagavad Gita 422
Bhaskara 419, 452, 530; Aryabhatiya, Commentary on 414-415,
420, 439
Bhaskaracharya (Bhaskara II) 414, 418, 431, 452, 562;
multiplication method 573-574
Bhattiprolu writing 377, 383, 385
Bhoja 414
Bible
Old Testament 134, 253-254; Daniel 137; Deuteronomy
254; Esther, Book of 137; Exodus 73; Ezekial 239; Ezra
137; Genesis 134, 253-254; Leviticus 254; Nehemiah
137; Pentateuch 137; Prophets, Books of the 137; Psalms
213, 217; Samuel 73; Zechariah 137 see also Torah
New Testament 257; Gospels 243, 257; Matthew 257;
Revelation (Apocalypse) 256-257, 260
Bihar 526
bijection 10
Billard, R. 406-407, 414-418, 431-432, 443, 529
billion 427-428
binary principle 6, 9, 89, 139, 166
binary system 40-41, 59, 592
binomial formula 528
Biot, E. 282, 336
Birman numerals 438
Birot, M. 134, 138
birth-date 313
al-Biruni, Muhammad ibn Ahmed Abu’l Rayhan 513-515,
519, 524, 588; Indian numerals 418, 438; Kitabfi tahqiq i
ma III hind 363-365, 368, 409, 426-429, 530, 534; Tarikh ul
Hind 251
Bisaya writing 383, 385
ibn Bishr, Sahl 514, 521
Bistami 522
biunivocal correspondence 10
Black Stone, The 146
blackboard 566-567
black-letter Hebrew (modern) 212-213, 215, 233
Bloch, O. 365, 427
Bloch, R. 190
boards: checkerboard 283-288; columnless 560-563; dust
555-563; wax 207-209, 563; wooden 64, 535-536 see also
abacus; tally sticks
Bodhisattva see Buddha
body counting systems 5, 12-19, 23, 214; and base 44-46
see also finger counting
Boecius 580
Boethius 48, 578-579
Bokhara 513, 522
Bolivia 69-70, 543
Boncompagni, B. 207, 362, 365, 533
bones 62-63, 269 see also tally sticks
Bons, E. 518-519
Book of Animals 521
Book of Kings 525
Boole, George 598
Boorstin, D. J. 91
Borchardt 55
Borda 42-43
Borneo 375, 383
Botocoudos 5, 72
Bottero, J. 80-81, 160
Bouche-Leclerq, A. 360
Bourdin, P. 21
Boursault 206
boustrophedon writing 186, 219
Brahma 376, 418-419, 422, 427, 441
Brahmagupta 414, 419, 439, 453; Brdhmasphutasiddhanta 420,
439, 520, 530, 573-576; multiplication method 573-576
Brahmi numerals 378-379, 382, 384-395, 402, 420, 433-436,
453
Brahmi writing 375-378, 397-399
Brasseur de Bourbourg 300
Brazil see Botocoudos
de Brebeuf, Georges 206
Breton 33-35, 38
Brice, W. C. 109
bride, price of 72
Brieux 205
Britain, Great 92, 146, 170-171, 176, 214; Treasury 590
British Honduras 299
British New Guinea 13-14
Brockelmann, C. 589
Brooke 18-19
Brothers of Purity (Ikhwan al-safa) 524
Bruce Hannah, H. 26
Le Brun, Alain 101, 109
Bubnov, N. 358, 400
Buddha 71, 408-409, 418, 420-425, 428
Buddhism 11, 71, 407-408, 443, 513
Bugis 14, 383, 385
Biihler 386-387, 389, 438
Buhturi 522
al-Bujzani, Abu’l Wafa’ 523, 548
Bulgaria 528
bullae 97, 99-105, 122, 234
ibn Bunan, Salmawayh 513, 522
Bungus, Petrus 199, 260
al-Buni 553-554
Buonamici, G. 190, 197
Bureau des Longitudes (Paris) 43
Burgess 411
Biirgi, Jost 595
Burma: numerals 374-375, 384-385, 388; writing 382-383
Burnam, R. L. 363, 579-580
Burnell, A. C. 389, 438
Burnham 57, 200
Bushmen 5, 72
Ibn Butlan 525
Byzantine Empire 222, 240, 360, 518, 523, 537; arithmetic 334
Cabbala 217, 554
Cadmos 219
Caesar, Julius 7
Cagnat 199
Cai Jiu Feng 279
La Caille 43
Cairo 513, 523
Cajori, F. 356-357, 434
Cakchiquels, Annals of the 301
calamus reed 539, 553
calculation 132, 541, 563-566; Babylonian 154-156; Egyptian
39, 174-176, 334; Mayan 303-305, 308, 321-322; North
Africa 559; tables 127-130, 146, 203-206, 283-288,
555-563 see also abacus; body counting; calculi; notched
bones; string; tally sticks
calculator, pocket, first 209-211
calculi 96-105, 118-119, 125-126, 139, 168; Elamites 103,
140-141; Roman abacus 203-205; Sumerian 121-124, 131
calculus 598
619
INDEX
calendar 18, 50, 239, 525; ciphers 195-196; Hebrew 215, 217;
lunar 19, 297, 407; Mayan 36, 297, 308, 311-322; Roman 7;
Shaka 407
Callisthenes 256
Calmet, Dom 359
Cambodia 407, 419; inscriptions 404-405, 431; numerals 375,
403, 413, 438 see also Khmer
Cambridge Expeditions 12-14
Campeche 299, 303
Canaan 228, 239
candela 43
Canossa, Darius vase 200-201
Cantera 216
Canton 272
Cantor, Georg 598
Cantor, Moritz 91
caoshu writing 267-268
Capella, Martianus 207
cardinal numeration 20-22, 24-26, 193; Attic system
182-183; reckoning devices 15-19, 96; Yoruba 37
Carib 36
Carolinas Islands 70
Carolingian script 586
Carra de Vaux, B. 358, 364, 400
cartography 526
Casanova, P. 549
Catalonia 223
Cataneo 361
Catherwood, Frederick 300
Cato 194
cattle 72
Cauchy 598
Cayley, Arthur 598
Ce Yuan Hai Jing 282
Celebes Islands 375
Celtic numbers 38
censuses 68
centesimal-decimal system 144-145
Central America 300-302, 308; counting 10, 303; numbers
162, 313; trading methods 72-73 see also Maya
Ceylon 332 see also Vedda
cha lum numerals 374
Chalcidean alphabet 190
Chalfant 269
chalk 566-567
chalkos 182, 200-203
Chalmers, J. 14
Chameali numerals 381, 384
Champa 404-407, 418; inscriptions 420-421, 431; numerals
383, 385, 413, 421
Chanakya 522
Chandra, Hema 425-426
Changal, Stela of 413
Chapultepec 301
Charlemagne 521
Charles III, king of Spain 248
Charpin, Dominique 88
Chassinat 176
de Chavannes 267
Chelebi, Evliya 529
Chermiss, tally sticks 66
chess 323-324
Chevalier, J. 437, 443, 553, 555
chevrons 148-149
Chhedi 454
Chiapas province 299, 303
Chichen Itza 300
Chilam Balam, Books of 301
children 4-5, 10, 214
chimpu 70 see also string, knotted
China 51, 263-273, 276-296, 381; abacus 283-294, 556;
counting 39, 49-50, 61, 66, 70, 343, 428; high numbers
276-278, 333, 429; monetary system 73, 75-76; number
mysticism 554; number system 162, 168, 263-296, 332,
336-343, 353-354, 375; outside influences 408-409, 512,
516, 520, 526
Chinassi 529
Chinese Turkestan writing 382, 385, 420
Chodzko, A. 543, 545
Chogha Mish 101
Chorem 513
chou 125, 283-288
Christ 251
Christianity 513; Arabic 240, 513; Central America 300-301;
demonised Arabic numerals 588; isopsephy 259-261 see
also Crusades
chronograms 250-252, 553 see also codes and ciphers
Chuquet, Nicolas 427-428
Chuvash 66
Cicero 47, 51, 194, 203
circle 92
City of God, The 257
Clandri, Philippi 589
Claparede, E. 365
classification of sciences 517, 523, 525
clay objects: accounting 78, 80, 109-111; tokens 96, 99 see also
calculi; tablet
clock-making 518
Coatepec 302
Coatlan 302
Code Napoleon 66
codes and ciphers 158-161, 248-262, 553-554 see also
mysticism; numerology
Codex Aemilianensis 579-580
Codex Mendoza 36, 302-303, 306
Codex Morley 298
Codex Selden 298
Codex Telleriano Remensis 307
Codex Tro-Cortesianus 298, 301, 312
Codex Vigilanus 362, 580
Codices, Hebrew 217
Codrington, M. 6, 19
Coe, M. D. 299, 320
Coedes, G. 403, 406-407, 413
Cohen, M. 185, 242-244, 376, 385-386, 533
Cohen, R. 238
coins 75-76, 183, 190, 520
Colebrooke, H. T. 573
Colin, G. S. 244, 250, 252
columnless board 562-563
Comte, Auguste 528
Conant, L. L. 19, 45
concrete numeration 21, 23, 167-168
La Condamine 42
Congo, early money 73
conic sections 522-523
Conrady, A. 66
Constantinople 520, 528
Contenau, G. 159
contracts 66, 70
Coomaraswamy 419
Copan 297, 313, 320
Coptic 168, 224
Copts 55
Cordoba 513, 523, 525, 587
Cordovero, Moses 253
Corinth 219
correspondence 21-23; biunivocal 10; one-for-one 10-12,
16-17, 19, 96, 191, 194
Cortez 302
Cos 183
Cottrell, L. 533
de Coulanges, F. 366
Coulomb 43
counting 10, 19-22, 76; cuneiform ideogram meaning 131;
methods 62-63, 68-71, 99; rhymes 214; systems see under
body counting; correspondence; mapping; see also under
specific race/ country
cowrie shells as currency 72
Crafte of Nombrynge, The 361
Creation 217, 251, 364; Mayan Long Count 316, 320
Cremonensis, Geraldus, Liber Maumeti filii Moysi Alchoarismi
de algebra et almuchabala 531
Crete 178-180, 521, 523; Linear A and Linear B 229, 326;
number system 9, 178-180, 326, 348
Crimea 226, 528
Cro-Magnon man 62-63
Crusades 525-526, 586-588
cryptography 158-161, 248-250, 259-261, 554 see also
mysticism; numerology
Ctesibius 518, 594
cubes 363, 524; roots 285, 293, 596
THE UNIVERSAL HISTORY OF NUMBERS
620
cubit 141
cufik see kufic
cuneiform notation 87-88, 135-138, 142, 180-181; numerals
84, 89-90, 100, 125, 145; codes and ciphers 158-161;
decimal 137, 139-140; script 107, 121, 148-149; tablets
130-134
Cunningham 386-387
Curr 6
currency 41, 72-76, 182-184, 308
Curtze 361
curviform notation 125, 130
Cushing, F. H. 15, 196
Cuvillier, A. 366
Cyclades 219
cylinders 522
cypher, etymology of 589-590
Cyprian 257
Cyprus 523
Cyril of Alexandria, Saint 56
Cyrillic alphabet 212
Cyrus of Persia 135
Czech, number names 33, 35
da zhuan writing style 280
Dadda III 419
al-Daffa, A. A. 519
DAFI (French Archeological Delegation in Iran) 101-103,
105-107, 109, 140, 248
DAGR 222
ad Dahabi, Ahmad 252
Dahomey 19
Daishi, Kobo 296
Dalmatia 194-195
Damais 405-406
Damamini, Ad 528
Damascus 513, 519-520, 525-526
Damerov 92-93
Dammartin, Moreau 588
Dan 36, 253
Dantzig, T. 6, 22-23, 36, 46, 334
Daremberg, C. 221, 428
al-Darir, Abu Sa’id 521
Darius, King 70, 201
Darwin, Charles 519
Das, S. C. 26
Dast, Zarrin 525
Dasypodius, Conrad 358-359, 590
Datta, B. 356, 364, 386-388, 399-400, 414, 419, 422, 434,
438, 562, 568, 573
d’Auxerre, Remy 207
da’wa 261-262
Dayak 18-19
De pascha computus 257
De ratione temporum 49-50, 52, 56
Dead Sea Scrolls 213, 234
decimal system 24-36, 39-44, 354-355, 542; Ben Ezra 346;
Chinese 263, 278-283; counting 68-69, 96, 139-142,
192-193, 208-209; Cretan 178, 180; Egyptian 39, 162;
fractions 282, 528, 595; hieroglyphs 167; Mesopotamia
138-146; metric system 42-43; proto-Elamite 120; Semitic
136; Sumerian 93-95
Decourdemanche, M. J. A. 248-249
Dedron, P. 48, 221-222, 428
Deimel, A. 82, 84, 89, 121, 131-132
Delambre 43
Delhi 526-528
Demetrius II 234
demotic writing 171
denarius 210
Dendara, temple of 176
Dene-Dindjie Indians 46
denier, French unit 92
Denmark 33, 38
Dermenghem, E. 519
Descartes, Rene 42, 199, 597-598
Destombes, M. 360
Devambez, P. 182-183
Devanagari numerals see Nagari numerals
Dewani numerals 543-545, 550
Dharmaraksha 425
Dharmashastra 419
Dhombres, Jean 43
Dibon Gad 212
Dickens, Charles 65
dictionary of Indian numerical symbols 440-510
Diderot 519
Diener, M. S. 443
digital 59
Diibat 138
Dingzlnu suanfa 343
Diocletian 260
Diophantus of Alexandria 221, 513, 522-524, 598
Diringer 270
disability, spatio-temporal 5
divination 159, 269, 549-556 see also mysticism
Divine Tetragram 218, 254 see also Yahweh
division: a la fran^aise 566; abacus 127-130, 206, 285, 287;
calculi 121-124; Egypt, Ancient 174-176
Dobrizhoffer, M. 6
Dodge 364
Dogon 72
Dogri numerals 370-371, 384, 421
Dols, P. J. 49
Donner 229
Dornseiff, F. 256
Doutte, E. 553-555
dozen 41, 92, 95
drachma 182-183, 201-203
Dravidian numerals 373-374, 383
Dresden Codex 301, 308, 310
duality 32
Duclaux, J. 367
Duke of York’s Island 6
Dulaf, Abu 523
Dumesnil, Georges 356
Dumoutier 272
dung (counting device) 12
duodecimal system 41-43, 92-95
duplications, abacus 206
Dupuy, Louis 42
Durand, J-M. 145, 336
dust-board calculation 555-563
Dutch, number names 33-35
ibn Duwad 523
Dvivedi, S. 414-415, 439, 573
dyadic principle see binary principle
e, number 597
Ea 161
Easter, determining 50
Ebla tablets 135, 145
eclipse 529
Ecuador 69, 543
Ed Dewachi, S. 519
Edesse 512
Edfu, temple of 176
Edomites 212
Egine 183
Egypt, Ancient 73-74, 162, 166, 236, 259, 389; and Arabs 519,
522-523, 525-529, 545; sign language 55
calculation 39, 174-176, 334; abacus 541-542; fingers 51-52,
61, 94
number system 9, 91, 162-177, 325-329, 342-350, 390;
alphabetic numerals 232, 238, 243; Arabic numerals 356;
Indian numerals 368, 534; number mysticism 554
writing 162, 212, 392; hieratic 170-171, 236-239; secret 248
Egyptian Mathematical Leather Roll (EMLR) 176
eight 34, 396; Chinese 269; Egypt 176; Hebrew 215; Indian
410; Japanese 273
eight hundred, Hebrew 216-217
eight thousand: Aztec 305; Mayan 308
eighteen, Egypt 177
eighty: cryptographic 248; Hebrew 215, 235
eighty-eight, Japanese 295
Eisenstein 96
El Obeid 134
El Salvador 299, 303
Elam 102, 134-135; accounting 99, 101-107, 111-120; calculi
140-141; cryptograms 159-160; numerals 9, 39, 96-120,
146; proto-Elamite script 107-120
Elema’s body counting system 13-14
Elephantine 213, 227, 229-233, 235, 390
621
INDEX
eleven, base 41
Eliezer of Damascus 253-254
Elogium of Duilius 189
Elzevir script face 588
EMLR (Egyptian Mathematical Leather Roll) 176
end of the world 426
engineering 518
England: clog almanacs 195-196; number names 31, 33-35,
428; score (twenty) 37-38; tally sticks 65
English, Old 72
English script 586
Englund 92-93
Enlil 161
Ephesus 199
Ephron the Hittite 73
Epicurus 522
Epidaurus 185
epigraphy 399-400, 402-407, 419
Epistle of Barnabas 257
Equador 308
equations 283, 287, 511, 525, 530
equivalence between sets 3-4
Erhard, F. K. 443
Erichsen, R. W. 342
Erpenius 359
Erse, Old 33-35
Eskimos 36
Essene sect 234-235
Essig 41-42
estranghelo 240
Ethiopia, number system 137, 238, 246-247, 353
Ethiopian numerals 387
Etruscans: abacus 125, 203; alphabet 212-213; number
system 9, 39, 189-190, 327, 349; Roman numerals 196-197
Eubeus 219
Euboea 183
Euclid 512-513, 522, 527, 588, 598; Elements 521, 523, 525
Euclidian geometry 598
Europe 42, 519-520, 571; Arabic numerals 577-582, 586-591
European numerals 392-396
Evans, Sir Arthur 178-179
Eve 254
evolution theory 519
Ewald 363
Exaltation oflshtar 159
exponential powers 528, 594
Ezra, Rabbi Abraham Ben Men ibn 346, 362, 514, 526, 589
al-Fadl 520
Fahangi Dijhangiri 52
Fairman, H. W. 176
Falkenstein 81-82
al-Faqih, ibn 523
Far East 70, 272-276, 278-283, 294-296 see also individual
countries
V
Fara 87, 101 see also Suruppak
al-Farabi 514, 517, 523
Faraut, F. G. 407
al-Farazdaq 58, 520
al-Farghani 521
al-Farid, Ibn 527
al-Farrukhan, ‘’Umar ibn 521
Farsi language 518
Fath, Abu’l 523
al-Fath, Sinan ibn 524
Fatima (Mohammed’s daughter) 71, 542, 555
Fayzoullaiev, O. 519, 531
al-Fazzari, Abu Ishaq Ibrahim 513-514, 520, 530
al-Fazzari, Muhammad Ben Ibrahim 520, 529-530
feet and inches 92
Fekete, L. 543, 547
Feldman, A. 518-519
Fenelon 206
Ferdinand II 528
Fevrier, J. G. 64, 66, 70, 79, 185, 213, 219, 376, 382, 387
Fez 252, 513, 520, 525, 527, 539
Fibonacci 365, 523, 588, 598; Liber Abaci 361-362, 588-589
fifteen 161, 177, 218
fifty 184, 186, 215; cryptogram 93, 161, 248; Roman numerals
188, 192
fifty thousand 184, 197-198
fifty-three 305
fifty-two 315-316
Fihrist al alum, Al Kitab al 364, 531, 539
Fijians 19
Filliozat, J. 335, 386-387, 431, 438, 443
finger counting 22, 28, 47-61, 168, 578; and base 44, 93-95
see also Bede; body counting
Finkelstein 134
Finot, L. 406
Firduzi, Abu’l Qasim 58, 525
Fischer-Schreiber, I. 443
five 34, 176, 194, 394, 442, 554-555; Attic 182; base 36,
44-46, 62, 192-193; Chinese 269; Greek 184; Hebrew 215;
Indian 410; Mayan 308; Minaean 186; Roman 188, 192;
rule of 9; Sheban 186
five hundred 184, 188, 216-217
five thousand 184, 197-198
Fleet 438
floating-point notation 594
Fold, P. 518-519
Folge 364
Folkerts, M. 580-581
Forbes, W. 547
Formaleoni 91
Formosa 73
fortune-telling see mysticism
forty 93, 161, 215, 248
forty-nine 442
forty-two 276
Fossey 272
Foulquie, P. 365, 541
four 33, 176, 215, 394, 410; base 94; limit of 7-9, 19, 22, 391;
Chinese 269, 271; Japanese 273; mysticism 94, 276
four hundred 215, 305, 308
four thousand 276
Fournier 588
fourteen 161, 177
fractions 548-549, 594, 596; Babylonian 151, 153, 408;
decimal 282, 528, 595; Egyptian 168-170; Indian 424-425;
Maya 298
France 42 — 43, 51, 72, 577, 586; counting 32, 38, 65-66, 290;
French Revolution 42, 206, 590, 595; number system 31,
33-35, 92, 427-428, 585; metric 42-43 see also DAFI
Franz J. 182
Frederic, Louis 263, 273, 296, 367, 374, 376, 389, 408, 417,
425, 440, 443, 513, 519, 543
Freigius 198-199
French National Archives 43
Friedrichs, K. 443
Frieldlein 580
Frohner 55
Fuegians 5
Fulah 36
Fuzuli 528
von Gabain, A. M. 27
ibn Gabirol, Salomon (Avicebron) 514
Galba, Emperor 200
Galen 256, 512-513
Gallenkamp, C. 297, 311-312, 314
Galois, Evariste 596, 598
games 294-296
Gamkrelidze, T. V. 385
gan ma zi writing 268
Gandhara 228
Gandz, S. 542
Ganesha 568
Gani, Jinabhadra 399, 419
Ganitasarasamgraha 399, 421
Garamond, Claude 588
du Gard, Martin 541
Garett Winter, J. 157
Gauss, Karl-Friedrich 589, 598
Gautama Siddhanta see Buddha
Gautier, M. J. E. 138
Gebir 521
gematria 252-256, 554
Le Gendre, F. 566
Gendrop 297-298
Genjun, Nakane 289
de Genouillac 86, 88
THE UNIVERSAL HISTORY OE NUMBERS
622
geography 541, 555
Geometria Euclidis 579
geometry 92, 541, 548, 588, 598; base 91-92, 95; base 12 41;
Non-Euclidian 527
Georgia 212, 225, 528
Geraty 235
Gerbert of Aurillac 362, 518, 578-579, 581-586
Germany: counting 65, 70-71, 205-206; language 28, 33-35,
72, 586; number svstem 31, 33-35
Gernet, J. 269-270
Gerschel, L. 6, 64, 66-67, 194
Gerson, Levi Ben 158
gestures, number 14-19, 58-59 see also body counting systems
Gettysburg Address 36
al-Ghafur, "Abd ar Rashid Ben l ’Abd 528
al-Ghazali, Abu Hamid 525
Ghaznavid 58, 523-524
Ghazni 513
Gheerbrant, J. 437, 443, 553, 555
ghubar numerals 385, 534-539, 550, 556, 559, 579, 585
Gibil 161
Gibraltar 520, 527
Gideon 257
Giles 268, 278
Gilgamesh 81
al-Gili 549
al-Gili, Abu’l Hasan Kushiyar ibn Labban 363, 513, 524, 534,
549, 560-562
Gill, Wyatt 12, 14
Gille, B. 518
Gille, L. 519
Gillespie, C. 519
Gillings, Richard J. 175-176
Ginsburg, Y. 199, 207, 284, 361-362, 589
Girard 312
Glareanus 359
glyphs see Maya
glyptics 81, 84
Gmiir 64-66
gnomon 298
Gnosticism 258-259
Goar, Father 359
gobar numerals see ghubar numerals
Godart 179
Godri numerals 381
gods: God (Judaeo-Christian) 258-259, 552; Mayan 300,
311-314; names and numbers 160-161, 258-259; and
spirits 270 see also Allah
Godziher, I. 51
Goldstein, B. R. 158
Golius 359
Gondisalvo, Domingo 362
Goths 33-35, 212, 226; Gothic script face 588
Gourmanches 39
Govindasvamin 414, 418
Goyon, J. C. 176
Granada 513
Grantha writing 383, 385
Greece, Ancient 182-191, 256; and Arabs 512, 515, 518, 528;
currency 75-76, 183
science 515, 517-518; astronomy 82, 156-157, 408, 549;
Greek Myth 360, 366, 401; isopsephy 252, 256-259, 360
Greeks, Ancient: abacus 200-202, 208; counting 39, 96, 125,
220, 427-428
number system 9, 33-35, 157, 327, 345, 348-350;
acrophonic 182-187, 201-203, 214; alphabetic numerals
190-191, 218-223, 232-233, 238-239, 329; Arabic
numerals 356, 358-361; fractions 595; high numbers
333, 429
writing 32, 162, 179, 376; alphabet 212-213, 219;
papyri 157
Green 92
Greenland 36, 305
Gregory V, Pope 578
Griaule, M. 72
Grmek, M. D. 516
Grohmann, A. 243
gross 41, 92
guan zi writing 267
Guarani 36
Guarducci 182
Guatemala 313, 318-319 see also Maya civilisation
Gueraud, O. 220-221
Guide to the Writer’s Art 543-544
Guitel, Genevieve 214, 343, 347, 408, 437
Guitel, R. L. 432
Guitel, G. 182, 267, 276, 356, 400, 403, 428
Gujarati numerals 369-370, 381, 384, 421, 438
Gundermann, G. 182
Gupta dynasty 419
Gupta numerals 378, 381-382, 394, 397-398, 421, 460
Gupta writing 377, 384, 420
Gurkhali numerals see Nepali numerals
Gurumukhi numerals 369, 381, 384, 421
Guyard 542
Gwalior 380, 394, 396, 400-401, 418, 421
Haab, Mayan solar calendar 312-313, 315
Habuba Kabira 101, 103
Haddon, A. C. 6, 14
Hadiths 47
Hafiz of Chiraz 528
Haggai 137
Haghia Triada 178, 180
Haguenauer, C. 273-275
Hajjaj, Abu ’Umar ibn 525, 529
al-Hajjami 525
al-Hakam II, caliph 523
Halevi, Yehuda 514
Halhed, N. 50
al-Hallaj, Abu Mansur ibn Husayn 523
Hambis, L. 27, 72
al-Hamdani 523
Hamdullah 542
Hamid, al-Husayn Ben Muhammad Ben 523
Hamit, Avdiilhak 529
Hammurabi 81, 135, 142, 145; Code of 86
al Hanbali, Mawsili 55, 58
hand, counting with 47-61, 68 see also body counting; finger
counting
hangu alphabet (Korea) 275
Hanoi 405
Harappa 375, 385
al-Harb, Urjuza fi hasab al ‘'uqud 542
Haridatta 388, 414, 418, 432
Harmand, J. 64, 66
al Harran, Sinan ibn al Fath min ahl 364
Harris Papyrus, The 170, 390
Harsdorffer, Georg Philip 356
Hartweg, R. 593
haruspicy see mysticism
Hasan, Ali ibn Abi’l Rijal abu’l (Abenragel) 363
al-Hasib, Hasbah 521
al-Hassar 563
Hassenffantz 43
Hattusa 180-181
Ibn Hauqal 522, 538
Haiiy 43
Havasupai 125
Havell, E. B. 516
Hawaii 70, 125
Hawtrey, E. C. 15
Hayes, J. R. 516, 519
al-Haytham, Abu Ali al Hasan ibn al Hasan ibn 363, 514, 518,
524
ibn Hayyan, Jabir 521
Ibn Hazem 525
Hebrew number system 39, 136-137, 145, 214-218, 345;
accounting 236-238; alphabetic numerals 158, 215-218,
227, 233-236, 238-239, 241, 329, 346; and Arabic
numerals 359; Ben Ezra 346, 362; mysticism 239, 250,
252-256, 554
Hebrews: calendar 215, 217; language 72, 137, 212-213,
215-218, 236 see also Israel; Jews
Hejaz 528
Helen of Troy 51
Heliastes, tablets of 214
Henan 269-270
heqat (Egyptian unit) 169-170
Heraclius 519
Heraklion 178
herdsmen see shepherds
623
INDEX
Hermite 596
Hero of Alexandria 513, 518, 522, 594
Herodotus 70, 219
Herriot, E. 541
Hierakonpolis 164-165
hierarchy relation 20, 24
hieratic script 170-171, 236-239
hieroglyphs: Cretan 178-180; Egyptian 162-177; Hittite
180-181, 326; Mayan 298-301, 311-314, 316-322
high numbers 298, 333, 428-429, 594; China and Japan
276-278; India 421-429, 434, 440, 460-463; Roman
197-200
Higounet, C. 77, 86, 218
Hilbert 598
Hill 580, 587
Himalayas 390
al-Himsi 522
Hindi language 380
Hindi numerals 368, 511, 532, 536, 538-539, 560
Hinduism 376, 407, 419, 443; calendar 50
Hippocrates 512
Hippolytus 258
hiragana 273
Hisabal Jumal 252, 261-262
Hittite number system 33-35, 39, 180-181, 326, 348
Hiyya, Abraham bar 514
Hoernan, Ather 588
Hoffmann, J. E. 91, 519
Hofner, M. 185
Homer 72
Honduras 299, 303, 313
Hopital des Quinze-Vingts 38
Hoppe, E. 91-92
Horace 207
horoscope 549
Horus 169, 177
Houailou 36
Hrozny 181
Huang ji 279
Hiibner, E. 187
Huet, P. D. 359
Hugues, T. P. 555
Huitilopchtli 301
Hunan 272
hundred 25, 179, 194; Chinese 263, 265, 269; Greek 182, 184;
Hebrew 215; hieroglyphic 165, 168, 178, 181, 325; Japanese
274; Mesopotamian 137-139, 142-144, 186, 229-231;
Roman numerals 188, 192
hundred and eight 295
hundred and seven 177
hundred thousand 140, 165, 168, 197-198, 325
Hungary 528
Hunger, H. 154, 159
Hunt, G. 6
al-Husayn, Abu Ja’far Muhammad ibn 520
Husayn, Allah ud din 524, 526
Huygens, Christian 42
hybrid systems 330, 332, 334-335, 345; Aramaean-Indian
386; classification 351-353; Tamil 372
Hyde, Thomas 158
Hypsicles 522
/ Ching 70
Icelandic, Old 33-35
ideographic representation 79-81, 98, 107, 136, 145, 163-164;
Akkadian 159-160; Chinese 265, 271, 273; Linear A script
178 see also hieratic script; hieroglyphs
al-Idrisi 526
Iffah, G. 137, 368 , 369-375
Ifriqiya 521-523, 525, 528; ghubari numerals 534, 536
Ikhanian Tables 527
Iliad 72, 214
Iltumish 526
Imperial measurements 92
al-Imrani 523
Inca civilisation 39, 68-69, 125, 308
incalculable, Indian 422
inch 92
India 356-439, 512, 520, 523, 526-528; astrology 417, 463;
astronomy 409-411, 416-417, 431-432, 513-514; writing
212, 431-432
Indian number system 332, 341, 346-347, 361-439, 534;
calculation 346, 435-437, 568; chronograms 251; counting
39, 49-50, 94, 559; dictionary of numerical symbols
440-510; fractions 424-425, 595; high numbers 421-426,
434, 440; Indian numerals 367-383, 389-399; in Islamic
world 511-576; place-value system 334-335, 353, 399-409,
416-421
Indju, Jamal ad din Husayn 528
Indo-Aramaic 228
Indochina 49-50, 65-66, 407
Indo-Europeans 22, 29-32
Indonesia 368, 407
Indraji, B. 388-389
Indus civilisation 39, 162, 385
infants see children
infinity 362, 419, 421-422, 426, 440, 470-472, 597-598
Intaille 203
integers 597; aspects of 21-22
International Standards system (IS) 43
Inuit 36, 44, 305
invoices 78, 110
Iran 81, 135, 522; accounting 97-99, 101-102; counting 94,
290; number system 368, 534 see also DAFI
Iraq 52; accounting 101, 121; counting 49, 94; number system
251, 368, 534 see also Sumer
Irish 33-35, 38
irrational numbers 528, 596-597
’Isa, ’Ali ibn 525
Isabella I 528
Isaiah 258
Isfahan 513
Ibn Ishaq, Hunayn 513, 522
Ishtar 161
Isidore of Seville 56, 578
Isis 169, 258
Iskhi-Addu, King 74
Islam see Muslims
Islamic world see under Arab-Islamic civilisation
Ismail, Mulay 528
Isma’il, Sultan 252
Isme-Dagan, King 74
isopsephy 252, 256-261
Israel 97-98, 212, 239 see also Hebrews; Jews
al-Istakhri 524
Italic script 212-213, 586
Italy 51, 522; number system 31, 238; writing 216, 219, 586
Itard, J. 48, 221-222, 428
Itzcoatl 301
Ivan IV Vassilievich 96
Ivanoff, P. 298-299
Ivanov, V. V. 385
Iyer 434
Jacob, Francois 593
Jacob, Simon 206
Jacobites 240
Jacques 443
Jacquet 438
Jaggayyapeta 378
Jaguar Priests 301
al-Jahiz 364, 521, 541
Jainas 425-426, 440; Lokavibhaga 416-420, 430
Jalalabad writing 376
al-Jamali, Badr 525
Jamiat tawarikh (Universal History) 516
Ibn Janah 525
de Jancigny, Dubois 444
Janus (god) 47
Japan 305, 381; games 294-296; mysticism 554; number
system 36, 273-283, 289-290, 388, 542-543
Jarir 520
al-Jauhari 521
Jaunsari numerals 381, 384
Java 406-407, 418, 420; Kawi writing 383-384, 404; Sanskrit
413
Javanese numerals 375, 392-393, 395, 438
al-Jayyani, Ibn Mu’adh Abu ‘’Abdallah 518, 524
Ibn Jazia 525
al-Jazzari, Isma’il ibn al-Razzaz 518, 527-528
jealousy multiplication 567-571, 576
Jefferson 42
THE UNIVERSAL HISTORY OF NUMBERS
624
Jelinek 62
Jemdet 101
Jemdet Nasr 81, 110
Jensen 219
Jerome, Saint 56
Jerusalem 233-234, 236, 525-526, 587-588
Jestin 89, 121
Jesus 257-258
Jews 134, 239, 256, 512-513, 537-538; mysticism 250,
252-256; number system 71, 157-158, 238 see also
Hebrews; Israel
Jiangxian, Old Mann of 280
Jinkoki 278
Jiu zhang suan shu 287
John, St 256, 260
John of Halifax 361
John of Seville 362
Jonglet, Rene 65
Jordan 228, 368, 534
Jouguet, P. 220-221
Judaea 236, 239
Julia, D. L. 593
Julian calendar 50
Juljul, Ibn 524
Junayd 523
Jundishapur 512-513
Justinian, Emperor 512
Justus of Ghent 48
Juvenal 55, 203
Kabul 520
Kabyles 66
Kadman 233
Kaiyuan zhan jing 408, 418
Kairouan 513
kaishu writing 267-268
Kaithi numerals 370, 381, 384, 421
Kalaman, King 244
Kalidasa 419
Kalila wa Dimna 323, 419, 520, 556
kalpa 473-474
al Kalwadzani, Abu Nasr Muhammad Ben Abdullah 364
Kamalakara 414
Kamil, ’Abu 514, 516, 523-524, 530
Kamilarai people 5
Kampuchea 375
Kandahar 376
Kangshi, Emperor 343
Kanheri 401
kanji ideograms 273
Kan jo Otogi Zoshi 289
Kannada numerals 374, 385
Kannara numerals 383, 438
Kapadia, H. R. 562
al Karabisi, Ahmad ben ’Umar 364
al-Karaji 511, 514, 516, 524, 526, 530, 548
Karlgren, B. 272
Karnata numerals 374
Karoshthi numerals 386-387
Karpinski, L. C. 207, 356, 361-362, 364, 381, 386-387,
399-400, 538, 580
ibn Karram 522
Karystos 269
al-Kashi 516
al-Kashi, Ghiyat ad din Ghamshid ibn Mas’ud 513, 528, 561,
571
Kashmir 368, 370-371, 381, 420-421, 438
katakana 273
Katapayadi numerals 388
al-Kathi 525
Kawi writing 383-385, 404, 421
Kaye, G. R. 358, 400-402, 407, 434
Kazem-Zadeh, H. 543, 545
kelvin 43
Kemal, Mustafa 529
Kemal, Namik 529
Keneshre 366, 512
Kenriyu, Miyake 284
Kerameus, Father Theophanus 256
Kern, H. 406, 413, 418
ketsujo 542
Kewitsch, G. 91-93
ibn Khaldun, ’Abd ar Rahman 261, 363, 365, 514, 519, 525,
528, 553; Prolegomena 529, 542, 550-552, 593
Khalid 512
Khalifa, Hajji 528
Khaliji, ’Ala ud din 527
Khan, Genghis 382, 526, 556
Khan, Haluga 526-527
Khaqani 58-59, 526, 556
Kharezm Province 513
Kharoshthi writing 376-377, 386
Khas Boloven 65-66
KhaSeKhem, King 165
Khatra 228
Khayyam, Omar 513-516, 525
al-Khazini, Abu Ja’far 523
Khirbet el Kom 235
Khirbet Qumran 234
Khmer 383, 404-407, 420; number system 36-37, 388,
403-404
Khmer numerals 375, 385, 421, 438
Khorsabad 141, 159
khoutsouri 225
Khoziba 259
Khudawadi numerals 369, 381, 384
al-Khujani 524
Khurasan 523-524
ibn Khurdadbeh 522, 537-538
al-Khuwarizmi, Abu Ja’far Muhammad Ben Musa 364-365,
513-514, 516, 521, 523-524, 529-531, 533, 539, 548, 560,
562, 588; algorithms 531, 587
Khuzistan 519
kilogram 42-43
al-Kilwadhi 522
al-Kindi 364, 514, 519, 521, 556
king, ideogram for 159-160
Kircher, A. 210, 226
al-Kirmani 524
Kis 81, 101, 134
Kitab al arqam (Book of Figures) 363
Kitabfi tahqiq i ma li’l hind 363, 368, 426
Knossos 178-180
knot, meaning decimal system 542
Kobel 205, 358
Kochi numerals 381, 384
Kokhba, Simon Bar 233
Koran 514, 519, 521, 553-554
Korea 275, 278-283
Kota Kapur 404
al-Koyunlu 528
Kronecker 593
Kshatrapa numerals 397-398
Kufa, founded 519
Kufic script 243, 539-540
al-Kuhi, Ibn Rustam 523
Kulango 36
Kului numerals 381, 384
Kululu 181
Kumi 190
Kurdistan 528
Kushana numerals 397-398
Kutila numerals 381, 384
Kyosuke, K. C. 305
Labat, R. 84, 87, 99
Lafaye, G. 51
Lagas 81, 93
Lagrange 42-43
Lakhish 213, 236
Lalitavistara Sutra 420-425
Lalla 414
Lalou, M. 26
Lambert, Meyer 90, 137
Lampong writing 383
Landa, Diego de 300-301, 314
Landa numerals 381, 384
Landsberger, D. 130
Langdon, S. 109, 116-117, 385-386
Lao Tse 70
de Laon, Radulph 207
Laos 375, 383
625
INDEX
Laplace, P. S. 42-43, 361
Larfeld 182
Laroche, E. 180-181
Larsa 146
laser 43
Latin 52, 72, 96, 194; alphabet 212; number names 7, 31,
33-35
Laurembergus 359, 589
Lavoisier 42
Law of 10 Frimaire, Year VIII 43
Laws of 18 Germinal, Year III 42-43
ibn Layth, Abu’l Ghud Muhammad 524
LCM (lowest common multiple) 93
Lebanon 368
Leclant 52
Lehmann-Haupt 91
Leibnitz 550, 598
Lemoine, J. G. 49-50, 56, 541-542
Lengua people 15
Lenoir 43
Leonard of Pisa see Fibonacci
Leonidas of Alexandria 256
Lepsius 75
letter numerals see Abjad numerals; alphabetic numerals
Lettres of Malherbe 51
Leupold, Jacob 57, 357
Levey 434
Levias, C. 359
Levy-Bruhl, L. 5, 19, 45-46
Leydon Plaque 318-319
Li, J. M. 275
Li Ye 282-283
Liber de Computo 56
Liber etymologiarum 56
Libya 368, 520
Lichtenberg 593
Lidzbarski, M. 212, 390
Liebermann, S. J. 98-99, 130, 140
ligatures 170-171, 228-229, 246, 391, 434
Light of Asia, The 421
Lilavati 431
Limbu numerals 381
Lincoln, Abraham 37
L’lnde Classique 443
Lindemann 596
Linear A script 178-180
Linear B script 178-180
lines, grouping of 433-434
Liouville 596
lishu writing 266-268
Lithuania 33-35
Lives of Famous Men 55
Lobachevsky 598
Locke 68
Lofler 91
logarithms, natural 597
logic 598
Lokavibhaga 416-420, 430
Lombard, D. 265
Lombok 375
London, Royal Society of 42
Long Count 316-319
Lot of Sodom 253
Louis XI France 38
Louis XIV 528
Louvre 146
Lucania 189
Lull, Ramon 550
de Luna, Juan 362
lunar cycle 17-18, 50, 217; calendars 19, 297, 407; eclipse 529;
mansions 554; and numerology 93
ibn Luqa, Qusta 513
Luther, Martin 260-261
Lutsu 64
Lycians 9, 39
Lydian civilisation 9
Lyon 141
al Ma’ali, Abu 556
al-Ma’ari, Abu’l ’ala 525
Macassar writing 383
Maccabeus, Simon 234
MacGregor, Sir William 14
Machtots, Mesrop 224
Madagascar 125, 368, 534
Madura 375
Magadha 383
Maghreb 244, 252, 513, 520-521, 525-528; calculation 556,
560, 563; numerals 356, 385, 534-539, 559; writing
539-540
al-Maghribi, As Samaw’al ibn Yahya 55, 363, 511, 514, 516,
526, 534
Maghribi script 539-540
magic 248-262, 298, 302, 549-556; talismans 262, 522, 554
see also mysticism
Magini 595
Magnus, Albertus 515
Mahabharata 419
Mahajani writing 381, 384
al-Mahani 522-523
Maharashtri writing 380, 384
Mahaviracharya 399, 414, 418, 421, 562
Mahmud 523
Mahommed see Mohammed
Maidu 125
Maimon, Rabbi Moshe Ben see Maimonides
Maimonides 526
Maithili numerals 370, 381, 384, 421
Majami 55
al Maklati, Muhammad Ben Ahmed 252
Maknez, chronograms 252
Malagasy 39
Malay, Old 383, 385, 404, 406
Malaya 534
Malayalam numerals 332, 334-335, 342, 353, 373, 383, 385
Malaysia 39, 368, 406-407, 418, 420
Maldives 374
Malherbe, M. 36, 51, 273
Mali 72
al Malik, ’Abd 252
Malinke 36, 44
Mallia 178
Mallon 224
Malta 522
al-Ma’mun, Caliph 512, 521, 529, 531
Manaeans 9
Manchuria 272
Manchus 39
Mandeali numerals 381, 384
Manipuri numerals 381, 384, 421
Mann 37
al-Mansur, Caliph 512, 520, 529-530
al-Mansur, Sultan Abu Yusuf Ya’qub 526, 528, 550
Mansur, Yahya ibn Abi 513, 521
many, concept of 5-6, 32, 94
mapping 10-12, 16-17, 21, 23
al-Maqrizi 528
al-Maradini 542
Marathi numerals 369, 380, 384, 421, 438
Marchesinus, J. 588
al-Mardini, Massawayh 513, 525
Marduk 146-147, 159, 161
Mari 74, 81, 134, 142-146, 336, 352
Maronites 240
Marrakech 525, 527
al-Marrakushi, Abu’l Abbas Ahmad ibn al-Banna 363, 527,
568
Marre, A. 363, 568
Martel, Charles 512 , 520
Martial 203
Martinet, A. 385
Marwari numerals 381, 384, 421
al-Marwarradhi 521
ibn Masawayh, Yuhanna 519-520
Mashallah 359, 520
Ma’shar, Abu 521
Mashio, C. 305
Maspero, H. 74, 267, 269
ibn Massawayh 522
Massignon, L. 512, 514-519
Masson, 0. 179
al-Mas’udi 514, 523
THE UNIVERSAL HISTORY OF NUMBERS
626
Materialen zum Sumerischen Lexikon 130
Mathematical Treatise 515
Mathematics in the Time of the Pharoahs 175
Mathews 268, 278
Mathura numerals 397-398
Matlazinca 301
Matzusaki, Kiyoshi 289
Maudslay, Alfred 300
Le Maur, Carlos 357
al-Mawardi 525
Maximus, Claudius 56
Maya civilisation 72, 297-322; astronomy 315-316, 321-322;
calculation 303-305, 321-322; calendars 36, 300, 311-319;
mysticism 300, 311-314, 316-322; writing 298-301, 305,
311-314, 316-322
Mayan number system 9, 36, 44, 162, 303-312, 345; posi-
tional 322, 337, 339-340, 353-354, 430; zero 320-322,
341-342, 430
Mazaheri, A. 363, 519, 533, 539, 541-542, 549, 556, 561
Mead 541
measurement 82, 91, 153, 158
Mebaragesi 81
Mecca 519, 529, 537, 554
Mechain 43
de Mecquenem, R. 109, 116-117
Media 519, 522
mediating objects see model collections
Medina 519
Mediterranean 212, 222
Mehmed IV, Sultan 529
Mehmet II, Sultan 528
Mei Wen Ding 280, 284
Mejing 516
Melanesian Languages 6
Melos 219
Mendoza, Don Antonio de 303
Menelaus 513, 522-523
Menna, Prince 61
Menninger, K. 190, 276, 283, 336, 343, 356, 428
al-Meqi, al-Amuni Saraf ad din 363, 526
Merida, bishop of 300
meridian expedition 42-43
Merv 513
Mesha, King 212-213
Mesopotamia 94, 162, 239; Arabs 512, 519, 526, 528;
Babylonian era 134-161; India 376, 513; Mari 142-146;
mysticism 93-94, 554; writing 212, 539 see also
Akkadian Empire; Elam; Semites; sexagesimal system;
Sumerians
counting: abacus 130-133, 562-563; bullae 99, 101; calculi
97-98; clay tablets 84-89
number system 82, 96-108, 134-161, 325-329; Aramaic
numerals 228; decimals 138-146; letter-numerals 243;
zero 152-154, 341
Messiah (jewish) 253
la Mesure de la Terre 42
metal, as currency 73-74
Metaphysics 20
Metonic cycle 195
metric system 595; history 42-43
metrology 182
Mexico 36, 299-301, 303, 305, 307
Mexico City 301, 305
Micah III 256
Middle East: calculi 97-98; language 52, 212, 222, 248-250;
Semites 134
Midrash 253
Mieli, A. 519
Mikame 289
Miletus 219
Milik, J. T. 234
Millas 216
Miller, J. 42,273-274
milliard 428
million 140, 165, 168, 325, 427-428
Minaeans 185-186
Minoan civilisation 39, 178-180
Minos, King 178
minus (mathematical concept) 89
Mirkhond 528
Miskawayh 525
Mi-s’on 406-407, 413
Misra, B. 562
al Misri, Abu Kamil Shuja’ ibn Aslam ibn Muhammad al
Hasib 364
Misri, Dhu ‘an Nun 522
Mithat, Ahmet 529
Mithras 259
Mitsuyoshi, Yoshida 278
Miwok 125
Mixtecs 36, 305, 307-308
mkhedrouli 225
mnemonics 432, 537
Moab 212
model collections 10, 12, 17-18, 23
modern numerals 324-325, 343-347, 356-365, 368, 385,
426-439, 592-599
Modi numerals 380, 384
Mogul Empire 528
Mohammed 47, 50-51, 58-59, 514, 519, 553
Mohenjo-daro 375, 385
Mohini 50
mole 43
Moliere 38
Moller, G. 342, 390
Mommsen, T. 187
Mon writing 194, 383
money 41, 72-76, 182-184, 308
Monge 42-43
Mongolia 22, 27, 39, 49, 51, 556
Mongolian Empire 382, 526-527
Mongolian numerals 382, 385, 395-396
monks 70-71; Zen 295
Montaigne, Michel de 205-206, 577, 590
Monteil, V. 513-514, 516, 518-519, 593
Montezuma 301, 303
Montucla, J. F. 360-361
moon 217, 239, 411 see also lunar cycle
Moor 439
Moraze, Charles 345, 347
Moreh Nebukhim (Guide for the Lost) 514
Morley, S. G. 298, 316, 320
Morocco 51, 252, 555
Morra 51-52
Moses 254 , 553
Moss 75
Mota 19
Motecuhzoma I 301, 303
Mouton, Abbe Gabriel 42
Moya, Juan Perez de 55
Mozarabes (Arabic Christians) 513
al Mu’aliwi, ’Ali Ben Ahmad Abu’l Qasim al Mujitabi al
Antaki 364
Mu’awiyah 520
Mudara, Muhammedal 252
Muhammad, Abu Nasr 523
Muhammad of Ghur 526
al-Muhasibi 522
al Mulk, Nizam 58
Multani numerals 381, 384, 421
multiplication methods: abacus 127, 204-206, 208-209,
285-287, 292, 557-559, 582-585; calculi 122; fingers
59-61; tables 154-156, 220, 561, 578; written 154-156,
174-176, 567-576
multiplicative principle 229, 231, 246, 263, 270, 330-334
al-Mu’min, ’Abd 525
al-Muqaddasi 524
Muqaddimah 261, 363, 519, 529, 542, 552-553, 593
al-Muqafa, Abu Shu’ayb 521
al-Muqafa’, ibn 520
Murabba’at 236
Murray Islanders 5-6, 14
Muslims: finger gestures 47, 50, 52, 58-59; Hisabal Jumal 250,
252, 261-262; magic talismans 262, 522, 554; prayer 9, 50,
71 see also Arab-Islamic civilisation
al Mustadi 252
al-Mustawfi, Hamdallah 527
al-Mu’tamin 525
Mutanabbi 522
ibn Mu’tasin, Ahmad 518
Muwaffak, Abu Mansur 524
Mycenae 179
627
INDEX
My res 179
myriad 26, 221-222
Mysticae numerorum signification esopus 199
mysticism, number: Arabs 512, 553-555; China 270; fear of
numbers 214, 275-276; India 431, 543; Mayan 321-322;
sacred symbols 93-94, 162, 239; soothsayers 261-262, 269,
551-556 see also codes and ciphers; magic
’n’ roots 528
Nabataean numerals 212, 227-228, 390
Nabi 529
al-Nadim, Ya’qub ibn 364, 524; Fihrist 364, 531, 539
al-Nafis, Ibn 527
Nagari numerals 364, 368-369, 384, 400, 421, 438, 481, 532,
538
Nagari writing 364, 377, 379-380, 388, 420
al-Nahawandi, Ahmad 521
Naima 529
Nakshatra 417
names of numbers 14-15, 19-23, 33-35, 136-137; games 159;
gods’ names used 95; Indian 481-482; Mayan 303-304;
prayer words 214
Nana Ghat 379, 387-388, 391, 397-399, 420, 435-436
Napier, John 597
Narayana 562
Narmer, King 164-165
Nasik 379, 387-388, 397-398, 435
Naskhi script 539-540
Nasr 101
Nasr, Abu 524
Nasr, S. H. 516, 519
nastalik script 539-540
Nathan, Ferdinand 291
Natural History 47, 198, 200, 427
Nau, F. 366
Naveh 232, 234
al-Nawabakht 520
al-Nayrizi (Anaritius) 522
Nayshaburi, Hasan ibn Muhammad an 571
al-Nazzam 521
Nebuchadnezzar II 135, 236
Nedim 529
Needham, Joseph 51, 264, 268-269, 278-284, 293, 408
Nefi 529
negative numbers 278, 283, 287, 597
Negev, A. 73
Nemea 185
Nepal 377, 384, 388, 390, 420
Nepali numerals 371, 381, 384, 392-398, 438
Nergal 161
Nero 256, 260
Nesselmann, G. H. F. 218
Nestor, King 55
Nestorian sect 240
Neugebauer, 0. 91-92, 150, 153, 157, 414-415
New Guinea 13-14, 305
New Hebrides 36
New Mexico 196
Newberry 52
Newton, Isaac 42
Nichomachus of Gerasa 43-44
Nicobar Islands 375
Nicomacchus of Gerasa 578
Niehbuhr, Karsten 48-49
Nigeria 70
Nilakanthaso-mayajin 414
Nile 259
nine 35, 396; Chinese 269; Egyptian 177; Hebrew 215; Indian
410; Japanese 276
nine hundred 216-217
nineteen 177
ninety 215, 235, 248, 295
ninety-nine 295
ninety-three 58
Ninevah 101, 103, 135, 146
Ninni, A. P. 197
Ninurta 161
Nippur 81, 130, 239
Nisawi, Abu’l Hasan ’Ali ibn Ahmad an 363, 524, 530, 539,
548, 560, 562-563
Nisibe 512
Nissen 92
Nizami 526, 556
Nommo the Seventh 72
non-equivalence between sets 3-4
non-Euclidian geometry 598
notched bones 11 see also tally sticks
Nottnagelus 359
Nougayrot, J. 85, 146
nought see zero
Nubians 39
Numa, King 47
number systems: alphabetic numerals 156-157, 212-262,
483-484; Arab-Islamic 511-576; Chinese 263-296; Cretan
178-180; Egyptian 162-177; Europe 578-582, 586-591;
Greek 182-187, 218-223, 232-233; Hebrew 214-218,
233-236; historical classification 347-355; Hittite 180-181;
Indian 367-439; Dictionary 440-510; Mayan 297-322;
Mesopotamian 96-108, 134-161; modern 324-325,
343-347, 356-367, 592-599; Roman 187-200 see also
abacus; accounting; base numbers; body counting; calcula-
tion; decimal; mysticism; position, rule of; Sumerian; zero
numerology 93-94, 161, 360, 554 see also codes and ciphers;
mysticism
Nur ad din 526
Nusayir, Musa Ben 520
Nusku 161
Nuwas, Abu 521
Nuzi, Palace of 100-101
Oaxaca Valley 301, 305
Oaxahunticu see Maya calendars
obols 182-183, 201-203
Oceania 10, 12-14, 36, 44, 554
Odyssey 214
Oedipi Aegyptiaci 226
Oghlan, Karaja 529
Ojha 386-387
Okinawa 70
Olivier 179
Omayyad dynasty 520
Omri, King 236
one 33, 194, 392; Aztec 305; Chinese 269; Greek 179, 182,
184, 186; Hebrew 215; hieroglyphic 165, 168, 176, 178, 181,
325; Indian 409-410; Maya 308; Roman numerals 188,
192; Sumerian 84, 148
one hundred and eight 71
one-for-one correspondence 10-12, 16-17, 19, 96, 191, 194
Opera mathematica 91
Ophel, accounting 236
Oppenheim, A. L. 100-101, 131
Ora 216
oral numeration 25-26, 265-266, 303
Orchomenos 183
order relation 20
ordinal numeration 20-22, 24, 182, 193
Ore 276, 428
Oriental Research Institute (Baghdad) 100
Origin of Species 519
Orissl numerals 370
Oriya numerals 370, 381, 384, 421, 438
Orontes 55
Oscan alphabet 212
Osiris 169, 259
ostraca 213, 236, 238
Otman, Khalif 58
Ottoman Empire 527-529, 543; secret writing 248-250; Siyaq
numerals 547-548
oudjat 169-170
ounce 92
ownership, mark of 66
oxen 72
Ozgiif 181
Pacific Islands 72, 125
Pacioli, Luca 57, 567, 576
pairing 6, 21
Pakistan 94, 386, 520; numerals 368, 381, 534; phalanx-
counting 94
Palamedes 219
Palenque 297, 316-317, 320
palaeography 391, 401, 404^106, 419, 538-539, 579
THE UNIVERSAL HISTORY OF NUMBERS
628
palaeo-Hebraic alphabet 212-213, 233, 236, 238
Palestine 70, 236, 239, 519, 528; numerals 228, 236-238,
241, 246
Pali writing 374, 377, 383, 385
palindromes, numerical 399
Pallava dynasty 378; numerals 397-398
Palmyra 212, 227-228, 248, 533
Palmyrenean numerals 390
Pahchasiddhantika 414-416, 439
Panchatantra 323, 419, 520
Panini 388-389
Paniniyam 389
Pantagruel 51
paper making 516, 521, 566-567
Papias 207
Pappus of Alexandria 221-222, 523
Papuans 13-14
papyrus 533
Paraguay 15
Parameshvara 414
parchment, Maya 301
Pardes Rimonim 253
Paris, B. N. 361-362
Paris Codex 301
Paris (of Athens) 51
parity, concept of 6
Parmentier, H. 413, 418-419
Parrot, Andre 142
Pascal, Blaise 282, 594, 598
Pascal’s triangle 282, 511
Pasha, Ziya 529
pebbles 12, 15, 96-97, 125; counting 126
Peguy, Charles 365
Peignot 381
Peignot script face 588
Peking 272
Peletarius 358
Pellat, C. 55, 541-542
Peloponnese 75-76, 183
pendulums 42
Pepys, Samuel 578
perception, limits of 6-10
Perdrizet, P. 256, 258-259
Pergamon 256
Perny, P. 51, 268, 271
Persia 70, 259, 376, 512-528; abacus 556, 562-563; number
system 39, 58, 250-251, 545-547, 553; writing 240, 248, 539
Persian Gulf see Sumerians
Peru 69-70, 308, 543
Peruvian Codex 308
Peten, Lake 299-300
Peter, Simon 257
Peterson, F. A. 312-314, 317, 320
Petitot 46
Petra 228
Petruck 434
Petrus of Dacia 361
Phaestos 178
phalanx-counting 94-95
Pheidon, King 75
Philippines 383
Phillipe, Andre 65
philology 419
Philo of Byzantium 513, 518
philosopher’s stone 518-519
Philosophica Fragmenta 203
Phoenicians 359; alphabet 212-214, 219, 239; number system
9, 39, 137, 227-228, 351; writing 185, 232, 236, 390
phonograms 80, 136, 265 see also hieroglyphs
Phrygia 238
pi 596-597
Piaget 4-5
Picard 598
Picard, Abbe Jean 42
pictograms 78-81, 85, 97-99, 107-108, 306 see also
hieroglyphs
Pieron, H. 365
Pihan, A. P. 268, 271, 356, 381, 543-545, 547
Pingree, D. 91-92, 150, 153, 157, 414-415
Pinyin system 265
Pisa, Leonard of see Fibonacci
Pizarro 68
place-value system 324, 559-560, 588; abacus 287-288,
434—437, 561; discovery 287-288, 337-339, 399-407,
416-421 see also position, rule of; positional systems
Planudes, Maximus 361, 365, 562, 589
plates, lead 181
Plato 512-513, 517, 523, 598
Plaut 273-274
Plautus 194
Pliny the Elder 47, 198, 427
Plotinus 513, 523
plurality 32
Plutarch 47, 55
Po Nagar 404-406, 420
Poincare, H. 367
Polish 33-35
Polybius 200
Polynesia 6, 72
polynomials 283
Pompeii 256
Popilius Laenas, C. 189
Popol Vuh 301
Porter 75
Portugal 33, 35, 51
Posener, G. 52, 533
position, rule of 24, 143, 145-155, 334-340, 345-346 see also
place-value system
positional systems: Arabic 186; Babylonian 145-154; Chinese
278-283; historical classification 353-355; India 411-421;
Mayan 308-312, 322
Pott, F. A. 36-37
Powell, M. A. 82, 121
powers: abacus 285; cubed 363; exponential 528, 594;
negative 156, 278; squared 323-324, 363; ten 278,
426-429, 440, 594
Prah Kuha Luhon 404
Prasat Roman Romas 413
prayer-beads 11, 50-51, 99
Pre-Sargonic era 81, 87, 89
Prescott, W. H. 69
priests, Mayan 311-312
primitive societies: barter in 72-73; counting 5, 10, 12-18, 46
Prinsep, J. 386-387
Prithiviraj 526
Prolegomena 261, 363, 519, 529, 542, 552-553, 593
Prophet, the see Mohammed
Proto-Elamite number system 326
Psammetichus, King 52
Psammites, The 333
Pseudo -Callisthenes 256
Ptolemy 513, 520-525, 588, 598
Ptolemy 1 256
Ptolemy II 232
Ptolemy V 167
Pudentilla, Aemilia 55-56
Puebla region 301
Pulisha Siddhanta 427
Punjab 228
Punjabi numerals 369, 381, 384, 421, 438
Putumanasomayajin 414
puzzles, number 176-177
Pygmies 5, 72
Pylos 179
Pythagoras 256, 515, 596
Pythagoras’ theorem 151, 522
al-Qalasadi 539
al-Qalasadi, Abu’l Hasan ’Ali ibn Muhammad 363, 516, 528,
563, 567
Qasim, Abu’l 524
al-Qasim, Muhammad Ben 529
al-Qass, Nazif ibn Yumn 523
al-Qays, Imru’ 520
al-Qifti, Abu’l Hasan 527, 529-530
quadrillion 427-428
Quahuacan 36
Quauhnahuac 36
Qubbut al Bukhari 252
Qudama 523
Quetzalcoatl 300
quinary systems 9, 44-46, 94-95
629
INDEX
Quintana Roo 299, 303
Quintilian 47
quintillion 428
quipucamayoc 69, 308
quipus 64, 68-69, 308, 542-543 see also string, knotted
Quirigua 298, 316-317, 319-321
ibn Qurra, Thabit 514
Qutan Xida see Buddha
Qutayba 513
Qutb ud din 526
Rabban, ’Ali 522
Rabelais 51
Rachet, Guy 81, 135
Raimundo of Toledo 362
Rajasthani numerals 381, 384
Ramus 358
Ramz 250
Rangacarya, M. 414, 418
rank-ordering 16
Ras Shamra 137, 214
Rashed, R. 363, 511, 519
Rashi 253
al-Rashid, Harun 512, 520
Rashid ad din 514, 516, 527
rational numbers 596-597
Razhes 522
al-Razi, Fakhar ad din 363, 513-514, 526
al-Razi, Muhammad Abu Bakr Ben Zakariyya (Razhes) 522
ready-made mappings 12, 17, 19
real numbers 597
Rebecca, wife of Isaac 257
rebus 302-303, 306-307
receipts 68, 70
Recorde, Robert 358, 590
recurrence 20
Red Sea 49
Redjang writing 383
Reinach 182
Reinaud 364
Reisch, Gregorius 591
Relacion de las Cosas de Yucatan 300
Renaissance 529
Renou, L. 335, 386-387, 431, 438, 443
Rey 513
Reychman, J. 543, 547
Rhangabes 201
Rhind Mathematical Papyrus (RMD) 171
Richard Lionheart 586
Richer 42
Ridwan, ’Ali ibn 525
Ridwan of Damascus 518, 527
Riegl 196
Rif 252
right-angled triangles 151, 522
Rijal, ibn Abi’l 524
Rivero, Diego 47
de Rivero, M. E. 69
RMD (Rhind Mathematical Papyrus) 171
Robert of Chester 362
Robin, C. 186-187
Rodinson, M. 185
Rollig 229
romaji 273
Roman Empire 7, 51, 92, 521, 577-578; calculation 39, 70, 96,
333-334, 427; abacus 125, 202-207, 209-211, 578-580,
582; currency 55, 76
Roman numerals 9, 187-200, 327-328, 349; used in Europe
208, 578-579
Romance languages 31-32
Romanian 33-35
Rong Gen 269
roots, square and cube 156, 285, 293, 419, 560, 596-597
rosaries 70-71
Rosenfeld, B. A. 571
Rostand, J. 593
Rudaki 523
Ruelle, C. E. 222
Rumelia 528
Rumi, Ya’qub ibn ’Abdallah ar 527
Ibn Rushd (Averroes) 514-515, 526
Russia 33-35, 66, 72, 212, 290
ibn Rusta of Isfahen 523
Rutten, M. 93
Ryu-Kyu islands 70, 542-543
Ibn Sa’ad 58, 71, 522, 542
Sa’adi of Chiraz 527, 538
Saanen 195
Sabaeans 9, 512
Saccheri 527
Sachau, E. 227, 235
sacred symbols 93-94, 162, 239 see also mysticism
de Sacrobosco, Jean 358, 361
Sa’ddiyat 335
Saffar, Ibn al 524
Saffarid dynasty 521
Saglio, E. 221, 428
ibn Sahda 522
Sahdad 101
ibn Sahl, Sabur 522
Saidan, A. 364, 563
Sakhalin, Ainu of 305
Saladin 526
Salamis, Table of 201-203
Samanid dynasty 521-522
Samaria 213, 236
Samaritans 212, 233
Samarkand 520, 528
al-Samh, Ibn 524
Sanayi, Abu’l Majid 58
sangi 278-283
Sankheda 402
Sankhyayana Shrauta Sutra 422
Sanskrit 72, 433; number names 29, 32-35, 404-406,
411-420, 530; high numbers 427-429, 434; oral counting
426-431; Panini 389; Shiddhamatrika 381 see also Brahmi;
Nagari writing
ibn Sarafyun, Yahya 522
Sarapis 256
Sarasvati (goddess) 439
Sardinia 520
Sargon I The Elder 135
Sargon II 139, 141, 159
Sari 232
Sarma, K. V. 414-415, 419
Sarton, G. 519
Sarvanandin 416-418
Sastri, B. D. 414
Sastri, K. S. 414
Satan 554, 588
Satires 207
Satraps 378, 407
Saudi Arabia 368, 534
Saul 73
Saxon, Old 33-35
Scandinavia 65, 196
Scheil, J. 102, 109, 116
Scheil, V. 115
Schickard 594
Schmandt-Besserat, Denise 97-100
Schnippel, E. 195-196
Scholem, Gershon 217, 256
Schopenhauer 20
Schrimpf, R. 278
science: classification 517, 523, 525; Koran 514
scientific notation 594
Scots Gaelic 33
Scott 52
Scythians 377
seals, cylinder 103-104, 106-107
Sebokt, Severus 366, 407, 419
secret writing 248-250 see also mysticism
Sefer ha mispar (Number Book) 362
Semites 81, 134-136; alphabet 212-213, 377; number system
22, 136-146, 227-232, 351 see also particularly Akkadian;
Arab-Islamic; Assyrian; Babylonian; Hebrews; Phoenician
cultures
Senart 386-387
Seneca 47, 200
Senegal 305
Sennacherib 146
THE UNIVERSAL HISTORY OF NUMBERS
630
separation sign 149
septillion 428
Serere 36
Sessa, legend of 323-324
sestertius 210
Seth 169
sets, theory of 598
seven 34, 395, 442; Chinese 269; Egyptian 176; Hebrew 215;
Indian 410; Japanese 276
seven hundred 216-217
seventeen 177
seventy 215
seventy-seven 294
de Sevigne 206
Seville 514, 527
sexagesimal system 82-84, 90-95, 126, 139, 157; Akkadian
134, 138, 239; astronomy 91-92, 95, 140, 157-158,
548-549; Babylonian 134-161; calculation 126-133, 140,
528; proto-Elamite 120
sextillion 428
Sezhong, King 275
Shah Nameh 58
Shahadah, prayer of 47
ibn Shahriyyar, Buzurg 524
Shaka calendar 407, 494
Shamash 161
Shan 39
shang deng number system 277-278
shang fang da zhuan writing 268
Shankaracharya 418
Shankaranarayana 388, 414, 418, 432
al-Shanshuri 536
Sharada numerals 371, 381, 384, 421, 438, 494
Sharada writing 371, 377, 420
Shaturanja (early chess) 323-324
Sheba 185-187, 327
shekel 73
shells 24-25, 37
Shem, son of Noah 134, 254
Shen Nong 70
shepherds, counting methods 47, 191-193, 214; and base 10
24-25; bullae 101, 103; pebbles 12; quipus 69; tally sticks
11, 64
Sher of Behar, King 251
Shiite Islam 521
Shiraz 243
Shivaism 407
Shojutsu Sangaka Zue 284
Shook 317
Shridharacharya 414, 562
Shripati 414, 562
Shukla, K. S. 414-415, 419
shunya (zero) 412, 495-496
Shuri 70
Shushtari 527
Siamese numerals 375, 388, 403
Siamese writing 383
Siberia 71-72
Sibti, Abu’l ’Abbas as 550-551
Sicily 190, 219, 521, 587
Siddham numerals 384
Siddham writing 377, 381, 420
Siddhamatrika writing 381
siddhanta see India, astronomy
Siddim, Valley of 253
sign language 52-59
al-Sijzi, ’Abd Jalil 524, 534
Silberberg, M. 346, 362, 589
silent numbers 214
Sillamy, N. 4, 365
Simiand, F. 366
Simonides of Ceos 219
Simplicius 512
Sin 161
ibn Sina, Al Husayn see Avicenna
Sinan, Ibrahim ibn 243
Sindhi numerals 369, 381, 384, 421, 438
Singapore 272
Singer, C. 518-519
Singh, A. N. 356, 364, 386-388, 399-400, 414, 419, 422, 434,
438, 562, 568, 573
Singhalese numerals 342, 352, 374, 383, 385, 388
singularity 32
Sino-Annamite writing 272
Sino-Japanese numerals 273-276, 278
Sino-Korean number system 275
Sircar 438
Sirmauri numerals 381, 384
Sitaq 248
Sivaramamurti 387
six 34, 161, 395; Chinese 269; Egyptian 176; Hebrew 215;
Indian 410
six hundred 84, 216-217
six hundred and sixty-six 260-261
sixteen 218
sixty 91, 93; base 40, 82; Hebrew 215; Mesopotamian 84,
141-142, 148, 161
Siyaq numerals 545-548
Skaist 235
Skandravarman, King 378
Skarpa, F. 194-195
Slane, I. 542, 552-553
Slavonic Church 33-34
Smirnoff, W. D. 260
Smith, D. E. 199, 207, 284, 356, 361-362, 364, 381, 386-387,
399-400, 538, 580, 589
Smith, V. A. 386
Snellius 595
sol, French unit 92
solar cycle 49-50; calendar, Maya 297; eclipse 529
de Solla Price, D. 518
Solomon Islanders 19
Solomon’s ring, legend of 357
Solon 200, 206
Sommerfelt, A. 5
soothsayers 261-262, 269, 551-556 see also mysticism
soroban 288-289, 294
Soubeyran, D. 144-145, 336
Sounda 375
Sourdel 540
Soustelle, Jacques 36, 72-73, 239
South America: counting 5, 10, 36, 125; Inca civilisation
68-69, 308
South Borneo 18-19
Spain 51, 250, 525, 553; and Arabs 248, 513, 520-521, 528,
587; Central America 300-303, 308; number system 31,
33-35, 359, 534-537, 585; Spanish Inquisition 588-589;
writing 216, 539, 586
Spanish Inquisition 588-589
spatio-temporal disabilities 5
spheres 522
Spinoza 199
spirits, malign 275-276
square alphabet 212-213, 215, 233
square roots 156, 285, 293, 419, 560, 596-597
squares (power of two) 323-324, 363
Sri Lanka 5, 372, 374
Stars and Stripes 290
Steinschneider, M. 346, 362
Stele of the Vultures 86
Stephen, E. 19
Stephens, John Lloyd 300-301
sterling currency 41
Stevin, Simon 595
Stewart, C. 543, 545, 547
sticks as counting devices 15-16, 125 see also tally sticks
stone, as medium 162
string, knotted 64, 68-71
Su Yuan Yu Zhian 281
Suan Fa TongZong 61, 284, 293
suan pan 288-294 see abacus, Chinese
suan zi notation 278-283, 288, 408
Subandhu 418
subha (prayer) 50
subtraction 174; abacus 127, 204-206, 285, 292
subtractive principle 89, 328
succession 21-22
Sudan 97-98, 539
Suetonius 256
Sufi 521, 550-551
al-Sufi, Abu Musa Ja’far 364, 521
al-Sufin, ’Abd ar-Rahman 524
631
INDEX
Sulawezi 383
Suli, As 522, 541-542
sulus script 539
Sumatar Harabesi 232
Sumatra 64, 383, 404
Sumer 101-102, 135
Sumerians 77-91; bullae 99, 103-104, 109-111; calculation
82-83, 94, 121-133, 140-142; number system 9, 77-95,
99-100, 109-120, 122, 139, 142, 147-148, 325-326,
349-350; Sumerian-Akkadian synthesis 137-138, 142, 148;
writing 77-81, 86-90, 107 see also Mesopotamia
Sumerisches Lexikon 121
Sunni Islam 521
superstition see mysticism
Suruppak 87-90, 121-122
Surya Siddhanta 411
Susa 101-107, 112-115, 119-120, 140, 149, 155, 158-160
Susinak-sar-Ilani, King 159
Suter, H. 363-364, 519, 563, 568
Swedish 33-35
Switzerland 31, 65-66, 195, 205
Sylvester II see d’Aurillac, Gerbert
symbolism 78, 499-501
synonyms 409-421, 430-432, 438
Syria 52, 145, 526; Arabs 512, 519, 522-524, 526-528; Hittites
180; India 376, 513; writing 212-213, 232, 248 see also
Ugaritic people
Syrian number system 227-228, 246; alphabetic numerals
238-243, 329; calculation 49, 94, 97-98, 101, 541-542,
563; Indian numerals 365-368, 534; Mari 142-146
Sznycer, M. 213
al-Tabari, Marshallah 514
al-Tabari, Sahl 514, 521
at Tabari, ’Ali Rabban 519
Tabasco 299, 303
tables: astronomical 146, 157-159, 198, 521, 527;
mathematical 127-130, 146, 203-206, 283-288, 555-563;
multiplication 154-156, 220, 561, 578
tablet: clay 77-80, 84-89, 92-93, 98-125, 132-135;
accounting 79-80, 101-122, 134; Babylonian 134, 138, 140,
147-148, 159-160; calculation 122-125, 147-148, 151,
562-563; Cretan 178-179; Ebla tablets 135, 145; Heliastes
214; Hittite 181; proto-Elamite 102, 105; Sumerian 77-78,
98, 101-102, 107, 122, 134, 140, 562-563; Tablet of Fate
146; wooden (abacus) 132-133
Tabriz, Ghazan Khan a 516
Tadjikistan 522
Tadmor 248
Tafel 190
Tagala writing 383
Tagalog numerals 385
ibn Tahir, Mutahar 364
Tahirid dynasty 521
Takari numerals 370-371, 381, 384, 421
talent (money) 182-183, 200-203
talismans 262, 522, 554
Talleyrand 42
Tall-i-Malyan 101
tally sticks 11-12, 16-18, 62-67, 191-197
Talmud 253
Tamanas 36, 44, 305
Tamil numerals 332, 334-335, 342, 353, 372-374, 383, 385
ibn Tamin, Abu Sahl 364
Tammam, Abu 522
Tangier 252
Tankri numerals 370-371
Tao Te Ching 70
Taoism 443
al Tarabulusi, Ahmad al Barbir 58-59
Tarasques 301
Tarih 250
Tarikh ul Hind 251
ibn Tariq, Ya’qub 513, 520, 530
Tartaglia, N. 358
Tashfin, Yusuf Ben 525
Tashkdpriizada 528
Taton, R. 515-516
Tavernier, J. B. 50
Tawhidi 524
tax collection 64-65, 68, 70, 302-303, 306
Tayasal 300
Taybugha 528
Taylor, C. 386-387
Taylor, J. 573
ibn Taymiyya 527
Tchen Yan-Sun 276, 428
Tebrizi 527
Tel-Hariri excavation 142
Telinga numerals 373
Tell Qudeirat 236, 238
Tello 101
Telugu numerals 373, 383, 385, 421, 438
ten 35; Chinese 263, 269; decimal system 24-32, 39-44;
Greek 182, 184, 186; Hebrew 215; hieroglyphic 165, 168,
177-179, 181, 325; mysticism 43-44, 161; powers of
426-429, 440, 594; Roman numerals 188, 192; sexagesimal
system 82-84, 93-95; tally sticks 194
ten thousand: Aramaic 230; Babylonian 140, 145; Chinese
263, 265; Greek 182, 184, 221-222; Hebrew 137;
hieroglyphic 165, 168, 179-180, 325; Japanese 274-275;
Roman numerals 197-198
Ten Years in Sarawak 19
Tenochtitlan 301-303, 306
Tepe Yahya 101-102
ternary principle 89, 139, 166, 227
Tertullian 47
Tetrabiblos 520
Tetuan 252
Texcoco, Lake 301
ibn Thabit, Ibrahim ibn Sinan 523
Thai numerals 375, 383, 385, 392-393, 438
Thebes 52, 219
Theodoret 55
Theodosius 522
Theon of Alexandria 91
Theophanes 359, 590
Theophilus of Edesse 513, 521
Thera 219
Thespiae 183
Thibaut, G. 415
Thibaut of Langres 257
thirty 93, 161, 215
thirty-six thousand 84
Thompson, J. E. 312, 316-317, 320-321
Thot 169-170, 176
thousand 25; Aramaic 230; Chinese 168, 263, 265, 269; Greek
182, 184, 186; hieroglyphic 165, 168, 178-179, 181, 325;
Japanese 274; Mesopotamian 137-139, 142, 145, 231;
Roman numerals 188-189, 192, 197-198
Thousand and One Nights 521
three 33, 393; base 40; Chinese 269; Egyptian 176; Hebrew
215; Indian 410; many as 4, 32, 94; ternary principle 9, 89
three hundred 215, 257
three hundred and sixty five 257-258
three thousand, six hundred 84, 93, 141-142, 148
Thureau-Dangin, F. 82, 91-92, 139, 152, 159, 407
Thutmosis 166
Tiberius, Emperor 200, 257
Tibet: counting 39, 70-71; number system 26-27, 371, 373,
388, 422; writing 377, 382, 420
Tibetan numerals 371, 385, 392-393, 395-396
Tijdschrift 406
Tikal 297, 318-320
time 17-19, 28, 49-50, 68, 298, 311; and base 41, 82, 158
Timur 528
Tiriqan, King 81
Tirmidhi 522
al Tirmidhi, Abu Dawud 51
Tizapan 301
Tlatelolco 302
Tod, N. M. 182-185, 233
tokens see abacus; calculi; currency; tally sticks
Tokharian language 32-35
Tokyo 274, 276, 289-290
Toledo 251, 514, 525, 587
Toltecs 300
Toluca 36
Toomer, G. J. 519, 531
topology 598
Torah 215, 218, 239, 253-254, 256 see also Bible
Torkhede 401
THE UNIVERSAL HISTORY OF NUMBERS
632
Torres Straits 6, 12, 14
Trajan’s column 588
transcendental numbers 596-597
triangles, spherical 527
trigonometry 420, 523, 526-529
trillion 427-428
Tripoli 528
Tropfke, J. 537
Truffaut, Francois 4
Tschudi, J. D. 69
ibn Tufayl (Abubacer) 526
ibn Tughaj, Muhammad 522
Tughluq, Firaz Shah 527
Tula 301
Tulu numerals 383, 385
Tumert, Ibn 525
Tunisia 520-521, 523, 555
ibn Turk, Abu al-Hamid ibn Wasi 521
Turkestan, Chinese, writing 382, 385, 420
Turkey 512, 529; mysticism 248-251, 553; Russian abacus
290; writing 180, 248-250 see also Ottoman Empire
Turkish, Ancient 27-29
at Tusi, Nasir ad din 513, 527, 562, 571-573
twelve, base see duodecimal system
twenty 44; base see vigesimal system; Egyptian 177; Japanese
274; mysticism 93, 161, 248; Semitic 215, 228-229
twenty six 254
two 33, 393; base see binary system; Chinese 269; Egyptian
176; Hebrew 215; Indian 409-410
two hundred 215
Tyal tribe 73
Tylor, E. B. 5
Tyrol 195-196
Tzolkin, Mayan calendar 312, 315
Uaxactun 320
Uayeb 314
Ugaritic people 39, 137, 145, 214, 244
Ulrichen 195
al umam, Tabaqat 515
’Umar, caliph 515, 519
al-’Umari 527
'Umayyad dynasty 512
Umbrian alphabet 212
Umna 81
unciae (Roman ounce) 210
United Kingdom: Chancellor of the Exchequer 590 see also
England; Scots Gaelic; Welsh
United States 92, 428
Universal History (Jami’at tawarikh) 516
Untash Gal 102
Upasak 387
al-Uqlidisi, Abu’l Hasan Ahmad ibn Ibrahim 364, 523, 563
Ur 81, 87, 90, 135
Urartu 9, 39, 139
Urmia, Lake 240
Uruk 81-81 , 86, 106, 110; clay tablets 77-78, 98, 101, 150,
152; number system 92-93, 101, 159
’Uthman, Abu 523
’Uthman, Caliph 519-520
Utu-Hegal, King 81
Uyghurs 28
Vajasaneyi Samhita 425
Vakyapanchadhyayi 414
Valabhi numerals 397-398
Vallat, F. 109
de La Vallee-Poussin, L. 530
value, concept of 72-76
Vandel, A. 367
Varahamihira 414-416, 419, 439, 504, 530
Varnasankhya numerals 388
Vatteluttu numerals 383, 385
Veda, Mannen 294
Vedas 29, 425
Vedda people 5, 72
Venezuela 36, 305
Ventris, Michael 179
Venus 297, 311, 315-316
Vera Cruz 302
Vercors 592
Vercoutter, Jacques 162
verse 431-432, 436-437
Vervaeck, L. 365
Vida, Levi della 58, 359
Vieta, Franciscus 597
Vietnam 272-273, 383, 407 see also Champa
vigesimal system 36-39, 44, 303-316; Aztec 306-308; Mayan
303-304, 306-311, 313, 316; Sumer 82
Vigila 362, 579
Vikrama 505
de Ville-Dieu, Alexandre 361
Vishmvamitra 421
Vishnu 50, 444
Visperterminen 195
Vissiere, A. 279
Vitruvius 194
Vocabularium 207
Vogel, K. 365, 519, 531, 533
Voizot, P. 356
von Wartburg, W. 427
Vossius, I. 359
vulgar fractions 595
Vyagramukha (Fighar) 529-530
Waeschke, H. 434, 562
Wafa, Abu’l 514
al-Wafid, Ibn 525
al Wahab Adaraq, Abd 252
ibn Wahshiya 523
Walapai 125
Wallis, John 91, 361-362, 597
Wang Shuhe 516
Waqqas, Muhammedal 252
Warka 103
Warka, Lady of 81
wax calculating board 207-209, 563
Weber 425
wedge 148-149, 160
Weidler, J. F. 359
weights and measures 183, 239; International Bureau of 43
Welsh 33-35, 38
Wessely, J. E. 259
West Bank 97-98
Weyl-Kailey, L. 5
Whitney 411
Wiedler 356
Wieger 269
Wilkinson 52
William of Malmesbury 362, 586
Willichius 361, 590
wind, evokes numbers 442
Winkler, A. 549
Winter, H. J. J. 519
de Wit, C. 176
Woepcke, F. 242, 356, 362-364, 368, 409, 421, 423, 427-429,
433, 438, 529-530, 534, 537, 549, 562-563
Wolof 36
Woods, Thomas Nathan 289
Wright, W. 241
writing 81, 107-108, 272-275; styles 171, 186, 539 see also
under specific race/country, mysticism
writing materials 85, 390-391, 430, 434; chalk 566-567;
papyrus 301, 533; reeds 85-87, 539, 553
ibn Wuhaib, Abu ’’Abdallah Malik 550
Wulfila 226
xia deng number system 277
Xiao dun 269-270
xingshu writing 267-268
Yaeyama 70
Yahweh 71, 212, 218, 253-255
Yahya, Abu 520
Yamamoto, Masahiro 278, 294
al Yaman, Hudaifa ibn 58
al-Yamani, Yahya ibn Nawfal 58, 520
Yang Hui 285
Yang Sun 283
al-Ya’qubi 522
Yaxchilan 316
year, days of 91
Yebu 36-37, 70
633
INDEX
Yedo 305
Yehimilk 213
Yemen 368, 528, 534 see also Sheba
Yishakhi, Rabbenu Shelomoh 253
Yong-le da dian 264
Yoruba 36-37, 305
Youschkevitch, A. P. 243, 365, 512-513, 515-516, 519, 530,
533, 548, 560, 562, 571
Yoyotte, J. 52, 533
Yucatan 299, 300-301, 303, 312
yuga (cosmic cycle) 411, 420, 506-507
Yum Kax 312
ibn Yunus, Matta 513, 523-524
ibn Yusuf, Ahmad 522
ibn Yusuf, al-Hajjaj 521
Zahrawi, Abu’l Qasim az (Abulcassis) 522
za’irja 550-552
Zajackowski, A. 543, 547
Zapotec 36, 162, 305, 307-308
al-Zarqali 525
Zaslavsky, C. 37, 44, 305
Zayd, Abu 523
Zen 295, 443
Zencirli, Aramaic numerals 229
zero 25, 324, 340-346, 354-357, 365, 416, 507-510, 587;
abacus 366, 434-437, 559; absence 145, 149-151, 343, 366,
372-374, 559; Babylonian 152-154; Chinese 266, 280-281,
408; Europe 588-590; Greek 157; imperfect 341-342; India
371, 399, 410, 412, 415-416, 420, 433, 437-439; Islamic
world 533-534; Mayan 308-311, 320-322
zhong deng system 277
Zhu Shi Jie 281
Zimri-Lim 142
ibn Ziyad, Tariq 520
zodiac 92, 549-551, 553
ibn Zuhr (Avenzoar) 526
Zulus 5
Zumpango 301
Zuni 15, 196
THE UNIVERSAL HISTORY OF NUMBERS
GEORGES ifrah, now aged fifty, was the despair of his maths teachers at
school - he lingered near the bottom of the class. Nevertheless he grew up to
become a maths teacher himself and, in order to answer a pupil’s question as
to where numbers came from, he devoted some ten years to travelling the
world in search of the answers, earning his keep as a night clerk, waiter, taxi-
driver. Today he is a maths encyclopaedia on two legs, and his book has been
translated into fourteen languages.
david bellos is Professor of French at Princeton University and author of
Georges Perec: A Life in Words. E. F. harding has taught at Aberdeen,
Edinburgh and Cambridge and is a Director of the Statistical Advisory Unit
at Manchester Institute of Science and Technology, sophie wood is a
specialist in technical translation from French and Spanish, ian monk,
while skilled in technical translation, is better known for his translations of
Georges Perec and Daniel Pennac.
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"Ifrahs book amazes and fascinates by the scope of its scholarship. It is nothing less
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Now in paperback, here is Georges Ifrah’s landmark international bestseller — the
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ematical systems, Ifrah imparts a unique insight into the nature of human thought —
and into how our understanding of numbers and the ways they shape our lives have
changed and grown over thousands of years.
"Dazzling.” — Kirkus Reviews
"Sure to transfix readers.” — Publishers Weekly
GEORGES IFRAH is an independent scholar and former math teacher. DAVID
BELLOS, the primary translator, is Professor of French at Princeton University.
SOPHIE WOOD, cotranslator, is a specialist in technical translation from French. IAN
MONK, cotranslator, has translated the works of Georges Perec and Daniel Pennac.
Cover Design: Wendy Mount
Cover Photograph: Scala/Art Resource, NY
JOHN WILEY & SONS, INC.
ISBN 0-471-3334D-L
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